I.A. Bandos and A.A. Zheltukhin- D=10 Superstring: Lagrangian and Hamiltonian Mechanics in...

8/3/2019 I.A. Bandos and A.A. Zheltukhin- D=10 Superstring: Lagrangian and Hamiltonian Mechanics in Twistor-Like Lorentz Harmonic Formulation

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R i IC/92/422INTERNATIONA L CENTRE FOR

THEORETICAL PHYSICS

D=10 SUPERSTRING:LAGRANGIAN AND HAMILTONIAN MECHANICS

IN TWISTOR-LIKE LORENTZ HARMONICFORMULATION

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

I.A. Bandosand

A.A. Zheltukhin

MIRAMARE-TRIESTE


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IC/92/422

International Atomic Energy Ag encyand

United Nations Educational Scientific and Cultural OrganizationINTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

D=10SUPERSTRING:LAGRANG1AN AND HAMILTONIAN MECHANICSIN TWISTOR-LEKE LORENTZ HARMONIC FORMULA TION'

I. A. Bandos "International Centre for Theoretical Physics. Trieste, Italy

andA.A. Zheltukhin

Kharkov Institute of Physics and Technology, 3101 08 Kharkov, Ukraine.

M1RAMARE-TRIESTEDecember 1992

This w ork w as supported in part by the Fund for Fundamental Researches of the State Comm itteefor Science and Technology of Ukraine.Permanent address: Kharkov Institute of Physics and Technology, 3101 08 Kharkov, Ukraine.

ABSTRACTThe Lagrangian and Hamiltonian mechanics of a r ecently pr oposed tw istor-ltke

Lorentz harmonic formulation of the D=10, N=UB Green-Schwarz superstring arediscussed The equation of motion are derived and the classical equivalence of thisformulation to the standard one is proved

Presented are the comp lete set of the covariant and irreducible f irst c lassconstraints generating the gauge sym metries of the theory, including K-symm etry. Thealgebra of all gaug e symm etries and sym plectic structure characteriz ing the set ofsecond class constraints are derived Thus, basis for the covariant BRST-BFVquantization of D10 superstring in the twistor-like approach is built.


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it a.jM.

1. Introduction

Superstring in D=10 [1-3] are discussed as the possible basis for building theselfconsistent quantum theory of g ravity and the Unified theory of all the interactions.Howev e r , i ts cov enant quantiza t ion is hampered by the p rob lem o f K-sytnmetry covariantdescrip tion because this ferm ionic sym m etry [4] is infinitely reduc ible in the standardsuperstring form ulation [1-3 ] , Unfortunate ly , the ex isting m odem schem es [5 -7 ] o fcovar iant quantization have been deve loped only for the system s with the finite level ofthe constraint rcduc ibility. (Rem em ber, that such problem appears already in thesuperpart ic le theory [8 ,3 ]) .

The pr ogress in solving the prob lem of covariant quantization is necessary for thecorr ect choic e of the superstring ground state , among the infinitely m any solutions forD=10 sup erstring c om pac tification. As the result of this lack of the true uniqueness theinfinitely m any different eff ectiv e 4-dim ensional theories are appeared instead of theunique 10-dimensional one [9].

One w ay to solv e the problem of covariant superstring quantization uses the fact thatthe reducib ility leve l of the sym m etries is not invariant under the possible r eform ulationsof the theory [5-7]. In o ther w ords , two c la ssica l ly equ iva lent theorie s may have d i f fe rentlevel of the reduc ibility of their sym m etries. Thus some form ulation of superstring theory,including auxiliary variables and being classically equivalent to the standard one [1-3 ] ,may have f in ite le ve l o f the rcducib i l i ty o f K-sym metry and even irreducib le K-symm etry1) .

Now there are know n m any different form ulations of the (super)par ticle and(super)string theories, w hich are classically equiv alent to the original ones [8, 1-3] and

2) Another wa y c onsist in attemp ts to extend the quantization schem e, develop ed byBatal in and V ilkov isky [6 ] , to the case o f system s w ith in f inite ly reducib le sym m etrie s(see 19-11] and Refs. therein). Such extensions use an infinitely reducible gauge-fixingconditions and produce fr ee typ e effec tive actions including infinitely many fields forthe superparticlcs and superstrings.How eve r , the straightfo rward ex tension o f the BV pre scr ip t ion [6 ] fo r the system s w ithinfinitely r educib le constraints leads to the well known troub les [10,11]. So thecohom ologies of the superparticle BRST operator calculated in this way differs fr om thestate spectrum of the Brink-Schwarz superparticle , following fr om the quantization in thel ight-cone g auge (see [10 ,11 ]) . To a ch ieve the r ight BRST cohom ology ( i .e . sta te spe ctrum)it is necessary to m odify not only BV -quantization pr escrip tion, but also the initialsuperpar ticle or superstring form ulation. Howev er, after this step, the second w ay r educesto a variant of the first one.

use the auxiliary harm onic [13-23] , tw :. sto r [24 -40) o r ve cto r [3 ,41 ] va riab le s . A lo t o fw ays m ay b e used for the introduction of such variables b oth in the Lagrangian andHam iltonian app roaches. For instance, in Refs.[14-16] the phase spac e of D=10 G reen-Sc hw arzsuperstr ing w as ex tended by adding the ve cto r SO(1 ,9MSO(1 ,1 )SO(8 )] harm onic va riab le s(u^jijf*) [13] w ith two l igh t- l ike ve cto rs u^ 11 be ing rep la ced by the b i l inearcom binations o f the D=10 boson ic sp inors v : u^ ' = v o m a v - The Lagrangem ultipliers method has been used for the extension of the action in the Ham iltonianfo rma lism. The "harmonic" va riab le s (v , M'1*) and the mom entum degrees o f f reedomcanonically conjugated to them took plac e in the action principle only through theconstraints w hich had b een introduced in the extended Hamiltonian. The Gr assmannianconstraints was cov ariantly decom posed into the irreducib le first and second class ones,the second class fer m ionic constraints was transformed into the first class on es using 'th eintroduced auxiliary f erm ionic variab les and the covariant quantization of Brink-Schw arzsuperparticle and Green-Sc hwar z superstring theories w as carrying out [14-16].

Howev e r , the p ro je ctors , used in [14 -16] include the inve rse degrees o f the m omentumvariab le s. As i t w as noted in [11 ] , such nonpo lynomia l i ty on the m omentumvariables leads to the nonlocality in the vor texes for the cor resp ondingsuperstring form ulation. So it is clear that the further develop m ent of the harm onicapproach and the clarification of the group structure of the harmonic v ariables arenecessary . It seem s to be useful to establish the geom etric nature of the harm onic sectorin the fram e of the Lagrangian approach to the 10-dimensional Green-Sc hw arzsuperstr ing descr ip t ion . Such p r ogre ss may p re sent a natura l geometric m echanism fo r theconstraints decom position and for the D=10 superstring covar iant quantization free fromthe m entioned troubles. This is just the aim the present work seeks to.

