Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for...

arX

iv0

712

4278

v2 [

hep-

th]

23

Apr

200

8

CALT-68-2666

NI-07085

Algebra of transfer-matrices and Yang-Baxter equations

on the string worldsheet in AdS5 times S5

Andrei Mikhailovlowast and Sakura Schafer-Nameki

California Institute of Technology1200 E California Blvd Pasadena CA 91125 USA

andreitheorycaltechedu ss299theorycaltechedu

and

Isaac Newton Institute for Mathematical Sciences20 Clarkson Road Cambridge CB3 0EH UK

Abstract

Integrability of the string worldsheet theory in AdS5 times S5 is related to the existence of aflat connection depending on the spectral parameter The transfer matrix is the open-endedWilson line of this flat connection We study the product of transfer matrices in the near-flat space expansion of the AdS5 times S

5 string theory in the pure spinor formalism The naturaloperations onWilson lines with insertions are described in terms of r- and s-matrices satisfying ageneralized classical Yang-Baxter equation The formalism is especially transparent for infiniteor closed Wilson lines with simple gauge invariant insertions

lowastOn leave from Institute for Theoretical and Experimental Physics 117259 Bol Cheremushkinskaya 25Moscow Russia

Contents

1 Introduction 3

2 Summary of results 5

21 Definitions 5211 The definition of the transfer matrix 5212 Setup expansion around flat space and expansion in powers of fields 5

22 Fusion and exchange of transfer matrices 7221 The product of two transfer matrices 7222 Relation to Poisson brackets 10223 r- and s-matrices and generalized classical YBE 11

23 Infinite Wilson lines with insertions 12231 General definitions 12232 Split operators 13233 Switch operators 14234 Intersecting Wilson lines 15

24 Outline of the calculation 16241 Use of flat space limit 16242 Derivation of r 17243 Boundary effects and the matrix s 17244 Dynamical vs c-number 17245 BRST transformation 18

3 Short distance singularities in the product of currents 18

31 Notations for generators and tensor product 1832 Short distance singularities using tensor product notations 20

4 Calculation of ∆ 20

41 The order of integrations 2142 Contribution of triple collisions to ∆ 2143 Coupling of dx 2244 Coupling of dϑL 2445 The structure of ∆ 25

5 Generalized gauge transformations 25

51 Dress code 2552 Asymmetry between the coupling of xd+x and xdminusx 26

521 Coupling proportional to zminus4u xdx 26

522 Asymmetric couplings of the form zminus2u zminus2

d xdx 2753 Asymmetry in the couplings of ϑdϑ 28

2

6 Boundary effects 28

61 The structure of Gplusmn 28611 Introducing the matrix s 28612 Cancellation of field dependent terms 29

62 Boundary effects and the global symmetry 29

7 BRST transformations 31

8 Generalized YBE 33

81 Generalized quantum YBE 3382 Some speculations on charges 3683 Contours with loose endpoints 38

9 Conclusions and Discussion 39

A Calculation of the products of currents 40

A1 Collisions J3+J3+ and J1+J2+ 40A2 Terms xdx in the collision J2+J2+ 42A3 Short distance singularities using index notations 43

B Very brief summary of the Maillet formalism 44

Bibliography 46

1 Introduction

Integrability of superstring theory in AdS5 times S5 has been a vital input for recent progressin understanding the AdSCFT correspondence However quantum integrability of the stringworldsheet sigma-model is far from having been established The notion of quantum integrabil-ity is well developed for relativistic massive quantum field theories which describe scattering ofparticles in two space-time dimensions But the string worldsheet theory is a very special typeof a quantum field theory and certainly not a relativistic massive theory It may not be themost natural way to think of the string worldsheet theory as describing a system of particlesIt may be better to think of it as describing certain operators or rather equivalence classes ofoperators What does integrability mean in this case Progress in this direction could be key tounderstanding the exact quantum spectrum which goes beyond the infinite volume spectrumthat is obtained from the asymptotic Bethe ansatz [1 2]

The transfer matrix usually plays an important role in integrable models in particular inconformal ones [3] The renormalization group usually acts nontrivially on the transfer matrix[4 5] But the string worldsheet theory is special The transfer matrix on the string worldsheet

3

is BRST-invariant and there is a conjecture that it is not renormalized This was demonstratedin a one-loop calculation in [6]

In this paper we will revisit the problem of calculating the Poisson brackets of the worldsheettransfer matrices [7 8 9 10 11 12] The transfer matrix is a monodromy of a certain flatconnection on the worldsheet which exists because of classical integrability One can thinkof it as a kind of Wilson line given an open contour C we calculate T [C] = P expminus

int

CJ

Instead of calculating the Poisson bracket we consider the product of two transfer matrices fortwo different contours and considering the limit when one contour is on top of another

At first order of perturbation theory studying this limit is equivalent to calculating the Poissonbrackets ndash we will explain this point in detail We find that the typical object appearing in thiscalculation is a dynamical (=field-dependent) R-matrix suggested by J-M Maillet [13 14 15]The Maillet approach was discussed recently for the superstring in AdS5 times S

5 in [16 10 12]The transfer matrix is a parallel-transport type of object Given two points x and y on

the string worldsheet we can consider the tangent spaces to the target at these two pointsTx(AdS5times S

5) and Ty(AdS5times S5) The transfer matrix allows us to transport various vectors

tensors and spinors between Tx(AdS5 times S5) and Ty(AdS5 times S5) This allows to constructoperators on the worldsheet by inserting the tangent space objects (for example part+x) at theendpoints of the Wilson line

or inside the Wilson line

We study the products of the simplest objects of this type at the first order of perturbationtheory The results are summarised in Section 2 The subsequent sections contain derivationsthe main points are in Sections 45 and 6 In Section 8 we discuss the consistency conditions(generalized Yang-Baxter equations)

4

2 Summary of results

This section contains a summary of our results and in the subsequent sections we will describethe derivation

21 Definitions

211 The definition of the transfer matrix

Two dimensional integrable systems are characterized by the existence of certain currents Jawhich have the property that the transfer matrix

T [C] = P exp

(

minus

int

C

Jaea

)

(21)

is independent of the choice of the contour In this definition ea are generators of some algebraThe algebra usually has many different representations so the transfer matrix is labelled by arepresentation We will write Tρ[C] where the generators ea act in the representation ρ

For the string in AdS5 times S5 the algebra is the twisted loop algebra Lpsu(2 2|4) and thecoupling of the currents to the generators is the following

J+ = (J[microν]0+ minusN

[microν]0+ )e0[microν] + Jα3+e

minus1α + Jmicro2+e

minus2micro + J α1+e

minus3α +N

[microν]0+ eminus4

[microν] (22)

Jminus = (J[microν]0minus minusN

[microν]0minus )e0[microν] + Jα1minuse

1α + Jmicro2minuse

2micro + J α3minuse

3α +N

[microν]0minus e4[microν] (23)

Here ema are the generators of the twisted loop algebra We will use the evaluation representationof the loop algebra In the evaluation representation ema are related to the generators of somerepresentation of the finite-dimensional algebra psu(2 2|4) in the following way

eminus3α = zminus3t1α eminus2

micro = zminus2t2micro e1α = zt1α etc (24)

where z is a complex number which is called ldquospectral parameterrdquo Further details on theconventions can be found in Section 31 and in [6]

212 Setup expansion around flat space and expansion in powers of fields

The gauge group g0 sub psu(2 2|4) acts on the currents in the following way

δξ0J1 = [ξ0 J1] δξ0J2 = [ξ0 J2] δξ0J3 = [ξ0 J3]

δξ0J0 = minusdξ0 + [ξ0 J0] where ξ0 isin g0 (25)

In terms of the coordinates of the coset space

J = minusdggminus1 g isin PSU(2 2|4) (26)

5

The gauge invariance (25) acts on g as follows

g 7rarr hg h = eξ ξ isin g0 (27)

There are two versions of the transfer matrix One is T given by Eq (21) and the otheris gminus1Tg Notice that gminus1Tg is gauge invariant while T is not We should think of T [C] asa map from the (supersymmetric) tangent space T (AdS5 times S

5) at the starting point of C toT (AdS5 times S

5) at the endpoint of CThe choice of a point in AdS5 times S

5 leads to the special gauge which we will use in thispaper

g = eRminus1(ϑL+ϑR)eR

minus1x (28)

Here R is the radius of AdS space and it is introduced in (28) for convenience The actionhas a piece quadratic in x ϑ and interactions which we can expand in powers of x ϑ Thereare also pure spinor ghosts λ w All the operators can be expanded1 in powers of x ϑ λ wWe will refer to this expansion as ldquoexpansion in powers of elementary fieldsrdquo or ldquoexpansion inpowers of xrdquo Every power of elementary field carries a factor Rminus1 The overall power of Rminus1

is equal to twice the number of propagators plus the number of uncontracted elementary fieldsA propagator is a contraction of two elementary fields

The currents are invariant under the global symmetries up to gauge transformations Forexample the global shift

Sg0x = x+ ξ +1

3R2[x [x ξ]] + (29)

results in the gauge transformation of the currents with the parameter

h(ϑ x eξ) = exp

(

minus1

2R2[x ξ] +

)

(210)

To have the action invariant we should also transform the pure spinors with the same parameter

δξλ = minus

[

1

2R2[x ξ] λ

]

δξw+ = minus

[

1

2R2[x ξ] w+

]

(211)

and same rules for wminus λ

1 The expansion in powers of elementary fields is especially transparent in the classical theory where it canbe explained in the spirit of [17] We write

x =

Nsum

a=1

ǫaeikaw+ikaw ++

sum

ab

Gab(ka kb)ǫaǫbei(ka+kb)w+i(ka+kb)w +

where ǫa a = 1 2 N are nilpotents ǫ2a = 0 for every a The nilpotency of ǫa implies that the powers of xhigher than xN automatically drop out

6

22 Fusion and exchange of transfer matrices

221 The product of two transfer matrices

Consider the transfer matrix in the tensor product of two representations ρ1 otimes ρ2 There aretwo ways of defining this object One way is to take the usual definition of the Wilson line

P exp

(

minus

int

Ja(z)ea

)

(212)

and use for ea the usual definition of the tensor product of generators of a Lie superalgebra

ρ1(ea)otimes 1 + (minus)F a otimes ρ2(ea) (213)

where a is 0 if ea is an even element of the superalgebra and 1 if ea is an odd element of thesuperalgebra

Another possibility is to consider two Wilson lines Tρ1 and Tρ2 and put them on top of eachother In other words consider the product Tρ2Tρ1 In the classical theory these two definitionsof the ldquocompositerdquo Wilson line are equivalent because of this identity

eα otimes eβ = eαotimes1+1otimesβ (214)

But at the first order in ~ there is a difference The difference is related to the singularities inthe operator product of two currents

Consider the example when the product of the currents has the following form

Ja+(w)Jb+(0) =

1

wAabc J

c+ + (215)

where dots denote regular terms Take two contours C1 and C2 and calculate the product

Tρ2 [C2] Tρ1 [C1] (216)

where the indices ρ1 and ρ2 indicate that we are calculating the monodromies in the represen-tations ρ1 and ρ2 respectively For example suppose that the contour C1 is the line τ = 0 (andσ runs from minusinfin to +infin) and the contour C2 is at τ = y (and σ isin [minusinfin+infin]) Suppose thatwe bring the contour of ρ2 on top of the contour of ρ1 in other words y rarr 0 Let us expandboth Tρ2 [C2] and Tρ1 [C1] in powers of Rminus2 and think of them as series of multiple integrals ofJ Consider for example a term in which one

int

J comes from Tρ2 [C2] and anotherint

J comesfrom Tρ1 [C1] We get

int int

dσ1dσ2 Ja+(y σ2)(ea otimes 1) J b+(0 σ1)(1otimes eb) =

int int

dσ1dσ21

σ2 minus σ1 + iyAabc J

c+ (ea otimes 1)(1otimes eb) (217)

7

The pole 1σ2minusσ1+iy

leads to the difference between limyrarr0 Tρ2 [C+y]Tρ1[C] and Tρ2otimesρ1 [C] Indeedthe natural definition of the double integral when y = 0 would be that when σ1 collides withσ2 we take a principle value

VP

int int

dσ1dσ2 Ja+(0 σ2)(ea otimes 1) J b+(0 σ1)(1otimes eb) (218)

Here VP means that we treat the integral as the principal value when σ1 collides with σ2Modulo the linear divergences which we neglect the integral (218) is finite This is becauseea otimes 1 commutes with 1 otimes eb But such a VP integral is different from what we would get inthe limit y rarr 0 by a finite piece Indeed

int

dwJa+(w + iǫ)J b+(0) = VP

int

dwJa+(w)Jb+(0) + (219)

+πiAabc Jc+(0) (220)

The second row is the difference between the VP prescription and the limyrarr0

prescription The

additional piece πiAabc Jc+(0) could also be interpreted as the deformation of the generator to

which Jc+ couples in the definition of the transfer matrix

Jc+(ec otimes 1 + (minus)F c otimes ec) 7rarr Jc+

(

ec otimes 1 + (minus)F c otimes ec + πiAabc ea(minus)F b otimes eb

)

(221)

We have two different definitions of the transfer matrix in the tensor product of two represen-tations Is it true that these two definitions actually give the same object There are severallogical possibilities

1 There are several ways to define the transfer matrix and they all give essentially differentWilson line-like operators

2 We should interpret Eq (221) as defining the deformed coproduct on the algebra ofgenerators The algebra of generators is in our case a twisted loop algebra of psu(2 2|4)There are at least three possibilities

(a) The proper definition of the transfer matrix actually requires the deformation of thealgebra of generators ea and the deformed algebra has deformed coproduct

(b) The algebra of generators is the usual loop algebra but it has a nonstandard co-product limyrarr0 Tρ2 [C + y]Tρ1[C] is different from Tρ1otimesρ2[C] the difference being theuse of a nonstandard coproduct We are not aware of a mathematical theorem whichforbids such a nontrivial coproduct

(c) The coproduct defined by Eq (221) is equivalent to the standard one in a sensethat it is obtained from the standard coproduct by a conjugation

∆0(ec) = ec otimes 1 + (minus)F c otimes ec (222)

∆(ec) = ec otimes 1 + (minus)F c otimes ec + πiAabc ea otimes (minus)F ceb =

= eπi

2r(ec otimes 1 + (minus)F c otimes ec)e

minusπi

2r (223)

8

We will argue that what actually happens (at the tree level) is a generalization of 2c The de-formation (223) is almost enough to account for the difference between limyrarr0 Tρ2 [C+y]Tρ1[C]and Tρ1otimesρ2 [C] but in addition to (223) one has to do a field-dependent generalized gaugetransformation2 The correct statement is

for a contour C going from the point A to the point B

limyrarr0

Tρ2 [C + y]Tρ1 [C] = eπi

2r(A)Tρ1otimesρ2 [C]e

minusπi

2r(B) (224)

where r is field dependent (ldquodynamicalrdquo) In fact r is of the order ~ This paper is all aboutthe tree level Therefore all we are saying is

limyrarr0

Tρ2 [C + y]Tρ1[C] = Tρ1otimesρ2 [C] +πi

2( r(A) Tρ1otimesρ2[C]minus Tρ1otimesρ2 [C] r(B) ) + (225)

where dots stand for loop effects The hat over the letter r shows that this is a field-dependentobject We will also use a field-independent r-matrix which will be denoted r without a hat ris the leading term in the near-flat-space expansion of r which is the expansion in powers ofelementary fields explained in Section 212

r = r minusπi

2

(

((zminus21 minus z

21)t

2)otimes [t2 x]minus [t2 x]otimes ((zminus22 minus z

22)t

2))

minus

minusπi

2

(

((zminus31 minus z1)t

1)otimes t3 ϑL minus t3 ϑL otimes ((zminus3

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

3)otimes t1 ϑR minus t1 ϑR otimes ((zminus1

2 minus z32)t

3))

+

+ (226)

Here r is given by Eq (233) and dots stand for the terms of quadratic and higher orders inx and ϑ The pure spinor ghosts do not enter into the expression for r only the matter fieldsx and ϑ

The special thing about the constant term r is that it is a rational function of the spectralparameter with the first order pole at zu = zd The coefficients of the x ϑ-dependent termsare all polynomials in zu zd z

minus1u zminus1

d The field dependence of the r matrix in this exampleis related to the fact that the pair of Wilson lines with ldquoloose endsrdquo is not a gauge invariantobject3

Eq (224) is schematically illustrated in Figure 1 A consequence of (224) is the equivalencerelation for the exchange of the order of two transfer matrices see Figure 2

limCuցCd

TCu(ρzuu )TCd

(ρzdd ) = exp(πi r)

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

exp(minusπi r) (227)

2Generalized gauge transformation is J 7rarr f(d + J)fminus1 If f isin exp g0 then this is a usual (or ldquoproperrdquogauge transformation as defined in Section 212 If we relax this condition we get the ldquogeneralized gaugetransformationrdquo see Section 5

3We use the special gauge (28) therefore in our formalism the lack of gauge invariance translates into thelack of translational invariance

9

=

Figure 1 Fusion of transfer matrices

=

Figure 2 Exchange of transfer matrices

222 Relation to Poisson brackets

At the tree level the calculation of the fusion of transfer matrices is equivalent to the calculationof the Poisson brackets This follows from the definition of the Poisson bracket

Tρ1 Tρ2 = lim~rarr0

1

i~

(

limyrarr0+

Tρ1 [C + y]Tρ2 [C]minus limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

Tρ1 [C + y]Tρ2[C] + limyrarr0+

Tρ2 [C + y]Tρ1[C] = 2Tρ1otimesρ2 [C] +O(~2) (229)

which holds to the first order in ~ These two equations and Eq (225) imply

Tρ1 [C] Tρ2 [C] = π ( r(A) Tρ1otimesρ2 [C]minus Tρ1otimesρ2 [C] r(B) ) (230)

and therefore the calculation of r is actually equivalent to the calculation of the Poisson bracketsTo derive (229) we expand the product T [C + y]T [C] as normal ordered product plus

contractions At the tree level only one contraction is needed schematically we get

J(w)J(0) = J(w)J(0) +F (w w)

where F (w w) is 1wor 1

w2 or 1wor 1

w2 times some expression regular at w rarr 0 see Section 3Then eq (229) follows from the relation

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

applied to the singular part of F (w w)The rdquostandardrdquo calculation of the Poisson bracket of two transfer matrices involves the

equal time Poisson brackets of the currents J(σ) J(σprime) This is proportional to δ(σ minus σprime) orpartσδ(σ minus σ

prime) This is equivalent to what we are doing because

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

(nminus 1)partσδ(σ minus σ

prime) (232)

10

We conclude that the difference between our approach based on the notion of rdquofusionrdquo andthe rdquostandardrdquo approach to calculating the Poisson brackets is a matter of notations (But webelieve that our notations are more appropriate for calculating beyond the tree level)

223 r- and s-matrices and generalized classical YBE

The open ended contours like the ones shown in Figures 1 and 2 are strictly speaking not gaugeinvariant In our approach we fix the gauge (28) and therefore it is meaningful to consider theseoperators as operators in the gauge fixed theory Nevertheless we feel that these are probablynot the most natural objects to study at least from the point of view of the differential geometryof the worldsheet

Figure 3 An infinite Wilson line with an operator insertion

The natural objects to consider are infinite (or periodic) Wilson lines with various operatorinsertions see Figure 3 How to describe the algebra formed by such operators What is the

relation between and We will find that the description of this algebrainvolves matrices r and s which have the following form

r =Φ(z1 z2)

z41 minus z42

(z1z32t

1 otimes t3 + z31z2t3 otimes t1 + z21z

22t

2 otimes t2) + 2Ψ(z1 z2)

z41 minus z42

t0 otimes t0 (233)

s = (zminus11 zminus3

2 minus z31z2)t

3 otimes t1 + (zminus21 zminus2

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

Φ(z1 z2) = (z21 minus zminus21 )2 + (z22 minus z

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

The notations used in (233) (234) are explained in Section 31 In section 8 we will study theconsistency conditions for r and s which generalize the standard classical Yang-Baxter algebraAt the tree level we will get a generalization of the classical Yang-Baxter equations

[(r12 + s12) (r13 + s13)] + [(r12 + s12) (r23 + s23)] + [(r13 + s13) (r23 minus s23)] = t123 (235)

where the RHS is essentially a gauge transformation the explicit expression for t is (87) Notethat neither r nor s satisfy the standard classical YBE on their own and even the combinationrplusmns satisfies an analogue of the cYBE only when acting on gauge invariant quantities Thereforewe have a generalization of the classical Yang-Baxter equations with the gauge invariance builtin

11

23 Infinite Wilson lines with insertions

To explain how r and s enter in the description of the algebra of transfer matrices we have tointroduce some notations

231 General definitions

Consider a Wilson line with an operator insertion shown in Fig 3 For this object to be gaugeinvariant we want O to transform under the gauge transformations in the representation ρprimeotimesρlowast

of the gauge group g0 sub psu(2 2|4) We will introduce the notation H(ρ1 otimes ρ2) for the spaceof operators transforming in the representation ρ1 otimes ρ2 of g0 With this notation4

O isin H(ρprime otimes ρlowast) (236)

Here ρlowast means the representation dual to ρFor example we can take ρ the evaluation representation of the loop algebra corresponding

to the adjoint of psu(2 2|4) with some spectral parameter z and take O = J2+

J2+ isin H(adz otimes (adz)lowast) (237)

In other words consider

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

This is gauge invariant because ad sub adotimes adlowast as a representation of psu(2 2|4) and thereforealso as a representation of g0 Of course we could also pick O = ad(J1+) or ad(J3+) Theseoperators have engineering dimension (1 0) Geometrically they correspond to part+x or part+ϑ

We want to study the objects of this type in the situation when two contours come close toeach other For example consider a Wilson line in the representation ρu with some operatorO inserted at the endpoint Let us take another Wilson line an infinite one carrying therepresentation ρd and put the Wilson line with the representation ρu on top of the the onecarrying ρd In the limit when the separation goes to zero we should have a Wilson line carryingρu otimes ρd at minusinfin and ρd at +infin

This defines maps Fplusmn see Figure 4 If O is inserted inside the contour (rather than at theendpoint) we get Gplusmn To summarize

F+ H(ρlowastu)rarr H(ρlowastu otimes ρlowastd otimes ρd) (239)

Fminus H(ρlowastd)rarr H(ρlowastu otimes ρlowastd otimes ρu) (240)

G+ H(ρlowastu otimes ρprimeu)rarr H(ρlowastu otimes ρ

lowastd otimes ρ

primeu otimes ρd) (241)

Gminus H(ρlowastd otimes ρprimed)rarr H(ρlowastu otimes ρ

lowastd otimes ρu otimes ρ

primed) (242)

4If ρprime is a trivial (zero-dimensional) representation then the Wilson line terminates In thiscase O isin H(ρlowast)

12

Figure 4 Fusion operations F+ Fminus and G+

232 Split operators

We also want to be able to insert two operators Oiup into the upper line and Ojdn into thelower line such that they are not separately gauge invariant but

sum

iOiupO

idn is gauge invariant

For example for a gauge invariant operator O we can insert Cmicroνt2micro otimes t2ν O where Cmicroν =

Cmicroν(xup xdn ϑup ϑdn) is some kind of a parallel transport This will be gauge invariant Wewill use a thin vertical line to denote such a ldquosplit operatorrdquo

In the tensor product notations for example when we write Cmicroνt2micro otimes t2ν O we assume that

the first tensor generator in the tensor product (in this case t2micro) acts on the upper Wilson lineand the second (in this case t2ν O) on the lower line We will need such operators in the limitwhere the upper contour approaches the lower contour Strictly speaking the split operatorwill depend on which parallel transport is used even in the limit of coinciding contours by themechanism similar to what we described in Section 221 We will not discuss this dependencein this paper because it is not important at the tree levelThe exchange map R acts as follows

R Hsplit(ρout1 otimes (ρin1 )lowast ρout2 otimes (ρin2 )lowast)rarr Hsplit(ρ

out2 otimes (ρin2 )lowast ρout1 otimes (ρin1 )lowast) (243)

The pictorial representation of R is

13

233 Switch operators

Given ρ a representation of psu(2 2|4) we denote the evaluation representation ρz Considerρu = ρz

inu ρprimeu = ρz

outu and ρd = ρzd where zinu zoutu and zd are three different complex numbers

Take O = 1 This is gauge invariant because ρzinu and ρz

outu are equivalent as representations of

the gauge group g0 We can think of such O as ldquothe operator changing the spectral parameterrdquoor the ldquoswitch operatorrdquo

For abbreviation we write ρinu = ρzinu and ρoutu = ρz

outu Let us first consider the operation G+ in

Figure 4 with O = 1 In Section 61 we will show that G+(1) is given (at the tree level) bythis formula