Moreove r , th is pape r is devo ted to a further deve lopm ent o f the approach put fo rwardin Refs .[22,23 ], wher e the action functional for the tw istor-like Lorentz harm onicfo rmula tion o f the D-1 0 , N- nB superstr ing w as p re sented . On the le ve l o f ideas , theform ulation [23] dev elops the result of Ref.[28 ], that the choic e of a twistor-like ac tionfor the superparticle is just enough for the covariant and irreducible description ofK-symmetry 3 ) . Thus the task of the covar iant constraint division m ay be r educed to the

3 ) Moreover , it was dem onstrated, that the ic-sym m etry can b e realized as the localrap enym m etry of the superpar ticle w orld-line and that the bosonic sp inor variables(twister com ponents) app ears as the superpartner of a Grassmannian target spac e coor dinates[28-3 0]. Such w orld-line (w orld-sheet) superfield version of twistor appr oach wasgeneralized to the case of sup erpar ticles and superstrings in m ultidimensional sp ace-tim e(D-4 .6 ,10 ) in Re fs . [30 -32 ,34 -40] .


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problem of the construction of the twistor-like form ulation of the action principle.For the theory of null super p-branes (i. e. m assless superp articles (|M) ), null

substrings (p=l) , null supermembranes (p-2)) in 4-dimensional space-time such pr oblem w assolved in Refs. [42,43,19]. In these works a twistor-like Lorentz-haimonic for m ulation fo rthe null super-p-branes w as constructed, the covariant constraint decomp osition w as carriedout, the conversion [7] of the second class constraints into the Abelian first class one isrealized and the covar iant BRST-BFV quantization w as carried ou t using the BFF scheme [7]

In Refs. [42,43,19] w e have noted that the spinor harmonics v "(t/J*1), v + (t,O M)[18], which ar e used in them as the auxiliary twistor-like variables, coincide with theNewm an-Penrose dyads, introduced in Ref.[44]. They are used fo r building the vector fields

of the Cartan mov ing repere (an isotropic tetrade [44]) attached to the worldof (null) super p-brane. And the twistor-like null super-p-bnne action is the

first order form functional constructed using the comp osed vector f rom this moving f rameset. This observation leads to the generalization of the Lorentz-harmonic app roach[118,19,42,43] to the description of superstring and other extended supersym m etric object(lor examp le, supermem brane) in higher dimensions D [2223 ,45 ,46]. Th e proposedgeneralization im plies the necessity of the consideration of the D-dimensional spinorharmonics as a generalized "dyads". Therefore, if the f irst order form action withauxiliary vector variables is known, the problem of the twistor-harm onic descrip tion of thesuperstring (and super-p-b ranes) im bedded into a D-dimensional space-tim e is reduced toconstructing of the realization of the Cartan repere (moving frame sy stem) u w (t > oM )au ^{fh in terms of spinor 2 [ D / 2 W D / 2 ] harm onic m atrix

(1.1)a - Spin{\m)

a = 1 2 V ; a - l , . . . , 2v

with v= [D/2] or (D-2)/2 for Majorana-Weyl spinors in D10(mod 8 ) [18-23,42,43,45,46].But such task ca n b e solved easily. The orthonormal repere

f ,..,A) , (1.2)discussed as DxD m atrix, belongs to the SCK1.D-1) group. Th e double covering of this group

4

is Spin(l,9). Thus the connection of the r cp e re u^B) , with harmonic variable matrix vdefined b y means of the "square root" type universal relation

(1.3)

As the result of E q . ( l . l ) , the relation (1 .3 ) m a y b e rewritten in the fo l lowing forms(1.4a)(1 4 b )

This is possible because, in general case, the follow ing identities

( v C T

Sp (v CT v

0 , ( when l o l ) ,

0 , ( wten l o l )

(1.5a)(1.5b)

are satisfied for the matrix v a * e 5pin(lX>-l) (1.1).The relations (l.I)-(1.3) are the basis of the twistor-like Lorentz harmonic appr oach

to super-p-brane theories.The discussed approach has been named harm onic one, because the condition (1.1) is not

realized b y expressing the m atrix v Q ' as an exponential function of the spin(l,9) Li ealgebra generators; it is realized by the requirement, that v a * matrix should satisfy a setof the so-called harmonicity c onditions

_ ( v ) " 0 (I - These conditions provide the satisfaction of all the relations (1.5), as well as of therelations (1.2), b y definition. And the use of them is m ore convenient, than the use of thestraightforward exponential param eterization (this fact w as been evident already in thecase of the com pact space SU(2VU(1) [47]).

For the case o f D - 1 0 superstring, the matrix v o * has one $0(1.) Majorana-Weyl spinor


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constraints. Thus w e g et all necessary inform ation for the conv ersion of the second classconstraints into Abelian first class ones (see [7]), forthcoming construction of theclassical BRST charge and covanant quantization, which are the subjects of future works.

The paper is organized as follows.To make clear the forthcoming descr iption of supentring in twistor-like formulation

w e consider the boson ic string form ulation with auxiliary vec tor var iables in detail. Thisis done in Section 2, w here the derivation of the m otion equation and the construction ofthe Hamiltonian form alism for sy stems w ith harmonic variables are discussed using thissimp le exam ple. For the reader convenience, the description is closed in this section.

In Section 3 w e describe the twistor-like Lorentz harm onic superstring form ulation[22,23] and discuss its equivalence to the standard one [1,3]. Here we der ive all theequations of m otion for the discussed superstring formulation.

The section 4 is devoted to the construction of the Hamiltonian form alism.The prim ary c onstraints are derived and the so-called cov anant m omentum densities for

the harmonic variab le are introduced in the Subsection 4.1. It is demonstrated m at thesem om entum var iables generate the current algebra extension of the SO(1,9) Lie algebr a on thePdisson brackets.

In the Subsection 4.2 the Dirac prescription of the checking the constraintconservation during evolution is carried out, the covanant and irreducible first classconstraints are derived.

In Section 5 the first class constraints are redefined. This redefinition leads to thesimp lification of the algebra generated by them on the Poisson brackets. Such algebra ispr esented in the Subsection 5.1. The sym plectic structure of the second class constraintsystem is derived in the Subsection S.2. The relation betw een the w ell-known V irasoroconstraints and the reparam etrization sym m etry generators of the twistor-like form ulation[:2,23] is discussed in the Subsection S.3.

Our notations for the Majorana-Weyl spinor indices in EMO coincide w ith ones fromRefs.[14,15] except for another choice of the metric signature (see Eq.(1.2)).

1 B O S O N I C S T R I N G IN THE C A R T A N M O V I N G F R A M E F O R M U L A T I O N .

2.1. Aaion principle and equations of motionTo make m ore clear the forthcom ing description of superstring in twistor -like

Lorentz-harm onk app roach we consider the bosonic string form ulation w ith thefollowing action functional

s - J i% L - \


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chosen to be c om posed f rom two l igh t- like ve cto rs w ith the def in ite and oppositew e ights under the SO (U ) g roup . Thus i t is conven ient to work in te rm s o f thel ight- l ike zwe inb e in comp onents

(2.4)and light-like vectors

(2-5)n 1* 21 n - ' ^ L 0 n 1" 2' [+2)

(com pr ise w ith Eqs. ( l .Sb )) 6Xwith [22 ,23]) .