G+(1) = 1+πi

2

[

(r + s)|ρinu otimesρd minus (r + s)|ρoutu otimesρd

]

+ (244)

Here the r matrix appears from the diagrams involving the interaction of currents in the bulk ofthe contours It comes from the deformed coproduct see Eq (223) The matrix s comes fromthe diagrams which are localized near the insertion of O These are the additional diagramsexisting because we inserted the impurities

The corresponding exchange relation is

where

R(1switch otimes 1) = 1 + πi r+(zinup zdn)minus πi r+(z

outup zdn) + (245)

r+ = r + s

Similarly if we lift the switched contour from the lower position to the upper position weshould insert R(1otimes 1switch)

14

R(1otimes 1switch) = 1 + πi rminus(zinup zdn)minus πi rminus(z

outup zdn) + (246)

rminus = r minus s

It is useful to write down explicit formulas for rplusmn = r plusmn s following from (233) and (234)

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

(z2d minus zminus2d )2(zuz

3dt

1 otimes t3 + z2uz2dt

2 otimes t2 + z3uzdt3 otimes t1)+

+z2uz2d(z

2u minus z

minus2u )(z2d minus z

minus2d )t0 otimes t0

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

(z2u minus zminus2u )2(zuz

3dt

+z2uz2d(z

2u minus z

]

(248)

We will use the notation

R+ = R(1switch otimes 1) (249)

Rminus = R(1otimes 1switch) (250)

234 Intersecting Wilson lines

In this paper we mostly consider exchange and fusion as relations in the algebra generated bytransfer matrices with insertions It is also possible to think of these operations as definingvertices connecting several Wilson lines in different representations For example the fusion canbe thought of as a triple vertex

Such vertices will become important if we want to consider networks of Wilson lines We wantto define this triple vertex so that the diagram is indepependent of the position of the vertexjust as it is independent of the shape of the contours At the tree level we suggest the followingprescription

15

The subscripts ldquogo-aroundrdquo and ldquoVPrdquo require explanation They indicate different prescrip-tions for dealing with the collisions of the currents coupled to totimes 1 with the currents coupledto 1otimes t Suppose that we consider the integral

int

dw Ja ta otimes 1 and the integration contour has

to pass through several insertions of Jb 1otimes tb The prescription is such that to the right of the

point V we treat the collision as the principal value integral while to the left of V the contourforint

dw(Jata)otimes 1 it goes around the singularity in the upper half-plane

The insertion of 1+ r2is necessary to have independence of the position of the vertex V Notice

that in defining the worldsheet fusion we use r rather than r+ s or rminus s This is different fromthe formula (244) for G+ which uses r + s

24 Outline of the calculation

241 Use of flat space limit

We will use the near flat space expansion of T [C+y]T [C] see Section 212 For our calculationit is important that the transfer matrix is undeformable The definition given by Eqs (21)(22) and (23) cannot be modified in any essential way More precisely we will use the followingstatement Suppose that there is another definition of the contour independent Wilson line ofthe form

T new = P exp

(

minus

int

C

Iaea

)

(251)

where the new currents I have ghost number zero and coincide with J at the lowest order inthe near flat space expansion In other words

I0plusmn = 0 + I1plusmn = minus1

RpartplusmnϑR + I2plusmn = minus

1

Rpartplusmnx+ I3plusmn = minus

1

RpartplusmnϑL +

where dots denote the terms of the order 1R2 or higher Let us also require that T new is invariant

(up to conjugation) under the global symmetries including the shifts (29) Then

(T new)BA = exp(ϕ(A))T exp(minusϕ(B)) (252)

16

where ϕ(w w) is a power series in x and ϑ with zero constant term Eq (252) says that thetransfer matrix is an undeformable object

242 Derivation of r

We will start in Section 4 by calculating the couplings of dplusmnx and dplusmnϑ These are the standardcouplings of the form Rminus1dplusmnx

micro(t2microotimes 1+1otimes t2micro) plus corrections proportional to Rminus3dplusmnx arising

as in Section 221 These couplings are defined up to total derivatives ie up to the couplingsof dx In particular a different prescription for the order of integrations would add a totalderivative coupling It will turn out that with one particular choice of the total derivativeterms the coupling is of the form

exp

(

πi

2r

)

[

dxmicro(t2micro otimes 1 + 1otimes t2micro) + dθαL(t3α otimes 1 + 1otimes t3α) + dθαR(t

1α otimes 1 + 1otimes t1α)

]

exp

(

minusπi

2r

)

(253)where r is the c-number matrix defined in Eq (233) These total derivative terms are impor-tant because they correspond to the field dependence of r in (224) The same prescription forthe total derivatives gives the right couplings for [x dplusmnx] and [ϑ dplusmnϑ] (Sections 52 522 and53) The best way to fix the total derivatives in our approach is by looking at the effects ofthe global shift symmetry (29) near the boundary as we do in Section 62 deriving (226)

According to Section 241 Eq (253) implies that

limyrarr0

Tρ2 [C + y]Tρ1 [C] = exp(ϕ(A)) exp

(

πi

2r

)

Tρ1otimesρ2 [C] exp

(

minusπi

2r

)

exp(minusϕ(A)) (254)

The right hand side is eπi

minusπi

2r(B) the difference between r and r is due to the field

dependent gauge transformation with the parameter ϕ

243 Boundary effects and the matrix s

We then proceed to the study of the boundary effects and derive the exchange relations forthe simplest gauge invariant insertion mdash the switch operator see Eqs (245) and (246) Thematrix s given by Eq (234) arises from the diagrams localized on the insertion of the switchoperator

244 Dynamical vs c-number

The r and s matrices appearing in the description of the exchange relations are generallyspeaking field dependent and in our approach they are power series in x and ϑ These seriesdepend on which insertions we exchange although the leading c-number term in r given by(233) should be universal For the exchange of the switch operator we claim that r and sentering Eqs (244) (245) and (246) are exactly c-number matrices given by (247) and(248) In other words all the field dependent terms cancel out The argument based on theinvariance under the global shift symmetry is given in Section 61

17

245 BRST transformation

The action of Q on the switch operator is the insertion of (minus)F(

1zoutminus 1

zin

)

λ The consistencyof this action with the exchange relation is verified in Section 7

3 Short distance singularities in the product of currents

31 Notations for generators and tensor product

Recall that the notations for generators of Lpsu(2 2|4) is

micro = zminus2t2micro e1α = zt3α (31)

The collective notations for the generators of psu(2 2|4) are

tia i isin Z4 a isin α micro α [ρσ] (32)

The coproduct for superalgebra involves the operator (minus1)F which has the property (minus1)F t3α =minust3α(minus1)

F see (221) The origin of (minus)F can be understood from this example

eψ1(totimes1)eψ2(tprimeotimes1)eψ3(tprimeprimeotimes1) eψ1(1otimest)eψ2(1otimestprime)eψ3(1otimestprimeprime) |0 gt otimes|0 gt= (33)

= eψ1(totimes1+(minus)Fotimest)eψ2(tprimeotimes1+(minus)Fotimestprime)eψ3(tprimeprimeotimes1+(minus)Fotimestprimeprime)|0〉 otimes |0〉 (34)

When we write the tensor products we will omit (minus)F for the purpose of abbreviation Forexample

1otimes t3α 7rarr (minus)F otimes t3α (35)

t3α otimes 1 7rarr t3α otimes 1 (36)

1otimes 1otimes t3α 7rarr (minus)F otimes (minus)F otimes t3α (37)

1otimes t3α otimes 1 7rarr (minus)F otimes t3α otimes 1 (38)

t3α otimes 1otimes 1 7rarr t3α otimes 1otimes 1 (39)

t3α otimes t3β 7rarr t3α(minus)

F otimes t3β (310)

Generally speaking 1otimes 1otimes otimes 1otimes tja otimes 1otimes otimes 1 means

(minus)jF otimes (minus)jF otimes otimes (minus)jF otimes tja otimes 1otimes otimes 1 (311)

With these notations we have

(t3α otimes 1)(1otimes t3β) = minus(1otimes t3β)(t

3α otimes 1) = t3α otimes t

3β (312)

18

We also use the following abbreviations

eminus1α otimes e

2micro = (zminus1t3α)otimes (z2t2micro) = zminus1

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2(eminus1α otimes e

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

When we write Casimir-like combinations of generators we often omit the Lie algebra index

t1 otimes t3 = C ααt1α otimes t3α

t3 otimes t1 = Cααt3α otimes t1α

t2 otimes t2 = Cmicroνt2micro otimes t2ν

t0 otimes t0 = C [microν][ρσ]t0[microν] otimes t0[ρσ] (316)

We will also use this notation

ti otimes tj otimes tk = faprimebprimecprimeCaprimeaCbprimebCcprimec tia otimes t

jb otimes t

kc (317)

wherefabc = fab

cprimeCcprimec = Str([ta tb]tc) (318)

For example

t3 otimes t1 otimes t0 = fαβ[microν]CααCββC [microν][ρσ]t3α otimes t

1βotimes t0[ρσ] (319)

Using these notations we can write for example

[ti otimes t4minusi otimes 1 tj otimes 1otimes t4minusj ] = (minus)i+j+ijt(i+j)mod 4 otimes t4minusi otimes t4minusj (320)

19

32 Short distance singularities using tensor product notations

Short distance singularities in the products of currents were calculated in [18 6] Here is thetable in the ldquotensor productrdquo notations

J1+ otimes J2+ = minus1

wu minus wdt1 otimes t3 part+ϑL

wu minus wdt3 otimes t1 part+ϑR minus

wu minus wd(wu minus wd)2

t3 otimes t1 partminusϑR

wu minus wdt1 otimes [t3 part+x]

wu minus wdt3 otimes [t1 part+x]minus

t3 otimes [t1 partminusx]

wu minus wdt0 otimes [t0 part+ϑR]minus

12

(wu minus wd)2t0 otimes [t0 ϑR]

wu minus wdt0 otimes [t0 part+ϑL]minus

12

(wu minus wd)2t0 otimes [t0 ϑL]

J1minus otimes J2+ = minus1

wu minus wdt1 otimes t3 partminusϑL

J1+ otimes J2minus = minus1

wu minus wdt3 otimes t1 part+ϑR

wu minus wdt1 otimes t3 partminusx

wu minus wdt3 otimes t1 part+x

Such ldquotensor product notationsrdquo are very useful and widely used in expressing the commutationrelations of transfer matrices We will list the same formulas in more standard index notationsin appendix A3

4 Calculation of ∆

In this section we will give the details of the calculation which was outlined in Section 221

20

41 The order of integrations

As we discussed in [6] the intermediate calculations depend on the choice of the order ofintegrations We will use the symmetric prescription This means that if we have a multipleintegral we will average over all possible orders of integration For example in this picture

we have three integrations and therefore we average over 6 possible ways of taking the integralsAnother prescription would give the same answer (because after regularization the multipleintegral is convergent and does not depend on the order of integrations) but will lead to adifferent distribution of the divergences between the bulk and the boundary

42 Contribution of triple collisions to ∆

Triple collisions contribute to the comultiplication because of the double pole Let us forexample consider this triple collision

Of course this is not really a collision since only the lower two points collide But we still callit a ldquotriple collisionrdquo This has to be compared to

where the integrals are understood in the sense of taking the principal value We have toaverage over two ways of integrating (1) first integrating over the position of the zminus2

u d+x onthe upper contour and then zminus2

d d+x on the lower contour and (2) first integrating over theposition of zminus2

d d+x and then integrating over the position of zminus2u d+x The first way of doing

integrations does not contribute to ∆ and the second does Indeed the contraction 〈d+xd+x〉gives minus 1

(wuminuswd)2zminus2u zminus2

d t2 otimes t2 and after we integrate over wd we get

21

Then integration over wu gives the imaginary contributionint

(

minus dwu

wuminusw

)

= minusπi

The contribution from the contractions 〈d+ϑLd+ϑR〉 is similar and the result for the contribu-tion of triple collisions to ∆ is

∆triple(ema ) = πi1

2[C+ minus Cminus 1otimes e

ma minus e

ma otimes 1] (41)

where 12 is because we average over two different orders of integration and Cplusmn is defined as

C+ = (zminus1t3)otimes (zminus3t1) + (zminus2t2)otimes (zminus2t2) + (zminus3t1)otimes (zminus1t3) (42)

Cminus = (z3t3)otimes (zt1) + (z2t2)otimes (z2t2) + (zt1)otimes (z3t3) (43)