*) In such form the coincidenc e of the reper e variables (1 ) w ith the10

The variation of the action (2.1) with respec t to the inverse z w einb einsg iv e s d ie fo l low ing re la tion

(2.6)

w hich is the simp le expression for the sou ldary fo rm /( d^ .^ )* ^ u ' o f t hew orld-sheet, induced by em bedding of the w orld-sheet into the D-dim ensionalMinkowski sp ace -t ime . Taking in to a ccount Eq .(1 .6 ) , we m ay exc lude the aux ilia ryzw e inbe in f ie ld f rom the a ct ion (2 .1 )

- 2ce = - 2

n [ j ] ) ,(2.7)

(2 .8 )

The resulting action (2.8) coinc ides with one from Ref.[3 0], wher e the auxiliaryvec tor fields from Cartan mov ing repere had been introduced for the first tim e forbuilding string and superstring actions.

Thus the action (2.1) is the first order form repr esentation for the"antisymm etric" a ct ion f rom Re f . [30 ] .

Now let's discuss the relation of the discussed string form ulation (2.1) withthe standard Dirac-Nam bu-Goto and Polyakov ones. '

It should be pr oved below , that the variation of the action (2.1) w ithrespect to the auxiliary vector fields n'"(^) leads to the follow ing nontrivialequations

y" (2.9)w hich m eans that the vec tors n u) are orthogonal to the string w orld-sheet.Eqs. (2 .6 ) , (2 .9 ) . The comp le teness o f the mov ing f rame system

vecto r harmonics f rom Re f . [13] is e v ident Howeve r , the repe re va riab le s w asused for the first time for the string and superstring description in Ref.[ 30].

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g ive the possib i l i ty to express a xm () through

(2.10a)

amand v ise ve rsa

Taking into acc ount Eqs.(2.7) and (2.11), as w ell as the definitions

w e m ay rew rite the a ct ion S (2 .1 ) in the fol low ing fo rm

s p = - (2ccO l

w hich is the known string action introduced in Ref.[48], From the other hand ;Eq .(2.10) leads to the follow ing expr ession for the induced m etric i

v vw hich r esults in the relation

e =The substitution of Eqs.(2.10a), (2.10c) into the functional (2.1) leads to the Dirac-Naoibu-Goto action

At last, the variation of the action S (2.1) with respect to x " giv es theequation

(2.12)w hich m ay be rew ritten in the standard form (se e Ref.[48])

12

y /7 r (2 .13)using Eq .(2 .11 ) .

How eve r , the deriva tion o f the Eq .(2 ,9 ) , wh ich is crucia l fo r the conc lusionpr esented abov e, is not so sim ple task. First of all note that the v ariations]p rob lem w ith re spe ct to l f | f ie lds is the p rob lem w ith condit iona l ex treme due tothe necessity to take into account the orthonormality conditions (2.2). It m ay b ereform ulated into the variadonal pr oblem w ith absolute extrem e, if w e extend theaction (2.1) by m eans of adding the conditions Em) (2.2) with thecorre sponding Lagrange m ultip l ie rs (see [30 ]) . Another w ay fo r y ie ld ing the r ightm otion equations is to restrict the class of adm issible variations o n j 0 by thevariations whic h conserv e the orthonorm ality conditions (2.2). The use of thism ethod does not require the introduction of the Lagrang e m ultipliers and seem s tobe sim p le r fo r the so lution o f the va riadonal p rob lem s characte rized by thesophisticated structure of constraints. The same m ethod will be used below forstudying D10 superstiing dynam ics in the twistor-likc form ulation [ 2223] .

2 2 . Admissible variations for repere variables.

Let's discuss arbitrary set of D independent vector variables nj 1 in D-dim ensional space. The condition of the independence has the form tkiinj* ) * 0.Thus the set of the variables nj1 1 discussed as DxD m atrix belong s to the GL(D.R)group. An arbitrary v ariation with respect to n 0)

5- BnJ" Sl dnm ay b e rewritten in the fo rm

5 - (/f'fiB),

(2.14a)

(2 .14b)

In E q . ( 2. 1 4b ) ( " ' S " ) ^ * " ( " \ " ^ i s " C a r t a n dif ferentia l fo rm , w h ichis invariant under the left GL(D.R) transformations. The differential op eratorsB J " a/On , appeared in (2.14b), may b e discussed as the covariant derivatives(see [47 ]) fo r the GL(D,R) group .

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Let's r estrict the right GL(D,R) transformations (acting on the numb ers (1)o f the ve cto rs n n> ) to be on ly f rom d ie Lorentz g roup S OO JM ). Then theinvariant m etric tensor

app ears and we achiev e the possibility to lover and to rise the indices in thebr ackets. After this step w e may transform Eq.(2.14b) into the form

* ( " * l 8 i l W " - w ^ " V < 2 1 4 c )and to dec om pose the GL(D,R) covar iant derivatives n m d/dn onto the symm etricand antisym m etric pa rts

(2.15a)

(2 .15b)

The corr esponding dec om position of the Cartan differential form is defined by therelations

Taking into account Eqs.(2.15) and (2.16), the expression (2.14) forarbitrary variation may be p resented in the form

14

5 = i fax w (2.17)It is easy to show that A o w ) and Km) operators generate the gl(D,R)

Lie a lgebra2 V / w ' ai8a)

= 2 / 2 a i 8 b )

= 2 K 2 - 1 8 c )

with the suba lgebra (2 .18a ) o f the Lorentz g roup p roduced by D(D-l) /2 Af l xk )operators. The operators K are related with the factor space

G L ( D ,R y S O ( l ,D - l >

and the num ber of them coinc ides with the numb er of the orthonorm ality conditions3 O X k )Now it is evident that the adm issible variation m ay include A operators

only. T his statement is true due to the fact that the Lorentz r otations are theonly transformations w hich conserve the orthonorm ality of repere. Let's, how ever ,arrive this statem ent in a m ore form al way . This help us to understand a m orecom plica ted case o f sp inor moving f rame variab le s ( i .e . Lorentz harm onics [20 -2 3 ] ) .

The action of A and K operators on the variables - /" m ay b e easilydeterm ined (see Eqs.(2.15))

(2.19a)

So we have2 r | * ,XCl () + ^ ft

(2 .19b)

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Eq s.(2.20) justify the statement thatorthononnality conditions (2.2)

operators conserve the

At the same t imeA S ' 0 ,

Henc e the operators K destroy the rep ere orthononnality. Moreover, theKdifferential form (2.16b ), rotated to the operator K (see (2.17)), isWreduced to the com plete differential of the orthononnality condition S , onthe surface (2.2)

I s - o (2 .21c)So the follow ing v ariations

IS-0 9)are adm issible (i.e . conser ve the repere orthonorm aUty c onditions (2.2)).

In Eq.(2.14d) the covariant SO(1 J> 1) derivatives has the form (2.15a) andthe expr essions for the Caitan form s (2.16a) m ay be reduced to the follow ingones

(2.22)

16

on the surface defined by the orthonorm ality conditions (2.2).It is interesting to note , that Eq.(2.14a) may be discuss ed as the de finition

of the covariant derivatives &m) . So , &m) m ay be understood as thederiva tive s w ith re spe c t to the Cartan fo rm s Q (8 ) .

If the reper e variables becom e the fields living on the wor ld-sheet

then the variational analogs A*() of the operators A should b e used(2.23)

and w e should use the follow ing f orm of the admissible v ariation(2.24)

instead of one defined by Eq.(Z14d).Now w e are ready to discuss the derivation of Eq.(2.9).Taking into account Eq .(2.24), it is easily to see , that the variation of theaction (2.1) w ith respect to the repere fields n lj] is defined by the relation

6S c)

W e stre ss, that the simple covariant derivative (2.15a) is used in the last partof Eq.(2.25). This is the result of application of the variational derivative(2.23) included in the previous part of this equation.

Hence , w e m ay conclude that:i) the right equations of m otion for the repere fields have the form s of thevariations of the action (2.1) with respect to the Cartan form s (2.22)

5S/8no )(k)(5)- 0 (2-26)

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(These eq uations take into acc ount the orthonorm ality c onditions (2.2)automatica l ly );

ii)thcse equations m ay be pr esented in terms of the Lagrang ian density andord inary covariant de riva tive s a s fo l low s

- 0 (2.27)

This statement is true for case s, whic h are sim ilar to the discusse d one, w herethere are no time deriv atives of reper e fields in the action;

iii) the equations of m otion are defined by the result of the action of theordinary covariant derivatives (2.15a) on the fields n \

So the equations of m otion for the n(i ) fields have the form

We m ay spe c i f y them as fo l lows , using the Eq .(2 .19a) ,(2 .28)

y m (2.29)Thus it is ev ident, that motion equations for the fields n(l > g ive nontriv ia lconsequences on ly fo r the cases (k )Hf) o r (\)~{f) . The equations (2.29) aresa tisf ied identica l ly w hen (k)# ( f } and OWH - This is the consequence o f thegauge SO(8) sy m m etry o f the d iscussed a ct ion (2 .1 ) . The ope rators Aa) gener ates these transform ations.

The Eq.(2.29) r educes to the follow ing relatione e? a,.x-" n

m ill]0 , (2.30)

w hen the both indices (k) and 0) b elong to the {f}- set Eq.(2,3O) is satisfiedidentically if Eq.(2.6) is taken into account This fact corresponds to theSO(1 ,1 ) gaug e sym m etry o f the discussed act ion (2 .1 ) .

Hence the unique nontrivial consequence of Eq.(2.28) corresponds to thevar ying of the action w ith respect to the Cartan form Slim describ ing thevaria t ions f rom the cose t SO(l ,D-l) / tSO (l , l )xSO (D-2)] . u has the fo rm o f therelation

18

= 0 (2.31)and is equivalent to the Eq.(2.9).

Henc e, the equations of m otion for the discussed b osonic string for m ulation(2.1) are derived using the variational principle based on the concept ofadm issible variation (2.22),(2.24) of the reper e fields. It is a simp le task toderiv e the same eq uations of m otion using arbitrary variations and the extendedaction functional comp leted by the products of the orthonormality c onditions (2.2)S**" on the Lagrange m ultipliers (see [30] for this app roach app lied to these cond order fo rm act ion) .

Howev e r , fo r the case o f tw isto r- like Lorentz harmonic fo rmula tion o fsuperstring (see Section 3)) the describ ed form of var iational princ iplesim plifies the calculation significantly.

2 3. Hamiltonian formalism and covariant momentum densities.

Now let's discuss the Ham iltonian form alism for the bosonic stringform ulation (2.1). The first order form of the action princip le results in thefo l low ing f a ct All the expressions fo r mom entum density va riab le s

(2.32)

(2.33)result in som e constraints. For the discussed form ulation of the b osonic stringthese prim ary constraints have the form

canonically conjugated to the configurational space coordinates of the theory

e e) n } - 0 , (2 .34a )(2 .34b )

19


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H o we ve r , th e repere orthonorm ality c onditions (2.2) should b e discussed as theadditional primary constraints

n-0). (2.35)

TJ& for the repere variables n cl >( ) ar el > ( )the Hamiltonian m echanics

corresponding

if the canonical mom entum densitiesused. Such extension of the set of constraints makesmore complicated in the discussed case (see [30]). But thecom plication for the case of twistor-like form ulation o f D - 1 0 supentring[22,23) b ecom es drastic. Indeed, in the formulation [22,23 ] the comp licatedharmonicity conditions (U O) .(l .ll ) appears instead of the orthonormalityconditions (2.2).

Henceforth, it is significant to work out the m ethod, whic h allows to excludethe conditions similar to (2.2) from the set of constraints and to discuss them asthe strong relations. Such a m ethod w as used in fact in Refs.[13-17,20,21] a nd w a sgrounded shortly in Ref.[13] for the superparticle case (see also Refs. [49,18]).Here w e justify this method in details for the case of bosonic string form ulation(2.1). Such justification m akes more clear the forthcoming discussion for the caseof twistor -like superstring form ulation.Let's return to the primary constraints (2.3 4). Th e first of them (2.34a) m aybe decom posed into the tw o relations, using the orthonorm ality conditions (2.2),

0 ,(2.36a)(2 .36b)

Eqs.(2.36) m eans, that the repere variables m a y b e discussed as the m atrix of theLorentz transformations, w hich c onnects an arbitrary coordinate frame w ith thefixed one, where the string mom entum density P has only tw o nonvanishingcom ponents (which coincides with the i-comp onents of the zw einbein density e e **)

p . (p p ^Similar interpr etation of the Cartan-Penrose representation rewritten in terms of

20

D-4 Lorena harmonic matrix w as given in Ref.[18],Let's extend such interpretation to the case of harmonic sector and form the

SO (1 J> 1 )L invariant m om entum densitiesa U a ( V i

After the division of F f t M ) into the symmetric

and antisymm etricTo*

" * ) ^ 1 ) ' " m O )

(2.38)

(2.39)

(2.40)parts, w e g e t D(D+1V2 symm etric and D(D-l)/2 antisymm etric constraints equivalentto (2.34b)

\m " ' (2-4U)n


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It may be justified that the variables \ m & and \ m & realize avector representation of the gl(D,R) current algebra on the Poisson brackets(2.42)

[ n w ( a ) >(2.43a)

(2.43b)- 2

(2.43c)= - 2

- 4 i 2M)

or , in the week sense,

- 4 (2.47)

So it is natural to consider the com binations of the phase var iables P" andi t, presented by I and E**", as a new canonically conjugated variablesdescribing D (D+ l)/2 degrees of fr eedom . Due to their v anishing in the w eek sense,the phase variables 5** and m ay be excluded from the string dynam icsby the transition from Poisson b rackets to the Dirac one (see [50])

[F , G ] D - [F, G]p + I J da [F. S ) , G] p -

The constraints n ^ (2.40) form the representation of the so(l, D-l ) currentalgebra and, consequently, do not change the constraints (2.35) in the w eek sense

V (2.44)

= " 2 ( n nSo it is naturally to consider n ^ as the (covariant) m omentum variables forthe degrees c onnected with the Lorentz subgroup SO(1JD-1) of the GL(D,R) group( i . e . to the orthonorm al rep ere).