The expression (41) for ∆trpl should be added to ∆dbl which is generated by the double collisionsWe will now calculate ∆dbl and ∆prime = ∆dbl +∆trpl

43 Coupling of dx

We have just calculated the contribution of triple collisions now we will discuss the contributionof double collisions and the issue of total derivativesEffect of double collisions

Collision contributes πi times

J1+J1+ minuszminus3u zminus3

d t1 and [t3 d+x] +

J1minusJ1minus +2zuzd t1 and [t3 dminusx] + zuzd t

1 and [t3 d+x] +

J3+J3minus +2zminus1u z3d t

3 and [t1 d+x] +

J3minusJ3minus +z3uz3d t

3 and [t1 dminusx]minus

J3+J3+ minus2zminus1u zminus1

d t3 and [t1 d+x]minus zminus1u zminus1

d t3 and [t1 dminusx]minus

J1minusJ1+ minus2zuzminus3d t1 and [t3 dminusx] +

J0plusmnJ2plusmnprime +3

2(z2d minus z

minus2d )[dx t2] and t2 (44)

22

In the calculation of the contribution of J0plusmnJ2plusmnprime we take an average of first taking an integralover the position of J0plusmn and then taking an integral over the position of J2plusmnprime To summarize

1

πi∆dbl(dx) = (minuszminus3

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(zminus1u z3d + z3uz

minus1d minus 2zminus1

u zminus1d )t3 and [t1 d+x] +

+(minuszuzminus3d minus z

minus3u zd + 2zuzd)t

1 and [t3 dminusx] +

+(z3uz3d minus z

minus1u zminus1

d )t3 and [t1 dminusx] +

+3

2(z2u minus z

minus2u )t2 and [t2 dx] (45)

Effect of triple collisions

1

πi∆trpl(dx) = [C+ minus Cminus 1 and (zminus2d+x+ z2dminusx)] =

= (zminus3u zminus3

d minus zuzd)t1 and [t3 d+x] + (zminus1

u zminus5d minus z

3uz

minus1d )t3 and [t1 d+x] +

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

+(zminus3u zd minus zuz

5d)t

1 and [t3 dminusx] + (zminus1u zminus1

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

This leads to the following expression for the total ∆prime

1

πi∆prime(dx) =

1

2((z2u minus z

minus2u )2 + (z2d minus z

minus2d )2)zminus1

u zminus1d t3 and [t1 d+x]minus

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

The calculations of this section can only fix the coupling of dplusmnx up to total derivatives ieterms proportional to dx = d+x + dminusx Only the terms proportional to lowastdx = d+x minus dminusx arefixed To fix the terms proportional to dx we have to either study the couplings of xdx or lookat what happens at the endpoint of the contour We will discuss this in Sections 5 and 6 Theresult it that the following additional coupling

1

2(z2u minus z

should be added to (46)

23

44 Coupling of dϑL

Similar to the dx terms we can discuss the dϑ coproductEffect of double collisions Here is the table

d t1 and t3 d+ϑL+

J1minusJ2minus +2zuz2d t

1 and t3 d+ϑL+ 4zuz2d t

1 and t3 dminusϑL minus

J1minusJ2+ minus2zuzminus2d t1 and t3 dminusϑL minus

J1+J2minus minus2zminus3u z2d t

1 and t3 dminusϑL+

J0J3 +3

2((z3 minus zminus1)t3) and t1 dϑL

Contribution of triple collisions

1

πi∆trpl(dϑL) = [C+ minus Cminus 1 and (zminus1d+ϑL + z3dminusϑL)] =

= zminus3u zminus2

d (1minus z4uz4d) t

1 and t3 d+ϑL+ zminus2u zminus3

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

+zminus3u z2d(1minus z

4uz

4d) t

1 and t3 dminusϑL+ zminus2u zd(1minus z

4uz

4d) t

2 and t2 dminusϑL+

+zminus1u (1minus z4uz

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

d + zminus2u zminus3

d )(1minus z4uz4d) t

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

+(zminus3u z2d + zminus2

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

Just as in case of the couplings of dx we observe that only the couplings proportional tod+x minus dminusx are fixed by the calculation in this section In fact the analysis of Section 5 willshow that we have to add the following total derivative coupling

(12)((z3 minus zminus1)t3) and t1 dϑL (48)

Adding this to ∆dbl +∆trpl we get

1

πi∆prime(dϑL) = minuszuz

2d [(z

2d minus z

minus2d )2 + (z2u minus z

minus2u )2] t1 and t3 dminusϑL+

+(2z3u minus zminus1u minus z

4dz

3u)t

minus(2zminus1u minus z

3u minus z

minus1u zminus4

d )t3 and t1 d+ϑL (49)

24

45 The structure of ∆

At the first order of perturbation theory ∆ = ∆0 +∆prime where ∆0(t) = totimes 1+ 1otimes t is the trivialcoproduct It follows from Sections 43 and 44 that ∆prime is given by the following formula

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

1 otimes t3 + z3uzdt3 otimes t1 + z2uz

2dt

2 otimes t2) + 2Ψ(zu zd)

z4u minus z4d

t0 otimes t0 (411)

We used the notations

Φ(zu zd) = (z2u minus zminus2u )2 + (z2d minus z

minus2d )2

Ψ(zu zd) = 1 + z4uz4d minus z

4u minus z

4d

The following identities are useful in deriving (410)

[zuz3dt

2 otimes t2 + z3uzdt3 otimes t1 (zminus1

u t3α)otimes 1 + 1otimes (zminus1d t3α)] = 2z3ut

3 bull t1 t3α

[t0 otimes t0 (zminus1u t3α)otimes 1 + 1otimes (zminus1

d t3α)] = minus2(zminus1t3) bull t1 t3α

[zuz3dt

u t2micro)otimes 1 + 1otimes (zminus2d t2micro)] = 2zminus1

u z3d[t1 t2micro] bull t

3

+ 2z2d[t2 t2micro] bull t

2 (412)

Here bull denotes the symmetric tensor product it is the opposite of and The minus sign in thelast line of (412) is because C αβ = minusCβα So in particular 2z3ut

3 bull t1 t3α = (z3ut3)otimes t1 t3α minus

t1 t3α otimes (z3dt3)

5 Generalized gauge transformations

51 Dress code

The coupling of fields to the generators of the algebra is strictly speaking not defined unam-biguously because of the possibility of a ldquogeneralized gauge transformationrdquo

J 7rarr f(d+ J)fminus1 (51)

where f is a group-valued function of fields depending on the spectral parameter z A ldquoproperrdquogauge transformation would not depend on z and would belong to the Lie group of g0 whilef in (51) belongs to the Lie group of g and does depend on z Therefore it would perhaps beappropriate to call (51) ldquogeneralized gauge transformationrdquo or maybe ldquochange of dressingrdquo Ifthere is some insertion A into the contour then we should also transform A 7rarr fAfminus1

25

One of the reasons to discuss the transformations (51) is that different prescriptions for theorder of integrations are related to each other by such a ldquochange of dressingrdquo A similar storyfor log divergences was discussed in [6] Different choices of the order of integration lead todifferent distribution of the log divergences between the bulk and the boundary

We agreed in Section 41 to use the ldquosymmetric prescriptionrdquo for the order of integrations Itturns out that with this prescription limyrarr0 Tρ2 [C+y]Tρ1[C] comes out in the ldquowrong dressingrdquoin the sense that the limit cannot be immediately presented in the form

P exp

(

minus

int

Ja∆(ta)

)

(52)

In particular xmicropart+xν couples to a different algebraic expression than xmicropartminusx

ν while in (52) theyshould both couple to ∆(t0[microν]) However it turns out that it is possible to satisfy the ldquodress

coderdquo (52) by the change of dressing of the type (51)We will now stick to the symmetric prescription for the order of integrations and study the

asymmetry between the couplings of xd+x and xdminusx and the asymmetry between the couplingsof ϑd+ϑ and ϑdminusϑ Then we will determine the generalized gauge transformation needed tosatisfy (52) and this will fix the total derivative couplings discussed in Section 43 It turnsout that in the symmetric prescription we will have to do the generalized gauge transformation(51) with the parameter

f = 1minusπi

2

(

(zminus2 minus z2)t2) and [t2 dx] + ((zminus1 minus z3)t3) and t1 dϑL+ ((zminus3 minus z)t1) and t3 dϑR)

+

(53)In the next Sections 52 and 53 we will show that the gauge transformation with this parameterindeed removes the asymmetry In Section 62 we will derive (53) using the invariance underthe shift symmetries

52 Asymmetry between the coupling of xd+x and xdminusx

521 Coupling proportional to zminus4u xdx

The most obvious asymmetry is that there is a term with zminus4u xd+x but no term with zminus4

u xdminusxThe term with zminus4

u xd+x comes from this collision

zminus2d+x zminus2d+x12 [x d+x]

The result is

πi

[

(zminus2d+x)otimes 1 1

4(zminus2t2)otimes [x t2]

]

(54)

This is unwanted so we want to do the generalized gauge transformation with the parameter

minusπi

2(zminus2t2) and [t2 x] (55)

26

which removes this coupling and adds instead a total derivative coupling to dx

minusπi

2(zminus2t2) and [t2 dx] (56)

We will now argue that the change of dressing with the parameter (55) also removes theasymmetry between the coupling of xd+x and xdminusx

Also the coefficient of zminus2u zminus2

d xd+x is different from the coefficient of zminus2u zminus2

d xdminusx Let usexplain this

d xdx

There is a contribution from a double collision and from a triple collision The double collisionis

zminus2J2+zminus2J2+

and we have to take into account the interaction vertex in the action

minus S 7rarr1

6πstr[x part+x][x partminusx] (57)

The calculation is in Section A2 and the result is

1

2πi Cmicroν(zminus2[t2micro [x dminusx]]) and (zminus2t2ν) (58)

There is also a triple collisionzminus2d+x (12)[x d+x]

zminus2d+x

It contributes1

4πi(zminus2[[d+x x] t

2] and (zminus2t2) (59)

The sum of equations (58) and (59) amounts to the following asymmetry of the form zminus2u zminus2

d

minus1

2]) and (zminus2t2) (510)

We see that (54)+(510) is[

(zminus2d+x)otimes 1 + 1otimes (zminus2d+x) 1

2πi (zminus2t2) and [x t2]

]

(511)

This is undone with the generalized gauge transformation with the parameter 12πi (zminus2t2) and

[x t2] which adds an additional total derivative coupling

1

2πi (zminus2t2) and [dx t2] (512)

This is the ldquoadditional couplingrdquo of Eq (47)

27

53 Asymmetry in the couplings of ϑdϑ

The situations with the couplings of ϑdϑ is similar There are asymmetric couplings of theform zminus4

u ϑLd+ϑR which are removed by the generalized gauge transformation This generalizedgauge transformation should also remove the asymmetry in the couplings of zminus2

u zminus2d ϑd+ϑ and

zminus2u zminus2

d ϑdminusϑ but we did not check this

Terms of the form zminus4u ϑLd+ϑR come from zminus3d+ϑRharrzminus1d+ϑL

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

They are similar to (54)

πi

[

(zminus3d+ϑR)otimes 1

(

minus1

4

)

(zminus1t3)otimes t1 ϑL

]

(513)

This should be removed with the generalized gauge transformation which simultaneously intro-duces the total derivative coupling

minusπi

2(zminus1t3) and t1 dϑL (514)

This is the ldquoadditional couplingrdquo of (48)

6 Boundary effects

61 The structure of Gplusmn

611 Introducing the matrix s

Here we will derive Eq (244) in Section 233 We inserted the switch operator on the upperline which turns zinu into zoutu Naively Eq (410) implies that

But this is wrong because there is an additional boundary contribution related to the secondorder poles in the short distance singularities of the products of currents Notice that thesesecond order poles correspond to the δprime terms in the approach of [13 14 15] (see Appendix B)At the first order in the x-expansion the contributing diagram is this one

28

and similar ones This turns 1plusmn πi2r + into 1plusmn πi

2(r + s) + where

s = C+ minus Cminus (61)

and Cplusmn are given by (42) and (43) Therefore G+ of the switch operator is the following splitoperator

G+(1switch) = 1 +πi

2(r(zinu zd) + s(zinu zd))minus

πi

2(r(zoutu zd) + s(zoutu zd)) + (62)

612 Cancellation of field dependent terms

Dots in (62) denote the contribution of the higher orders of the string worldsheet perturbationtheory Those are the terms of the order ~