Contrary to II , the symm etric constraints don't conserve theorthonorm ality c onditions (2.35) . Indeed,

(2.48)

The m omentum variables, remaining after the exclusion of ^ (lK]). are thecovariant mom entum densities n . So it is im portant to express the Diracbrackets (2.48) in term s of II . With that end in view let us discuss thechange of variables from P* m om entum densities to Fl and ones. Itis based on the evident relation

)

(2.49)Using (2.49) w e find thatJ d o 7 S P ( ' ) n i ( a V 5 n ( r ) a ) ( o )

22 23


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(2.50)

J d & o / ^

and consequently(2.51) ;

(2.52)Using the representation (2.S2), the change of the m omentum densities m a y b e donein the Poisson and Dirac brackets. So Eq.(2.48) m a y b e presented in the form

[F, G]D= Jdo (oF/8Paj) 8Q/ox"(o) - 6F/Sx"(a) 5G/8PJ

+ J do (SF/ o T I^c ) P^oyG -do

+ I J do [F, S*W) (o)]p [ I J o ) , G]p -

- I J do [F, G] p ,where the variational covariant derivatives S are defined b y Eq.(2.23).

Let us discuss som e functionate F , G which ar e Independent on thevariables

F * FExn((i), P_(or),

(2.53)

L-X(2.54)

24

G- G[x"(o),The Dirac brackets (2.53) coincide with (the "covariant" version of) the Poissonones on the class of such functions

[ F , G ] D J ( o ) 2 ( r W ) ( o ) G - [F , G ] pS o , the ordinary Poisson brackets (2.55), together w ith the strong relations

(2.2), m a y b e used for the function with the properties (2.54).There fore w e a r e f ree f rom th e necessity of the inclusion of the

orthononnality condition* {12) into tb e Hat of the Ham illonian constr aints ifthe phase space include only th e covariant m omentum densit ies n^.C a) (2 .40)fo r tb e repere variables. Such m omentum densit ies ar e characterized b y t h ep ro p e r t y

(2.56)

The similar pr escription should b e used below for the investigation of theHamiltonian m echanics for the twistor-like supem ring formulation [22,23], Itg i v e tb e possibility to take into account the com plicated harmonicity conditions(1.10), (1.11) as the "strong" relations and exclude them from the list ofHam iltonian c onstraints.

To clarify the nature of the covariant mom entum densities n^^Ca) let'sprove that they m a y b e defined as the derivatives of the Lagrangian density w ithrespect to the x-components of the Cartan differential form (2.22)

n * , W (0)f,

The comp onents a " * - (Q \(2.57)

Q* W% ) of the Cartan25


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differential form n m \d ) (2.22) with respect to the holonomic basis dp- (dt,da ) ar c defined b y the relation

= dp dx da (2 .58) IIndeed, using the com pleteness of the set of differential form s OP^Sjand S f tx l )(5) (2.17), w e m a y decom pose the derivative with respect to 8j iiX> as

follows , a/afi ' * > a / a s < k * > ,

(2.59) IMultip ly ing Eq .(2 .59 ) on n and taking the antisym m etric p an of the resultingexpression , w e g e t the re lat ion

(2.60)

Henceforth, the expression (2.57) coincides with (2.40) (see also (2.35)), a nd weconclude, that the covatiant mom entum density characterizing b y the property 1(2.56) is defined by the follow ing expression !

n (k ) (1) (2.61)Finally, w e should note, that the cov aiiant m omentum density (2.61) is the

"classical analog" for the variational cov aiiant derivative (2.23). This statementmeans, that the Poisson bracket of n with any admissible functional, defined jon the configurations] space (2.33), coincides with the action of the variational 'covaiiant derivative (2.23) on the sam e functional

F[ ]p (2.62)The discussed properties (2.56). (2.61), (2.62) of the covaiiant m omentum

density n ^ should help us to find the corresponding variable for the26

twistor-like supcrstring formulation [22,23] and, thus, to simp lify thei n v e s t i g a t i o n of i t s H a m i lt o n i a n m e c h a n i c s ( s e e S e c t io n 4 ) .

27


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3 . D = 1 0 SUPERSTRINGIN TW ISTOR-LKE LORENTZ HARMONIC FORMULATION

3,1, Action functional

The twistor-like action functional for the D 10, N -IIB superstring hasthe fonn [22 , 2 3]

V z Co c ) -

(3 .1 )

J dutc e [ - u lj + + c ]

J dida e ( c + ^ (o V * Z1*21 co" (v ; o j; )

\l KV k m k' ' ' (3.1a)

S w z 1 J


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0) = -(5 o f - 5 * ,

In Eqs.(3 .5 ) -y^ a re the a -matrice s fo r SO(8) g roup (see [3 ]) . 7^ y j -The p re sence o f the Wess-Zumino te rm (3 .1b ) in the a ct ion (3 .1 ) leads to

the invariance o f th is a ct ion under the asym m etry transfo rmations whic hexp l ic i t fo rm w as p re sented in Re fs . [23 ,46 ] , There a re a lso e v identreparamctr iza t ion sym m etry and the gauge sym m etry t inder the r ight p roduct o fSO(8) and SO(1 ,1 ) g roups.

The SO(8 ) gaug e sym m etry transform ations result in the arbitrary rotations ofthe e ight spac e l ike com posed ve cto rs u (see Eq .(1 .8 )) among them se lve s. And theSCK1.1) ones result in the pseudorotan'ons of the vectors u"1 . To a ch ieve theinvariance of the action functional (3.1), they should be identified w ith thew orld-sheet Lor entz group transform ations acting on the "flat" indic es of thez w i e n b e in s e ' (see Eq.(2.4)).

The relations (3.4) together with the gaug e sym m etry under the the rightproduct o f 8 0 (1 , 1 ) and SO(8) g r oups pe rmit us to identif y the space o f harm onic ;va riab le s { (v ^ , v j )} w ith the cose t space SO(1 ,9V[SO(1 ,1 )SO(8)] [22 ,23 ,46 ] . Westress that the so called " boost" sym m etry is absent in the discussed super stringform ulation (3 .1) in distinction with the form ulations of the D- 10 Green-Schw arzheterotic superstring presented in [38, 39]. The cause of such distinction shall bed iscussed be low .

3.2. Harm onic variables, composed moving frame vectorsand admissible variations

The relation (3.4) is realized by the requirem ent that the variables


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3" A f t0 ,

(256-25 6=0, c onsequently additional degr ees of freedom arc not included in thetheory) .

Note that the distinction in the 80 (1, 1) w eights for the sam e SO(8 ) ((s) or(c) sp inor) index structure shal l he lp us to dist ingu ish the harm onics v ^ , v ^(3 .4 ) f rom the (3 .7 ) ones v~ a t v ^ in the expressions where the S0(1 .9 ) sp inorind ice s a re contracted and omitted (see , fo r example , E qs .Q.l l ) ) .