2 and higher The terms with 12(r + s) are of the

order ~ Remember that we are also expanding in powers of elementary fields It turns outthat all the terms of the order ~ (ie tree level) in G+(1switch) are c-number terms written in(62) there are no corrections of the higher powers in x and ϑ This is because such correctionswould contradict the invariance with respect to the global shifts (29) Indeed suppose thatR(1switch otimes 1) contained x and ϑ For example suppose that there was a term linear in xsomething like x t otimes t Then the variation under the global shift (29) will be proportionalto ξ t otimes t and there is nothing to cancel it5 This implies that R(1switch otimes 1) is a c-numberinsertion ie no field-dependent corrections to (245) (246) (247) (248)

62 Boundary effects and the global symmetry

We explained in Section 3 of [6] that the global shifts act on the ldquocapitalrdquo currents by the gaugetransformations (normal gauge transformation not generalized)

SξJ = minusdhhminus1 + hJhminus1

h = 1minus1

2Rminus2[x ξ] + (63)

Suppose that the outer contour is open-ended then this is not invariant under the global shifts

The infinitesimal shift of this is equal to

5 If we inserted some operator O which is not gauge invariant for example O = t2micro the variation under theglobal shift will give [t2micro [ξ x]] This is linear in x but x will contract with d+x in

int

(zminus2J2+dτ+ + z2J2minusdτ

minus)resulting in the x-independent expression of the form zminus2t2otimes [t2 ξ] which will cancel the ξ-variation of the fielddependent terms(226)

29

Therefore because of this contraction

We have the imaginary contribution

Using the terminology from Section 23 we should say that F+(1) is such that

SξF+(1) = minusπi1

2[t2 ξ]otimes (zminus2

d minus z2d)t

2 (64)

There are similar considerations for the super-shifts Therefore

F+(1) = const + πi1

2[x t2]otimes (zminus2

d minus z2d)t

2 +

+ πi1

2ϑL t

1 otimes (zminus1d minus z

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

The relation between this formula and the generalized gauge transformation with the parameter(55) is the following Part of (65) comes from (55) and another part from the followingdiagrams

These two diagrams contribute

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

2 otimes [x t2] (66)

And the generalized dressing transformation with the parameter (55) gives the boundary termπi1

d minus z2d)t

2 minus πi14(zminus2u minus z

2u)t

2 otimes [x t2] which in combination with (66) gives

πi1

d minus z2d)t

2 (67)

which is in agreement with (65) Similar diagrams with fermions give terms with ϑ in (65)

30

Notice that the constant terms in (65) are essentially the same as in Section 61

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

The difference between F+(1) and G+(1switch) is that G+(1switch) is a c-number while F+(1) isfield-dependent That is because 1switch is invariant under the gauge transformations becauseρz

out

u and ρzin

u are the same as representations of the finite dimensional g0 sub Lpsu(2 2|4)

7 BRST transformations

Here we discuss the action of QBRST on the switch operators end verify that it commutes withG+ There are two BRST currents holomorphic Q and antiholomorphic Q They are bothnilpotent and QQ = 0 The total BRST operator is their sum

QBRST = Q+Q (71)

Here we will consider the holomorphic BRST operator Q The action of Q on the currents is

[ǫQ Jplusmn(z)] = D(z)plusmn (ǫzminus1λ) (72)

(see for example Section 2 of [6]) In other words

[ǫQ TBA (z)] =1

zǫλ(B) TBA (z)minus TBA (z)

1

zǫλ(A) (73)

The switch operator turns zin into zout We have

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

According to (244) the fusion of the switch operator on the upper contour is the split operatorπi2(minusrout+ + rin+ ) Therefore

QG+1switch =πi

2

[

(zminus1out(λotimes 1) + zminus1(1otimes λ)) (minusrout+ + rin+ )minus (75)

minus(minusrout+ + rin+ ) (zminus1in (λotimes 1) + zminus1(1otimes λ))

]

(76)

Now we have to caclulate G+Q1switch The action of G+ on(

1zoutminus 1

zin

)

λ is essentially the sameas the action on the switch

πi

2

[

minusrout+ (zminus1out(λotimes 1)minus zminus1

in (λotimes 1)) + (zminus1out(λotimes 1)minus zminus1

in (λotimes 1))rin+]

plus the contribution of this diagram

31

This diagram contributes

minus πi(

(zoutu )minus1 minus (zinu )minus1)

(1minus zminus4d ) t3 otimes t1 λ (77)

where we have used the short distance singularity

λ(wu)otimes (minusw+ λ(wd)) = minus1

wu minus wdt3 otimes t1 λ (78)

Therefore the condition [QG+] = 0 can be written as follows

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

t3 otimes t1 λ (79)

This can be verified using the identity

[

(zu)minus1λotimes 1 + (zd)

minus11otimes λ r+(zu zd)

2

]

=

(

1minus1

z4d

)

(zminus1u t3otimest1 λminusz3dλ t

1otimes t3) (710)

Notice that this identity can be used to derive equation (247) for r+ Also notice that the

second term on the right hand side minus(

1minus 1z4d

)

z3dλ t1otimes t3 is a gauge transformation and does

not contribute because the switch operator is gauge invariantAs another example let us consider the upper Wilson line terminating on λ In this case

the second term on the right hand side of (710) does contribute

minus

(

1minus1

z4d

)

z3d[λ t1 λ]otimes t3 (711)

But this cancels with the contribution of the contraction of λ with the integralint

dτminus(1minusz4d)Nminus

arizing in the expansion of the lower Wilson line

Indeed this contraction givest1 otimes (1minus z4d)t

3 λ (712)

32

and therefore there is an additional contribution to [QG]

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

Notice that (711) cancels with (713) because [λ t1 λ]otimes t3 minus t1 otimes [λ t3 λ] = 0

8 Generalized YBE

The r-matrix (411) does not satisfy the classical Yang-Baxter equation in its usual form butthe deviation from zero is a polynomial in z1 z2 z3 z

minus11 zminus1

2 zminus13 Using the notation of Eq

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

= t0 otimes t2 otimes t2(

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

+ t3 otimes t3 otimes t2(

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

minusz1z2z63 minus z1z

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

We will now explain why (81) is not zero and what replaces the classical Yang-Baxter equationWe will also derive a set of generalized YBE which we conjecture to be relevant in the quantumtheory

The consistency conditions follow from considering the different ways of exchanging theproduct of three Wilson lines with insertions We first consider the case of gauge invariant in-sertions in this case the R-matrices are c-numbers Then we will consider the case of non-gauge-invariant insertions namely loose endpoints In this case the R-matrices are field-dependentand the generalized Yang-Baxter equations are of the dynamical type

81 Generalized quantum YBE

To understand the quantum consistency conditions for the R matrices let us put the Wilsonline with the spectral parameter switch on top of two other Wilson lines the other two Wilsonlines having no operator insertions The equations of this section will not change if we puta constant gauge invariant operator at the point on the upper contour where we switch thespectral parameter (instead of just 1) For example Cmicroνt2microt

2ν is a constant gauge invariant

operator It is gauge invariant because commutes with g0

33

3

2

3

2

3

2

3

2

3

2

3

1

Figure 5 Generalized YBE 1

The generalized quantum Yang-Baxter equations (qYBE) are obtained from the exchanges

illustrated in figure 5 The notations are = R+ = Rminus1+ = Rminus = Rminus1

minus The insertion ofthe spectral parameter changing operator is marked by a black bar

Equating LHS and RHS in figure 5 yields

R23minusR13+Rminus123minusR23+R12+R

minus123+R23minusR

minus113+R

minus123minusR23minusR13+R

minus123minussim= R12+R13+R

minus112+R12+

(82)After cancellations of RRminus1

minus123+sim= R12+R13+ (83)

Here the sign sim= means that the ratio of the left hand side and the right hand side commutes

34

3

1

2

22

with O

Rminus113+R

minus112+R23minusR13+R

minus123minusR23+R12+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

At the first order of perturbation theory the left hand side of (84) is cf (247) and (248)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

And the right hand side of (84) is

= minus4 t0 otimes (z22 minus zminus22 )t2 otimes (z23 minus z

minus23 )[t0 t2]minus

minus4 t0 otimes (z2 minus zminus32 )t1 otimes (z33 minus z

minus13 )[t0 t3]minus (87)

minus33 )[t0 t1]

35

(the indices of t0 contract with the indices of another t0 so t0otimes t0 stands for C [micro1ν1][micro2ν2] t0[micro1ν1]otimes

t0[micro2ν2] similarly the indices of t2 contract with the indices of another t2 and t1 with t3) Our Ois just the spectral parameter switch it is a constant gauge invariant operator In particular[t0O] = 0 cf (85) In other words eq (8687) is the generalized classical Yang-Baxterequation modulo gauge transformation

Similarly putting the switch operator on the lower contour (see Figure 5) we get the fol-lowing consistency condition

R23+R13minusRminus123+R23minusR12minusR

minus123minus = R12minusR13minus (88)

Finally we turn to the exchange of R+ and Rminus which is derived in Figure 6 Equating theLHS and RHS of this graph we obtain

(R23minusR13+Rminus123minus)(R23+R12minusR

minus123+)(R23minusR

minus113minusR

minus123minus)R23+

=(R12+R13minusRminus112+)(R12minusR23+R

minus112minus)(R12+R

minus113minusR

minus112+)R12minus

(89)

Note that we equate of course only one side of the insertion at a time Note also that forR+ = Rminus this returns to a standard YBE Another way of writing it

R12minusRminus123+

= (adR12+(R13minus)adR12minus

(R23+)adR12+(Rminus1

13minus))minus1(adR23minus

(R13+)adR23+(R12minus)adR23minus

(Rminus113minus))

(810)In [19 20] another generalization of quantum YBE was proposed as the quantum version ofa more restricted set of classical YBE The main difference to the equations here is that theones in [19 20] impose the standard qYBE on R (and thus the standard YBE on the classicalr-matrix) and supplement these by equations of the type RSS = SSR However the mainproblem with this approach is that the case of principal chiral models and strings on AdS5timesS

5

do not fall in the class of models where r satifies the YBE separately from s

82 Some speculations on charges

Strictly speaking our derivation of equations like (83) only applies to the terms quadratic inr (ie tree level) Although the derivation outlined in Figures 5 and 6 seems to apply also atthe level of higher loops in fact there might be subtleties associated to overlapping diagrammsinvolving all three lines

Nevertheless let us for a moment take the proposed generalized qYBE (83) seriously andsee how it could be put to use in order to construct a quadratic algebra of RTT type Therelation (83) can be thought of in the following way Let us begin with the standard YBEwhich reads

R12R13R23 = R23R13R12 (811)

This can formally be thought of as ldquoR12 and R13 commute up to conjugation by R23rdquo orexplicitly

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

The relation (83) generalizes this version of the YBE naturally in that

R12+R13+ =(

R23minusR13+Rminus123minus

) (

R23+R12+Rminus123+

)

(813)

If we interpret this as RTT relations we obtain

T12+T13+ =(

R23minusT13+Rminus123minus

) (

R23+T12+Rminus123+

)

(814)

Naively one might then conclude that this equation is in fact of the type that has been discussedin [19] equation (14)

A12T1B12T2 = T2C12T1D12 (815)

where T1 = T12+ and T2 = T13+ and

A12 = Rminus123minus B12 = 1 C12 = Rminus1

23minusR23+ D12 = Rminus123+ (816)

However in [19] it is required that the matrices ABCD satisfy a set of equations in particularA and D have to separately satisfy the standard YBE as well as equations of the type ACC =CCA and DCC = CCD as well as [A12 C13] = 0 and [D12 C32] = 0 have to hold (note that it ispointed out in [19] that these are only sufficient conditions) We do not require these equationsbut only seem to be imposing the equation (83) This is in fact a much weaker equation buthas the vital advantage that it gives as a classical limit an the algebra of r minus s-matrices as werequire it

In view of the algebra (813) the standard argument of construction of commuting chargesdoes not go through namely [tr2(T12+) tr3(T13+)] is not obviously vanishing as the conjugationin this case is by R23+ and R23minus respectively which do not agree in the present case At thispoint a construction that appears in [19] is useful despite the fact that their transfer matrixalgebra is different from ours First let us simplify notation and suppress the physical spaceindex of the T -matrices so we consider the exchange relation of T2 and T3 Ti is an element ofEnd(ρa) equiv ρaotimes ρ

lowasta (at least for finite dimensional representations) Thus we can label them by

T(aa) where a denotes the dual representation The generalized RTT relations then become