It is easy to p rove that the com posed repe re ve cto rs < n) ( 3 . 3 ) m a y b eexpr essed trough the inverse harm onic matrix (3.7) as well as trough the ordinaryone (3.4) (see Eqs.(1.9))

A (3 .9 )In te rm s o f the harmonic va riab le s v * v ^ and v ' a , v j 0 1 E q s . ( 3 . 9) m a y b espec ified as follows (s ee Eqs.(1.8 ),(l,9) and the O-matrix repr esentation (3.5))

HI n m Z *2'- (3 .10 )

(3 .11a)

(3 .11b)

(3 .11c)

The orthonotmality conditions (1.2) m ay be spec ified as follows

< > < > " (3 .12 )32

] = 0 # (3 .12a .b )

( 3 . 1 2 c )

2 , (3 .12d ,c)

To justify them exp licitly the identity (1.12) and the conseq uences (1.4) of theharm onicity conditions (3.6). (3.8 ) should be used (see [46] for details). For thed iscussed D-10 superstr ing case the re la t ions (1.4 ) may be sp e ci f ied as f o l lows

( 3 . 1 3 )

Cr*.a ** * V_ , " f t i

,I- 21 2 va i

( 3 . 1 3 a )( 3 . 1 3 b )( 3 . 1 3 c )

(3 .14 )

2 v " 01 v -vA A

(3 .14a)( 3 . 1 4 b )

( 3 . 1 4 c )

33(3-15)


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(3.15a)(3.15b]|(3.15c)

where the conditions ( 3 .8 ) . used expliciUy. To specify Eq . (3.18) let's useEq.(3.11), the consequences (3.13)-(3.16) of the harmonicity conditions and theknown identities (see, for example, Refs.[14,15])

Te < 3 1 9 a>

(3.16)

(3.16rf(3.16b)'(3.16c)

VT I

A>

1 2

41 16

(3.19b)

(3.19c)

For the forthcoming derivation of the equations of m otion let's discuss theconcept of an admissible variation for the case of spinor harmonic variables. Thiiis the variation which doesn't destroct the harmonicity conditions (3.6), (3 .8](or, equivalently, the relation (3.4)). Such variation w as discussed in detail fo ithe case of the fundamental repere v ariables in Section (2) (see Eqs.(2.14a)-(2.14d. Thus w e m a y om it some evident steps in the discussion of the spinorharmonic case.

An arbitrary variation of the variables v " and (v" 1) m v.a 8

8v ( 3 . 1 7 J

m ay b e written in the form

(3.18)a

Indeed, varying Eq.{3.15)

g * - 2 < v H ' : - rand contracting the result (3.20) with the 10x16 matrix

( 5 < o V

5 b t a i n

(3.20)

1 0 (5 * > v " ' 6 v o < l t ) ) ( 3 - 2 U )

It is easy to see that the left hand side of Eq.(3.21) m a y b e presented in theform B " * ^ 5 B (I ) ( O J . This results from the vanishing of the expression

" * ^ ' " " Sm)

0 h i h i h f hn =' ( k X Ui1 000) which is the consequence of them (kXU 1 0 0 0 )orthonorm ality conditions (3 .12). The right hand side of Eq.(3.21) m a y b e

transform ed using the identities (3.19c) for v 'ov and the relations

34 35


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0-22)

Thus w e derive from Eq .(3.21). 10 (3.23)

+ I (o1"2 ) 5p(om v ' o v ) + 1 - (a " 1 " * 4 ) 5 p ( o v ' S v )1 2 "l*""4Contracting Eq.(3.23) with the matrices /, cm) , .follow ing relations

S p( v 'ov ) - 0 * v ; 5v+= - v * o v ; ,

, w e produce the

(3.24)

v W " )

Sp(v'6v o ) - 0 ,m m

Q*M)(8) , (3.23)

which are the straightforward consequences of the harmonicity conditions (3.6),(3.8). Taking into account Eqs.(3.24)-(3.26)), it is easy to derive from theidentity (3.19c) the follow ing expr ession for v'&V

(3.27)

Thus the adm issible variation which conserve the harmonicity conditions (3.6),( 3.80 has the form

O X * ) 1 (3.2*)

36

which coincides with Eq.(2.14d). However, in Eq.(3.28) the SO{1,9) Cartan form sQ* W) (5) aw defined b y Eq.(3.25) in term s of spinor harmonic var iables (3.4),(3.7) , and covanant derivatives A are defined as follows

. a a (3.29)

Taking into account the definition of the com posed rcpere vectors (3.3) (orEq.(3.9)) w e m a y obtain the action of the covanant derivatives A g ^ , o n th e

(3.30)

Eq.(3 .30) coincides with Eq.(2.19a). I t ma y b e also justified that Au,xyoperators g enerate the Lorentz group algebra (2.18a).

For the forthcoming discussion it is useful to specify Eqs.(3.28), (3.25),(3.29) as follows (the contracted spinor indices are om itted)

o _ I *( ) A " + G tl2U (5 ) A^ - I a\$) A ( iXi)(3.31)

(-2 (3.32)

i ^ V

i v

37

(3.32a)

(3.32b)

(3.32c)


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. [ v yn 5 v + + v +

A = (-2 A . A[*IK , A )

- v" 8/av-k -

A - v :

A 0*" i (v 7 ^ a/sv + v" y ^ 8/avj

(3.33)

+ v * 8 / 8 v + , ( 3 . 3 3 a )

( 3 . 3 4 b )

(3.34c)

\ ), (3.34d)

A[-2Ki) (J) (3.38)

In the notations (3.33) the Lorentz group algebra (2.18a), generated by theoperators A , takes the form of the relations

[ A< \

[A*',

= 0 ,

2 g " ^ ' * - 2 8 PV* ,

= . f i" ' A l R l i +

( 3 . 3 9 a )

( 3 . 3 9 b )

( 3 . 3 9 c )

( 3 . 3 9 d )

( 3 . 3 9 e )

A (0 ) t 2 1 = 2 l f a l ,

Am2 H (

0 >

(3.35)

(3.36)

[.iKi) t*ii _ 0

(3.37)

The nature of the operators A*0' , A l* il u ' , A11*" be c om es e v ident f rom theEqs.(3 .35 )-(3 .39 ) . A 0 X i ) and A*" generate the SO(8 ) and SO (1,1)transform ations respec tively. The operators A1*21 generate the transform ationsf rom the cose t space SO(1 ,9 )[SO


23/44

(3.40)

This expr ession is similar to Eq.(2.6) since the iweinb ein variables are absent inthe expr ession for the additional Wess-Zum ino term (3.2) . j

In the straightforward analogy w ith the repere bosonic string f orm ulation(see Se c t ion 2 , Eq .(2 .28)) . the equations o f m otion fo r the harm onic va riab le s

vj. can be presented in the form

(3 .41 )

using the admissib le variation (3.28 ), whic h conser ve the harraonicity c onditions(3 .6 ) , (3 .8 ) .