T(22)T(33) = R23minusT(33)Rminus123minus

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

where Rab acts on the ρa part of T(aa) etc and we defined

R(33)(22) = R23minusRminus123minus

R23+Rminus123+

(818)

We require that R satisfies the YBE in order for the exchange algebra of T(aa) to be consistentAt this point the deviation from the construction in [19] is necessary Our Rabplusmn matrices obeythe generalized YBE (83) and the complete set of consistency conditions on Rplusmn should implyYBE for R This requires in particular additional relations for R12minus and R21minus Once the YBEfor R are established we define the dual RTT algebra as

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

Consider a matrix representation (scalar matrix) of (819) given by τ(22) and τ(33) There is anatural inner product between the representations and their duals in particular τaaTaa Thusacting with τ22τ33 on the generalized YBE in the form (817) we obtain

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

This allows for construction of a family of infinite commuting charges by expanding theseexpressions in powers of the spectral parameter

83 Contours with loose endpoints

The consistency condition for the exchange of contours with endpoints is more complicatedAgain we can compare (123) rarr (213) rarr (231) rarr (321) to (123) rarr (132) rarr (312) rarr (321)When we exchange (123)rarr (213) we get the insertion of the split operator Fminus1

minus (F+(1))

At the first order of perturbation theory

Fminus1minus (F+(1)) = 1+ r + s+ q (822)

where q are field-dependent (= dynamical) terms Indeed the main difference between theexchange of the switch operator and the exchange of the endpoint is that the endpoint is notgauge invariant and therefore the exchange matrix is field dependent The expansion of q inpowers of x and ϑ starts with

q =1

2

(

[x t2]otimes t2 + ϑ3 t1 otimes t3 + ϑ1 t3 otimes t1)

+ (823)

Then when we exchange (213)rarr (231)rarr (321) we get additional contributions coming fromthe contraction of q12 with the currents integrated over line 3 for example

(824)

38

On the other hand if we look at the field independent (leading) terms we will get an equationidentical to (84) but now T123 does not act as the identity on the endpoint because theendpoint is not gauge invariant But in fact the T on the right hand side of (84) cancels withthe terms arising from the contractions (824)

To summarize we have the following two types of consistency conditions

1 Consistency conditions for the exchange of gauge invariant operators In this case theright hand side of (84) does not spoil the consistency because of equation (85) whichexpresses the gauge invariance of the inserted operators

2 Consistency conditions for the insertions which are not gauge invariant In this case theright hand side of (84) cancels against the terms arising from the diagrams like (824)

9 Conclusions and Discussion

We have setup a formalism in which to compute the product of two transfer matrices using theoperator algebra of the currents In particular to leading order in the expansion around flatspace-time a structure reminiscent of classical r-matrices appears This is however modified inthat we require a system of r and s-matrices which satisfy a generalized classical YBE Thisis related in the approach of [15] to Poisson brackets being non-ultralocal

We consider it a first step towards constructing the analog of a quantum R-matrix whichsatisfies a generalized quantum YBE The situation is different from [19 20] because theclassical r-matrix in our case does not satisfy the standard classical YBE (which is one of theassumptions that goes into the construction in [19 20]) but the combined equation for r and s(86)

The most promising direction to extend this work is to construct the quantum conservedcharges from the T -matrices as outlined in section 82 It would also be interesting to test thegeneralized quantum YBE explicitly at higher orders in the 1R expansion

It would also be interesting to understand how the r-s-matrices found here relate to theclassical su(2|2) r-matrices found from the light-cone string theory and super-Yang Mill dual in[21 22 23] The connection if it exists would presumably be along the line of our speculationsin section 82

Acknowledgments

We thank Jean-Michel Maillet for very interesting discussions The research of AM is supportedby the Sherman Fairchild Fellowship and in part by the RFBR Grant No 06-02-17383 andin part by the Russian Grant for the support of the scientific schools NSh-806520062 Theresearch of SSN is supported by a John A McCone Postdoctoral Fellowship of Caltech Wethank the Isaac Newton Institute Cambridge for generous hospitality during the completionof this work

39

A Calculation of the products of currents

Here we will describe some methods for calculating the singularities in the product of twocurrents We will only discuss two examples The first example is the collision J3+J3+ and thecollision J1+J2+ The second is the singularities proportional to xdx in the collision J2+J2+which we needed in Section 522

A1 Collisions J3+J3+ and J1+J2+

Collision J3+J3+

(part+ϑL + [ϑR part+x])α(wa)larrrarr(part+ϑL + [ϑR part+x])

β(wb) (A1)

The cubic vertex ([ϑL partminusϑL]part+x) does not contribute to the singularity but the other cubicvertex does

minus S 7rarr1

π

int

d2v1

2str ([ϑR part+ϑR]partminusx) (A2)

After integration by parts the interaction vertex becomes

minus1

π

int

d2v1

2str ([partminusϑR part+ϑR]x+ str [ϑR part+partminusϑR]x) (A3)

Integrating by parts part+ in the second term we get

1

π

int

d2v str

(

minus[partminusϑR part+ϑR]x+1

2str [ϑR partminusϑR]part+x

)

(A4)

This implies that (A1) gives the same singularity as the following collision in the free theory

(

part+ϑL minus [part+ϑR x] +1

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

(wa minus wb)2([t1 x(wa)]otimes t

3 + t3 otimes [t1 x(wb)]) +

+1

2

1

wa minus wb([t1 part+x(wa)]otimes t

3 minus t3 otimes [t1 part+x(wb)]) (A6)

This is equal to2

wa minus wb[t1 part+x]otimes t

3 +wa minus wb

(wa minus wb)2[t1 partminusx]otimes t

3 (A7)

Collision J1+J2+

(part+ϑR + [ϑL part+x])larrrarr(part+x+ 12[ϑL part+ϑL] + ) (A8)

40

We have to take into account the interaction vertex in the action

minus S 7rarr1

π

int

d2v1

2str ([ϑL partminusϑL]part+x) (A9)

It is convenient to denote the contracted fields by using prime For example this notation

1

2str ([ϑprimeL partminusϑL]part+x

prime) (A10)

means that ϑL is contracted with the part+ϑR in J1+ and part+x with part+x in J2+ Therefore partminusϑLremains uncontracted There is another possible contraction

1

2str ([ϑL partminusϑ

primeL]part+x

prime) (A11)

In the interaction vertex (A10) let us integrate by parts partminus We will get

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

2str ([ϑprimeL ϑL]partminuspart+x

prime) (A12)

In the second expression let us integrate by parts part+ The result is

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

primeL ϑL]partminusx

prime) +1

2str ([ϑprimeL part+ϑL]partminusx

prime) (A13)

The first term coincides with (A11) and together with (A11) gives

minus str ([partminusϑprimeL ϑL]part+x

prime) (A14)

This is easy to contract and precisely cancels the ldquodirect hitrdquo [ϑL part+x]larrrarrpart+x The secondand thrid terms combine with the ldquodirect hitrdquo

part+ϑRlarrrarr12[ϑL part+ϑL] (A15)

to give the same contribution as the collision

part+ϑprimeRlarrrarr[ϑprimeL part+ϑL] (A16)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

A2 Terms xdx in the collision J2+J2+

Consider this collisionzminus2J2+

zminus2J2+

More explicitly we are looking at(

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

Couplings to xdx receive contributions from the quartic interaction vertex

minus S 7rarr1

6πstr[x part+x][x partminusx] (A19)

We denote the contracted fields xprime(wu) and xprimeprime(wd) When partminusx in the interaction vertex gets con-

tracted with part+x in one of the J2+ this contribution cancels the ldquodirect hitrdquo part+xlarrrarr16[x [x part+x]]

Let us study the diagrams in which partminusx in the interaction vertex remains uncontracted Thereare the following possibilities

16π

int

d2v str (2[xprime part+xprimeprime][x partminusx] + (A20)

+[x part+xprime][xprimeprime partminusx] + (A21)

+[x part+xprimeprime][xprime partminusx]) (A22)

Here prime and double prime mark the contracted elementary fields for example in the firstterm 2[xprime part+x

primeprime][x partminusx] the elementary field xprime contracts with part+x(wu) in zminus2u J2+(wu) and part+x

primeprime

contracts with part+x(wd) in zminus2d J2+(wd) while [x partminusx] remains uncontracted This gives

16π

int

d2v str

(

(minus2)

(v minus wu)(v minus wd)2Cmicroν [t2micro [x partminusx]]otimes t

2νminus

minus(minus1)

(v minus wu)2(v minus wd)Cmicroν [x [t2micro partminusx]]otimes t

2ν minus

minus(minus1)

(v minus wu)(v minus wd)2Cmicroνt2micro otimes [x [t2ν partminusx]]

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

wd minus wu(wd minus wu)2

Cmicroν [t2micro [x partminusx]]otimes t2ν (A24)

(A simple way to get the singularity of this integral is by considering partpartwu

) This contributes tothe current-generator coupling

1

2πi Cmicroν(zminus2[t2micro [x partminusx]])otimes (zminus2t2ν) (A25)

42

Taking into account that Cmicroν [t2micro t0] otimes t2ν = minusCmicroνt2micro otimes [t2ν t

0] we can rewrite (A25) using theand-product

1

2πi Cmicroν(zminus2[t2micro [x partminusx]]) and (zminus2t2ν)

A3 Short distance singularities using index notations

In the main text we gave the expressions for the short distance singularities in the tensorproduct notations Here we list the singularities using more ldquoconservativerdquo index notations

J α1minus(w1)Jmicro2+(w2) =

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

J α1+(w1)Jmicro2minus(w2) =

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

Jα3minus(w1)Jmicro2+(w2) =

1

R3

part+ϑγR

w1 minus w2fγαmicro (A28)

Jα3+(w1)Jmicro2minus(w2) =

1

R3

part+ϑγR

J α1+(w)Jβ1minus(0) = minus

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

Jα3+(w)Jβ3minus(0) = minus

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

J α1+(w1)Jmicro2+(w2) =

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

Jα3+(w3)Jmicro2+(w2) =

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

(w3 minus w2)2partminusϑ

γRfγ

αmicro +O

(

1

R4

)

(A33)

J α1+(wa)Jβ1+(wb) = minus

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

Jα3+(wa)Jβ3+(wb) = minus

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

wa minus wb(wa minus wb)2

partminusxmicrofmicro

αβ +O

(

1

R4

)

(A35)

J α1+(w1)Jα3+(w3) = minus

1

R2

1

(w1 minus w3)2C αα +O

(

1

R4

)

(A36)

Jmicro2+(wm)Jν2+(wn) = minus

1

R2

1

(wm minus wn)2Cmicroν +O

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

(w0 minus w1)2+part+ϑ

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

(w0 minus w2)2+part+x

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

B Very brief summary of the Maillet formalism

Let us briefly review the situation in Maillet et alrsquos work and how this connects to our presentanalysis In [15] a formalism was developed which generalizes the classical YBE to incorpo-rate the case of non-ultralocal Poisson brackets Consider the algebra of L-matrices (spatialcomponent of the Lax operator)

L(σ1 z1) L(σ2 z2) = [r(σ1 z1 z2) 1otimes L(σ1 z1) + L(σ1 z1)otimes 1]δ(σ1 minus σ2)

+ [s(σ1 z1 z2) 1otimes L(σ1 z1)minus L(σ1 z1)otimes 1]δ(σ1 minus σ2)

minus (s(σ1 z1 z2) + s(σ2 z1 z2)) δprime(σ1 minus σ2)

(B1)

The terms proportional to δprime are the so-called non-ultralocal terms The algebra (B1) is adeformation of the standard ultra-local one by terms depending on the matrix s which unliker is symmetric Jacobi-identity for yields a generalized dyamical YBE6

[r12 minus s12 r13 + s13] + [r12 + s12 r23 + s23] + [r13 + s13 r23 + s23] +H(r+s)123 minusH

(r+s)213 = 0 (B2)

6Note that the signs are slightly different in [15] from the equations we will be using

44

The dynamicity is due to the terms Hijk which arise if r+ s is field dependent and are definedby

L(σ1 z1)otimes 1otimes 1 1otimes (r + s)23(σ2 z2 z3) = H(r+s)123 (σ1 z1 z2 z3)δ(σ1 minus σ2) (B3)

In the case of s = 0 and r constant (field-independent) the relation (B2) reduces to thestandard classical YBE This formulation was applied to the O(n) model [15] and the complexSine-Gordon model [14] (where in both cases the r minus s-matrices are dynamical) as well as theprincipal chiral field [13] in which case the terms Hijk vanish Note that the field-dependenceof the r minus s-matrices seems to be due to the field-dependence of the non-ultralocal term

45

Bibliography

[1] N Beisert R Hernandez and E Lopez A crossing-symmetric phase for AdS(5) x S(5)strings JHEP 11 (2006) 070 [hep-th0609044]

[2] N Beisert B Eden and M Staudacher Transcendentality and crossing J Stat Mech0701 (2007) P021 [hep-th0610251]

[3] V V Bazhanov S L Lukyanov and A B Zamolodchikov Integrable structure ofconformal field theory III The Yang-Baxter relation Commun Math Phys 200 (1999)297ndash324 [hep-th9805008]

[4] C Bachas and M Gaberdiel Loop operators and the Kondo problem JHEP 11 (2004)065 [hep-th0411067]

[5] A Alekseev and S Monnier Quantization of Wilson loops in Wess-Zumino-Wittenmodels hep-th0702174

[6] A Mikhailov and S Schafer-Nameki Perturbative study of the transfer matrix on thestring worldsheet in AdS(5) x S(5) arXiv07061525 [hep-th]

[7] A Das J Maharana A Melikyan and M Sato The algebra of transition matrices forthe AdS(5) x S(5) superstring JHEP 12 (2004) 055 [hep-th0411200]

[8] A Das A Melikyan and M Sato The algebra of flat currents for the string on AdS(5) xS(5) in the light-cone gauge JHEP 11 (2005) 015 [hep-th0508183]

[9] M Bianchi and J Kluson Current algebra of the pure spinor superstring in AdS(5) xS(5) JHEP 08 (2006) 030 [hep-th0606188]

[10] N Dorey and B Vicedo A symplectic structure for string theory on integrablebackgrounds JHEP 03 (2007) 045 [hep-th0606287]

[11] A Mikhailov Bihamiltonian structure of the classical superstring in AdS(5) x S(5)hep-th0609108

[12] J Kluson Reduced sigma-model on O(N) Hamiltonian analysis and Poisson bracket ofLax connection JHEP 09 (2007) 100 [arXiv07073264 [hep-th]]

[13] J M Maillet Hamiltonian structures for integrable classical theories from gradedKac-Moody algebras Phys Lett B167 (1986) 401

[14] J M Maillet New integrable canonical structures in two-dimensional models Nucl PhysB269 (1986) 54

46

[15] J M Maillet Kac-Moody algebra and extended Yang-Baxter relations in the O(N)nonlinear sigma model Phys Lett B162 (1985) 137

[16] J Kluson Note about classical dynamics of pure spinor string on AdS(5) x S(5)background Eur Phys J C50 (2007) 1019ndash1030 [hep-th0603228]

[17] A A Rosly and K G Selivanov On amplitudes in self-dual sector of Yang-Mills theoryPhys Lett B399 (1997) 135ndash140 [hep-th9611101]

[18] V G M Puletti Operator product expansion for pure spinor superstring on AdS(5) xS(5) JHEP 10 (2006) 057 [hep-th0607076]

[19] L Freidel and J M Maillet Quadratic algebras and integrable systems Phys Lett B262

(1991) 278ndash284

[20] L Freidel and J M Maillet On classical and quantum integrable field theories associatedto Kac-Moody current algebras Phys Lett B263 (1991) 403ndash410

[21] A Torrielli Classical r-matrix of the su(2mdash2) SYM spin-chain Phys Rev D 75 105020(2007) hep-th0701281

[22] S Moriyama and A Torrielli A Yangian Double for the AdSCFT Classical r-matrixJHEP 0706 083 (2007) arXiv07060884 [hep-th]

[23] N Beisert and F Spill The Classical r-matrix of AdSCFT and its Lie BialgebraStructure arXiv07081762 [hep-th]

47

Introduction
Summary of results
- Definitions
- - The definition of the transfer matrix
  - Setup expansion around flat space and expansion in powers of fields
  - - Fusion and exchange of transfer matrices
    - - The product of two transfer matrices
      - Relation to Poisson brackets
      - r- and s-matrices and generalized classical YBE
      - Infinite Wilson lines with insertions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        Intersecting Wilson lines
        
        Outline of the calculation
        
        Use of flat space limit
        
        Derivation of
        
        Boundary effects and the matrix s
        
        Dynamical vs c-number
        
        BRST transformation
        
        Short distance singularities in the product of currents
        
        Notations for generators and tensor product
        
        Short distance singularities using tensor product notations
        
        Calculation of
        
        The order of integrations
        
        Contribution of triple collisions to
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        Generalized gauge transformations
        
        Dress code
        
        Asymmetry between the coupling of xd+x and xd-x
        
        Coupling proportional to zu-4 xdx
        
        Asymmetric couplings of the form zu-2zd-2 xdx
        
        Asymmetry in the couplings of d
        
        Boundary effects
        
        The structure of G
        
        Introducing the matrix s
        
        Cancellation of field dependent terms
        
        Boundary effects and the global symmetry
        
        BRST transformations
        
        Generalized YBE
        
        Generalized quantum YBE
        
        Some speculations on charges
        
        Contours with loose endpoints
        
        Conclusions and Discussion
        
        Calculation of the products of currents
        
        Collisions J3+J3+ and J1+J2+
        
        Terms xdx in the collision J2+J2+
        
        Short distance singularities using index notations
        
        Very brief summary of the Maillet formalism
        
        Bibliography

Page 2: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

Contents

1 Introduction 3

2 Summary of results 5

21 Definitions 5211 The definition of the transfer matrix 5212 Setup expansion around flat space and expansion in powers of fields 5

22 Fusion and exchange of transfer matrices 7221 The product of two transfer matrices 7222 Relation to Poisson brackets 10223 r- and s-matrices and generalized classical YBE 11

23 Infinite Wilson lines with insertions 12231 General definitions 12232 Split operators 13233 Switch operators 14234 Intersecting Wilson lines 15

24 Outline of the calculation 16241 Use of flat space limit 16242 Derivation of r 17243 Boundary effects and the matrix s 17244 Dynamical vs c-number 17245 BRST transformation 18

3 Short distance singularities in the product of currents 18

31 Notations for generators and tensor product 1832 Short distance singularities using tensor product notations 20

4 Calculation of ∆ 20

41 The order of integrations 2142 Contribution of triple collisions to ∆ 2143 Coupling of dx 2244 Coupling of dϑL 2445 The structure of ∆ 25

5 Generalized gauge transformations 25

51 Dress code 2552 Asymmetry between the coupling of xd+x and xdminusx 26

521 Coupling proportional to zminus4u xdx 26

d xdx 2753 Asymmetry in the couplings of ϑdϑ 28

2

Bibliography 46

1 Introduction

3

int

CJ

4

21 Definitions

T [C] = P exp

(

minus

int

C

Jaea

)

(21)

minus3α +N

[microν]0+ eminus4

[microν] (22)

1α + Jmicro2minuse

3α +N

5

minus1x (28)

Sg0x = x+ ξ +1

3R2[x [x ξ]] + (29)

h(ϑ x eξ) = exp

(

minus1

2R2[x ξ] +

)

(210)

δξλ = minus

[

1

2R2[x ξ] λ

]

δξw+ = minus

[

1

2R2[x ξ] w+

]

(211)

x =

Nsum

a=1

ǫaeikaw+ikaw ++

sum

ab

6

P exp

(

minus

int

Ja(z)ea

)

(212)

Ja+(w)Jb+(0) =

1

wAabc J

c+ + (215)

Tρ2 [C2] Tρ1 [C1] (216)

int

int int

dσ1dσ21

7

VP

int int

int

dwJa+(w)Jb+(0) + (219)

prescription The

(

)

(221)

= eπi

minusπi

2r (223)

8

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 3: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

Bibliography 46

1 Introduction

3

int

CJ

4

21 Definitions

T [C] = P exp

(

minus

int

C

Jaea

)

(21)

minus3α +N

[microν]0+ eminus4

[microν] (22)

1α + Jmicro2minuse

3α +N

5

minus1x (28)

Sg0x = x+ ξ +1

3R2[x [x ξ]] + (29)

h(ϑ x eξ) = exp

(

minus1

2R2[x ξ] +

)

(210)

δξλ = minus

[

1

2R2[x ξ] λ

]

δξw+ = minus

[

1

2R2[x ξ] w+

]

(211)

x =

Nsum

a=1

ǫaeikaw+ikaw ++

sum

ab

6

P exp

(

minus

int

Ja(z)ea

)

(212)

Ja+(w)Jb+(0) =

1

wAabc J

c+ + (215)

Tρ2 [C2] Tρ1 [C1] (216)

int

int int

dσ1dσ21

7

VP

int int

int

dwJa+(w)Jb+(0) + (219)

prescription The

(

)

(221)

= eπi

minusπi

2r (223)

8

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 4: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

int

CJ

4

21 Definitions

T [C] = P exp

(

minus

int

C

Jaea

)

(21)

minus3α +N

[microν]0+ eminus4

[microν] (22)

1α + Jmicro2minuse

3α +N

5

minus1x (28)

Sg0x = x+ ξ +1

3R2[x [x ξ]] + (29)

h(ϑ x eξ) = exp

(

minus1

2R2[x ξ] +

)

(210)

δξλ = minus

[

1

2R2[x ξ] λ

]

δξw+ = minus

[

1

2R2[x ξ] w+

]

(211)

x =

Nsum

a=1

ǫaeikaw+ikaw ++

sum

ab

6

P exp

(

minus

int

Ja(z)ea

)

(212)

Ja+(w)Jb+(0) =

1

wAabc J

c+ + (215)

Tρ2 [C2] Tρ1 [C1] (216)

int

int int

dσ1dσ21

7

VP

int int

int

dwJa+(w)Jb+(0) + (219)

prescription The

(

)

(221)

= eπi

minusπi

2r (223)

8

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 5: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

21 Definitions

T [C] = P exp

(

minus

int

C

Jaea

)

(21)

minus3α +N

[microν]0+ eminus4

[microν] (22)

1α + Jmicro2minuse

3α +N

5

minus1x (28)

Sg0x = x+ ξ +1

3R2[x [x ξ]] + (29)

h(ϑ x eξ) = exp

(

minus1

2R2[x ξ] +

)

(210)

δξλ = minus

[

1

2R2[x ξ] λ

]

δξw+ = minus

[

1

2R2[x ξ] w+

]

(211)

x =

Nsum

a=1

ǫaeikaw+ikaw ++

sum

ab

6

P exp

(

minus

int

Ja(z)ea

)

(212)

Ja+(w)Jb+(0) =

1

wAabc J

c+ + (215)

Tρ2 [C2] Tρ1 [C1] (216)

int

int int

dσ1dσ21

7

VP

int int

int

dwJa+(w)Jb+(0) + (219)

prescription The

(

)

(221)

= eπi

minusπi

2r (223)

8

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 6: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

minus1x (28)

Sg0x = x+ ξ +1

3R2[x [x ξ]] + (29)

h(ϑ x eξ) = exp

(

minus1

2R2[x ξ] +

)

(210)

δξλ = minus

[

1

2R2[x ξ] λ

]

δξw+ = minus

[

1

2R2[x ξ] w+

]

(211)

x =

Nsum

a=1

ǫaeikaw+ikaw ++

sum

ab

6

P exp

(

minus

int

Ja(z)ea

)

(212)

Ja+(w)Jb+(0) =

1

wAabc J

c+ + (215)

Tρ2 [C2] Tρ1 [C1] (216)

int

int int

dσ1dσ21

7

VP

int int

int

dwJa+(w)Jb+(0) + (219)

prescription The

(

)

(221)

= eπi

minusπi

2r (223)

8

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 7: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

P exp

(

minus

int

Ja(z)ea

)

(212)

Ja+(w)Jb+(0) =

1

wAabc J

c+ + (215)

Tρ2 [C2] Tρ1 [C1] (216)

int

int int

dσ1dσ21

7

VP

int int

int

dwJa+(w)Jb+(0) + (219)

prescription The

(

)

(221)

= eπi

minusπi

2r (223)

8

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 8: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

VP

int int

int

dwJa+(w)Jb+(0) + (219)

prescription The

(

)

(221)

= eπi

minusπi

2r (223)

8

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 9: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

limyrarr0

minusπi

2r(B) (224)

limyrarr0

r = r minusπi

2

(

((zminus21 minus z

21)t

22)t

2))

minus

minusπi

2

(

2 minus z2)t1))

minus

minusπi

2

(

((zminus11 minus z

31)t

2 minus z32)t

3))