Eq s.(3.41) are satisfied identically when (k)-( i) and GW j) (seeEq .(3 .10 )) . This re su lts f rom the SO(8) gauge sym m etry o f the a ct ion (3 .1 )genera ted by Am opera tors (3 .33d) . I f (kH + 2] , (1 )- t -2 ] (or v ise ve rsa ) ,then Eqs.(3.41) reduce to the relation (see Eq.(3.37))

(3.42)

Taking into ac count Eq.(3.4O) it may b e justified that Bq .(3.42) is satisfiedidentically. This fact is associated with the gaug e SO(1,1) sym m etry of thetheory . How eve r , the genera tor o f the co rre sponding sym m etry includes te rm s a ct ingon the z w einbein f ields in addition to the A*01 opera tor (3 .33a ) (see Se ct ion 4 ) .

Since , Eq s.(3.41) lead to the nontrivial results if and only if (k )-[2] ,(1) (i) (or v ise v ersa). In this c ase they reduce . to the eq uations (seeE q s . ( 3 . 3 7 ) , ( 3 . 3 8 ) )

(3.43)w hich can b e easily transformed into the form

a 0 ) = 040

(3.44)

(com prise w ith Eqs. (2 .31 ) , (2 .9 )) .Taking in to a ccount Eqs. (3 .40 ) . (3 .44 ) , and the com p le teness condit ions

(which fo l low s f rom the orthonormality ones (3 .12 ))

8 B - I am 2I a1'21 1 m

the coef ficien ts eo (3.2) of the (0-form pull-back m ay be decom posed on the u^l igh t- l ike ve cto rs

t2]

; - i c (a-)lf land vise versa

(3,45)

(3.46)

Thus the vectors u" 1*21 are tangent to the superstring w orld-sheet on the sh ell,defined b y the m otion equations. Contrary, the vectors umM are orthogonal to thewor ld-sheet on this shell.

Using Eqs. (3 .40 ) , (3 .45 ) , (3 .46 ) , the c la ssica l equ iva lence o f the d iscussedD- 10 super string form ulation with the standard Green-Schw arz one [1] can bejustified easily. Substituting Eq. (3 .46) into the functional (3 .1) and using thedefinition of the wor ld-sheet metric (2.4), w e get the standard action functional[1 ] ( com pr ise w ith Eqs.(2 .9>(2 .13)) .

The equation of m otion for the x" ^) f ield, 6S/oY "(4) = 0 , has the form

- a o , (3.47)

w hich is sim ilar to Eq.(2.12), exc ept for the last term containing Grassm anniandegr ees of freedom , and m ay be easily reduced to the standard form [1]

V (3.48)41

m m Hi f ff


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using Eq .(3 .48) (see a lso Eq .(2 .1 )) .The equations SS / f ie*1 = 0 have the form

" 2 (" 1

w hich m ay b e reduced to the standard one [1]

( 3 4 9 >

4. HAMILTONIAN FORMALISM FOR D=10 SUPERSTRINGIN TWISTOR-LIKE FORMULATION

To simp l i f y Hamilton ian fo rm a lism and to m ake the m eaning o f som e c onstra intsm ore c lea r , le t us re fo rmula te the a ct ion p rincip le (3 .1 ) in te rms o f the zw e inbe indensities

Vif Eqs .(3.46) are taken into acc ount How ever , it is interesting to use Eq.(3.45)and to exclude the fields v from the equation (3 .49 ) . S ince w e de rive thefo l low ing re la t ions

^\fiai v nl = 0 , (3.51a)

0 (3 .51b)

w hen the Eqs.(3.13a, b) are taken into accountTherefore the equations of motion for the D - l, NIIB superstring in twistor-

l ike fo rmula tion (3 .1 ) have the form o f Eqs. (3 .40 ) , (3 .44 ) - (3 .47 ) , (3 .52 ) . Therelations (3.4 7), (3 .52) are equivalent to the standard equations of m otion(3 .48) , (3 .50 ) [1 ] , howeve r they have m ore sim p le fo rm. Thus the tw isto r- l ikeform ulation (3 .1) are equivalent to the standard one [1] on the classic al lev el[23 ] and sim plifies the equations of motion essentially.

In the next sections the Hamiltonian form alism for the twistor-like D -1 0superstring form ulation (3.1) is worked out This form alism is necessary for thecovar iant superstring quantization using the BFV -BFF schem e [7].

e -

? - ( y - P^21). &*' P*'21 )) - e e/ (4.1)

instead o f zw e inbe in e' , ej1 themse lve s

(4.3)

S, - - i J fftto ( P ^ * 21 " lJ c a'

- i J (v c a '

" J(4 .3a )

(4 .3b )Here (see Eq .(3 .2 ))

a," - i W -Of c ourse , the Wess-Zumino te rm (4 .3b ) is not modif ied . Howev e r , the tw isto r- like

42 43


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part of the action (4.3a) includes the term s whic h are dependent on the densitiesp i11*21 in a linear or bilinear way . At the sam e time , their dependenc e on the inversezwe inbe in va riab le s ^ 2 ) are the m ore com p lica ted ones (see Eq .(3 .1b )) .

4.1. Primary constraints and covariant mom entum density .

The canonica l m omentum densitie s

V V ' *a are conjugated to the configurational space (target space ) coordinates of thediscusse d superstring form ulation (4.3)

Mz e a i ea2 -a avw ith respec t to the standard Poisson b rackets

(4.5)

(4.6)

Here the m ultip l ie r ( - 1 ) *^ is equa l to (-1 ) . i f the bo th ind ice s M and N be long tothe ferm ionic v ariab les, and is equal (+1) in any other case.

The action functional (4.3) is the fust order one on the proper timederivatives (i.e on the velocities). Hence all the expressions (4.4) for thecanonical m om entum densities leads to the primary c onstraints. For the nonharmonicalvariables such constraints are

(p)X - 0 . (4.7c)

(4.7d)

For the sp inor harm onics

) - * * )T , * )


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The expressions a ( l > [20,21] included in Eq.(49a) vanish as the result ofm r " " jEq .(4 .9a ) [46 ] and m ay be sp e ci f ied as fo l lows

a1'21 -iv _ 0 .i1"21 = 5 VOA u . r j VYA "

(4.10a)

(4 .10b) I

(4 .10c)

The r elations (4.9), (4.10) are com plicated ones. So it is evident, that thecom putation of the c onstraint algebra is a bant task, if the Eqs.(4.9) areunderstood as the "week" equality.

Henc e, it is imp ortant to work out the method whic h allows to exclude theconditions (3 .6),(3 .8) fr om the set of constraints and to discuss them us the"strong" relations [50]. Such method can be devised in the straightforward analogyw ith one discussed in Section 2 for the case of bosonic string repere for m ulation.

This m eans that the conc ept of cov ariant m om entum density should be used. Noww e disc uss it for the case of twistor-like superstring form ulation (4.3) usingthe exper ience obtained in section 2 (and hence om itting som e technicaldetails).