+

+ (226)

minus1u zminus1

limCuցCd

TCu(ρzuu )TCd

[

limCuրCd

TCu(ρzuu )TCd

(ρzdd )

]

9

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 10: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

=

1

i~

(

limyrarr0+

Tρ2 [C + y]Tρ1 [C]

)

(228)

and the equation

limyrarr0+

J(w)J(0) = J(w)J(0) +F (w w)

w2 or 1wor 1

limǫrarr0+

(

1

(w + iǫ)n+

1

(w minus iǫ)n

)

= 2VP1

wn(231)

limǫrarr0+

(

1

(w + iǫ)nminus

1

(w minus iǫ)n

)

=2πi(minus1)n

prime) (232)

10

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 11: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

r =Φ(z1 z2)

z41 minus z42

(z1z32t

22t

z41 minus z42

t0 otimes t0 (233)

2 minus z31z2)t

2 minus z21z

22)t

2 minus z1z32)t

1 otimes t3 (234)

where

minus22 )2

Ψ(z1 z2) = 1 + z41z42 minus z

41 minus z

42

11

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 12: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

P exp

(

minus

int +infin

0

ad(J(z))

)

ad(J2+) P exp

(

minus

int 0

minusinfin

ad(J(z))

)

(238)

lowastd otimes ρ

primed) (242)

12

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 13: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

232 Split operators

sum

iOiupO

13

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 14: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

inu ρprimeu = ρz

G+(1) = 1+πi

2

[

]

+ (244)

where

outup zdn) + (245)

r+ = r + s

14

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 15: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

outup zdn) + (246)

rminus = r minus s

r + s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(247)

r minus s

2

∣

ρuotimesρd

=1

z4u minus z4d

[

3dt

+z2uz2d(z

2u minus z

]

(248)

15

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 16: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

int

T new = P exp

(

minus

int

C

Iaea

)

(251)

1

RpartplusmnϑL +

16

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 17: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

242 Derivation of r

exp

(

πi

2r

)

[

]

exp

(

minusπi

2r

)

limyrarr0

(

πi

2r

)

(

minusπi

2r

)

minusπi

17

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 18: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

1zoutminus 1

zin

)

F otimes t3β (310)

3β (312)

18

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 19: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

eminus1α otimes e

u z2d t3α otimes t

2micro (313)

eminus1α and e

2micro =

1

2micro minus e

2micro otimes e

minus1α ) (314)

eminus1α and e

1β=

1

1β+ e1

βotimes eminus1

α ) (315)

jb otimes t

kc (317)

wherefabc = fab

For example

19

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 20: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

12

20

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 21: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

21

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 22: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

(

minus dwu

wuminusw

)

= minusπi

ma minus e

ma otimes 1] (41)

43 Coupling of dx

d t1 and [t3 d+x] +

1 and [t3 d+x] +

3 and [t1 d+x] +

2(z2d minus z

22

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 23: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

1

u zminus3d + zuzd)t

1 and [t3 d+x] +

+(z3uz3d minus z

minus1u zminus1

+3

2(z2u minus z

1

= (zminus3u zminus3

u zminus5d minus z

3uz

+(zminus2u zminus4

d minus z2u)t

2 and [t2 d+x] +

5d)t

d minus z3uz

3d)t

+(zminus2u minus z

2uz

4d)t

2 and [t2 dminusx]

1

πi∆prime(dx) =

1

2((z2u minus z

minus2d )2)zminus1

minus1

2((z2u minus z

minus2d )2)zuzd t

1 and [t3 dminusx]

+(zminus2u zminus4

d minus z2u) t

2 and [t2 d+x] +

+(zminus2u minus z

2uz

4d) t

+3

2(z2u minus z

1

2(z2u minus z

23

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 24: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

44 Coupling of dϑL

d t1 and t3 d+ϑL+

1 and t3 dminusϑL+

J0J3 +3

1

= zminus3u zminus2

d (1minus z4uz4d) t

2 and t2 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

4uz

4d) t

4uz

4d) t

2 and t2 dminusϑL+

4d) t

3 and t1 dminusϑL

= (zminus3u zminus2

1 and t3 d+ϑL+

+zminus1u zminus4

d (1minus z4uz4d) t

3 and t1 d+ϑL+

u zd)(1minus z4uz

4d) t

1 and t3 dminusϑL+

4d) t

3 and t1 dminusϑL

1

2d [(z

2d minus z

4dz

3u)t

3u minus z

minus1u zminus4

24

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 25: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

∆prime =πi

2

[

r∆0]

(410)

where

r =Φ(zu zd)

z4u minus z4d

(zuz3dt

2dt

z4u minus z4d

t0 otimes t0 (411)

minus2d )2

4u minus z

4d

[zuz3dt

3 bull t1 t3α

[zuz3dt

3

2 (412)

51 Dress code

25

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 26: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

P exp

(

minus

int

Ja∆(ta)

)

(52)

f = 1minusπi

2

(

+

The result is

πi

[

]

(54)

minusπi

26

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 27: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

minusπi

d xdx

minus S 7rarr1

1

zminus2d+x

It contributes1

d

minus1

]

(511)

1

27

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 28: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

zminus2u zminus2

1

2[ϑLdϑR]+ 1

2[ϑRdϑL]

πi

[

(

minus1

4

)

]

(513)

minusπi

6 Boundary effects

28

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 29: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

2(r + s) + where

πi

h = 1minus1

int

29

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 30: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

d minus z2d)t

2 (64)

d minus z2d)t

2 +

+ πi1

2ϑL t

3d)t

3 +

+ πi1

2ϑR t

1d)t

1 + (65)

πi1

d minus z2d)t

2 + πi1

4(zminus2u minus z

2u)t

d minus z2d)t

2u)t

πi1

d minus z2d)t

2 (67)

30

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 31: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

F+(1) = 1 +πi

2(r(zu zd) + s(zu zd)) + (68)

out

u and ρzin

QBRST = Q+Q (71)

[ǫQ TBA (z)] =1

1

zǫλ(A) (73)

Q1switch =

(

1

zoutminus

1

zin

)

λ (74)

QG+1switch =πi

2

[

]

(76)

1zoutminus 1

zin

)

πi

2

[

31

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 32: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

minus πi(

0 =

[

r+(zoutu zd)

2

1

zoutu

λotimes 1 +1

zd1otimes λ

]

minus

[

r+(zinu zd)

2

1

zinuλotimes 1 +

1

zd1otimes λ

]

+

(

1

zoutu

minus1

zinu

)(

1minus1

z4d

)

[

2

]

=

(

1minus1

z4d

)

1otimes t3) (710)

1minus 1z4d

)

minus

(

1minus1

z4d

)

3 λ (712)

32

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 33: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

minus t1 otimes(

1minus z4d) 1

zd[λ t3 λ] (713)

8 Generalized YBE

minus11 zminus1

(317)

[r12 r13] + [r12 r23] + [r13 r23] = (81)

z22z23z

41 minus

z23z22minusz22z23

+1

z22 z23z

41

)

+

minusz31z23z

32 +

4

z1z23 z2minus

1

z51z23z2minus

1

z1z63z2minus

1

z1z23 z

52

)

+

minusz2z33 z

41 +

z33z32

+z2z3minus

1

z32z3z41

)

+

52z

23 minus z

51z2z

23 + 4z1z2z

23 minus

1

z31z32z

23

)

+

+ permutations

33

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 34: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

3

2

3

2

3

2

3

2

3

2

3

1

minus123+R23minusR

minus113+R

minus112+R12+

minus123+sim= R12+R13+ (83)

34

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 35: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

3

1

2

22

with O

Rminus113+R

minus123+ = T123 (84)

T123O1Tminus1123 = O1 (85)

[r23minus r13+] + [r13+ r12+] + [r23+ r12+] = (86)

minus33 )[t0 t1]

35

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 36: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

minus113minusR

minus123minus)R23+

minus113minusR

minus112+)R12minus

(89)

R12minusRminus123+

(Rminus113minus))

5

R12R13R23 = R23R13R12 (811)

R12R13 =(

R23R13Rminus123

) (

R23R12Rminus123

)

(812)

36

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 37: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

R12+R13+ =(

) (

R23+R12+Rminus123+

)

(813)

T12+T13+ =(

) (

R23+T12+Rminus123+

)

(814)

A12T1B12T2 = T2C12T1D12 (815)

23minusR23+ D12 = Rminus123+ (816)

R23+T(22)Rminus123+

= R(33)(22)T(33)T(22) (817)

R23+Rminus123+

(818)

T(22)T(33)R(33)(22) = T(33)T(22) (819)

37

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 38: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

(τ22T22)(τ33T33) = (τ33T33)(τ22T22) (820)

and thus[(τ22T22) (τ33T33)] = 0 (821)

minus (F+(1))

q =1

2

(

+ (823)

(824)

38

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 39: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

Acknowledgments

39

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 40: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

Collision J3+J3+

β(wb) (A1)

minus S 7rarr1

π

int

d2v1

minus1

π

int

d2v1

1

π

int

d2v str

(

)

(A4)

(

2[ϑR part+x]

)α

(wa)larrrarr

(

2[ϑR part+x]

)β

(wb) (A5)

The singularity is

1

+1

2

1

This is equal to2

3 +wa minus wb

3 (A7)

Collision J1+J2+

40

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 41: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

minus S 7rarr1

π

int

d2v1

1

prime) (A10)

1

primeL]part+x

prime) (A11)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime)minus1

prime) (A12)

minus1

2str ([partminusϑ

primeL ϑL]part+x

prime) +1

2str ([part+ϑ

prime) +1

prime) (A13)

prime) (A14)

which givespart+ϑ

αL

wa minus wbfα

αmicro (A17)

41

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 42: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

zminus2J2+

part+x+1

6[x [x part+x]]

)

(wu)otimesharr

(

part+x+1

6[x [x part+x]]

)

(wd) (A18)

minus S 7rarr1

16π

int

primeprime

16π

int

d2v str

(

(minus2)

2νminus

minus(minus1)

2ν minus

minus(minus1)

)

=

= minus 12π

int

d2v str1

2ν = (A23)

= minus1

2

1

42

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 43: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A26)

1

R3

partminusϑγL

w1 minus w2

fγαmicro (A27)

1

R3

part+ϑγR

1

R3

part+ϑγR

1

R3

partminusxmicro

wa minus wbfmicro

αβ (A30)

1

R3

part+xmicro

wa minus wbfmicro

αβ (A31)

43

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 44: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

1

R3

part+ϑγL

w1 minus w2

fγαmicro +O

(

1

R4

)

(A32)

2

R3

part+ϑβR

w3 minus w2fβ

αmicro +1

R3

w3 minus w2

γRfγ

αmicro +O

(

1

R4

)

(A33)

1

R3

part+xmicro

wa minus wbfmicro

αβ +O

(

1

R4

)

(A34)

2

R3

part+xmicro

wa minus wbfmicro

αβ minus1

R3

αβ +O

(

1

R4

)

(A35)

1

R2

1

(

1

R4

)

(A36)

1

R2

1

(

1

R4

)

(A37)

J[microν]0+ (w0)J

α1+(w1) = minus

1

2R3

(

ϑβR(w0)

βR(w0)

(w0 minus w1)

)

fβα[microν] +O

(

1

R4

)

(A38)

J[microν]0+ (w0)J

α3+(w3) = minus

1

2R3

(

ϑβL(w0)

βL(w0)

(w0 minus w3)

)

fβα[microν] +O

(

1

R4

)

(A39)

J[microν]0+ (w0)J

λ2+(w2) = minus

1

2R3

(

xκ(w0)

κ(w0)

(w0 minus w2)

)

fκλ[microν] +O

(

1

R4

)

(A40)

(B1)

(r+s)213 = 0 (B2)

44

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 45: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

45

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 46: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

Bibliography

46

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

Page 47: Algebra of transfer-matrices and Yang-Baxter equations on ... · Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK Abstract Integrability of

(1991) 278ndash284

47

Introduction
Summary of results
- Definitions
        
        General definitions
        
        Split operators
        
        Switch operators
        
        
        
        
        Derivation of
        
        
        
        BRST transformation
        
        
        
        
        Calculation of
        
        
        
        Coupling of dx
        
        Coupling of dL
        
        The structure of
        
        
        Dress code
        
        
        
        
        
        Boundary effects
        
        The structure of G
        
        
        
        
        
        Generalized YBE
        
        
        
        
        
        
        
        
        
        
        Bibliography

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Documents