Le t us remind some p rope rt ies o f the covariant m omentum densit ie s , wh ich w asdiscussed in Section 2. First of all they should have the vanishing (in the weaksense, see Eq .(2.56)) Poisson brackets with the harm onicity conditions (3.6 ), (3.8 ).From the other hand, it is known that the harm onicity conditions (3.6), (3.8 ) arethe realization of the relation (3.4) [20,21,46] . Since , the covar iant mom entumvar iables should be canonically c onjugated to some param eters of SO(1.9) gr oupincluded in the spinor harm onics ( v a * , v ^ ) v a * , ( v ^ a , v ) m v ^ . There fo re ,the covariant m om entum densities should be associated with the Lorentz gr oup too,and hence they should generate the SO(1,9) group algebra on the Poisson brackets.

Another degrees o f f r eedom included in the sp inor harm onics v _ t , v ^ ! , v" a , v t are killed by the harm onicity conditions (3.6), (3.8 ). Henceforth, the harm onic

m omentum degrees o f f r eedom, wh ich cannot be reduced to the covariant ones , shouldbe conjugated to the harm onicity conditions in the week sense (se e Eq s.(2.46),(2.47)for the case of bosonic string reper e form ulation). Since we m ay understand thecondition of vanishing of these variables, together with the harm onicity c onditions(3.6), (3.8), as the "strong" equalities, if the corresponding Dirac brackets areused instead of Poisson ones (4.6). These Dirac brackets should be analogous to onespresented in Eq .(2 .48) . How eve r , i f w e d iscuss the space o f funct ions dependent onthe covariant harm onic m omentum densit ie s on ly , these Dira c b ra cke ts co incide w iththe Poisson ones (4.6).

The discussed situation is similar to the case , wher e the second c lassconstra ints a re so lved exp l ic i t ly ( i .e ., the supe rf luous m omentum degrees o f f reedomvanish and the coordinates conjugated to them are expressed through the "p hysical"ones) [5 0]. The unique distinction is that the 256+256 harm onic var iables areexpr essed through the 45 degrees of freedom associated with the SO(1,9) gr oup in animp lic i t way . Such imp l ic i t dependence is def ined by the harm onicity c ondit ions(3 .6 ) , (3 .8 ) . (See [22 , 2 3 , 46 ] fo r de ta ils) .

Hence , the harmonicity condit ions (3 .6 ) , (3 .8 ) can be excluded f rom the se tof Ham iltonian constraints w ithout changing the Poisson brackets if w e define theset of covar iant m om entum densities with the prop erties listed abov e and exc ludea ll o ther harmonic m omentum variab le s f rom the phase space .

The experience of studying the repere bosonic string form ulation (2.1) giv es usthe pr escrip tion for the extracting of the covariant mom entum densities from the setof canonical ones.

First of all, these densities are the classical analogs of the covariantderiva tive s (3 .31 )-(3 .3 4 ) appearing in the expression (3 .31 ) fo r the adm issib levaria t ion . I .e . , they m ay be de rived f rom the expressions (3 .31 )-(3 .3 4 ) b y thesimp le rep la cement o f the de riva tive s a /dv , 6/av by the canonica l m omentumdensities Pm , P a*.

From the other hand, it may be derived as the derivatives of the Lagr angiandensity L o f the a ct ion (4 .3 ) w ith re spe ct to Qzm) (where O t * W ) are thet-coefftoients of the pull-backs of the SO(1,9) Cartan differential form s (3.3 2) onthe world sheet) . The ir form m ay be de rived f rom Eqs.(3 .25 ) , (3 .32 ) a s fo l low s

46 47


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* ^ 1 ' - - ISp { 4 tor , equ iva lently , using the fo l low ing repr e senta tions

(4.11)

(4.12)

fo r the d iscussed pu l l-backs.Thus the general expression for the covariant mom entum densities in the wh ole

phase space (4.S), (4.4) has the form

>_v) - flL/fl(fl.v" 1)o* xV), (4 .13)The Poisson br ackets of the covariant mom entum density with any functional F,

l iv ing onthe configurational space ofthe d iscussed dynamica l system , m ay bepresented in the form

B (o ) . F[v ,v" ' ,x ,8 ] ] = 2 0 ) ( B ( o ) F [ v , v ' V , B ] , (4.14)Here ax k '(o) are the variational analogs ofthe covariant harm onic derivative s(3 .29 ) , (3 .33 ) . Thus these covariant de riva tive s p lay thesame ro te for thecovariant m omentum v ariab le s , asthe ordinary derivatives play for the canonicalones

[ P ^ o ) , F [ / ^ ] ] p = c75z(o) F[^]i M o r e o v e r , thecovariant mom entum densit ie s (4 .13) g enera te s thecurrent

algebra, assoc iated with the Lorentz group algebra (2.18a), (3 .39), on the Poissonbrackets (4.6)

[n n(o),nM((i0]p-[A",An ] 6(0-0-) (4.15)and have the vanishing Poisson br ackets with the harm onicity conditions (3.6), (3 .8)

S(oO, ] p 5=0 0 , (4.16)

w hen the sam e harm onicity conditions are taken into acc ountThus we should leave on ly the covariant harm onic m omentum densit ie s (4 .13) in

the phase space , w h ich is pa ramete rized now by fo l low ing va riab le s

, P B ( O ) ;

And only the prim ary constraints

- 0

n 0 ) f t ) ( o) ) (4.17)

(4-18)

should be taken into account besides ones presented in Eqs.(4.7). Eqs.(4.18) replacethe w hole set (4.8), (4.9) ofconstraints for harm onic variables inthe discussedappr oach. The harm onicity conditions (3.6), (3.8) are understood asthe strongequality.

The Poisson brackets are defined byEqs.(4.14), (4.15) or bythe basicrelations

, ]p = i

(4.19a)

(4 .19b)

which lead tothe Eqs.(4.15), when the Jacoby identities for the Poisson br acketsare taken into account.

Now let us discus s the form of the canonical Ham iltonian HQ density w h ich a reconsistent w ith the definitions ofthe Poisson brackets and ofthe Ham iltonianequations of motion

f ( o ) . (4.20)48 49


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The standard exp ression for the canonical Ham iltonian

H'1"*1 = - (-if* a/4 VM - L (4.21)has such consistency w ith Eqs ,(4 ,6 ) and (4.20). This may be ve ri f ied by fo l low ingform al m anipulation. The use the density (4.21) in the Eq.(4.20) written for thesimp le st funct ion f = ^ ( t . o ) l e a d s to the identity

] p

To ac h ieve such c onsistency w ith Eqs. (4 .14 ) , (4.19 ) , (4 .20 ) , we shoulddefine the canonical H am iltonian density in te rms of covariant harmonic m omentumvariab le s as f o l l ow s

Ho L(a) (4.22)I.e., instead of the standard com bination 6 ^ V^ (wh ich can be derived by therep la cem ent 8 - ax , 3 / f l r ^ -?M from theexpression for arbitrary variation 8 o / 'alBi^ ) the expression

n a x k ) ( a )( w h i c h can be derived by the rep la cement Q*W> (S) - n ' kW ) (a ) , ACXB. . (o ) f rom the expression for the adm issible variation (3.28 )) app earsin the canonical Ham iltonian.

For the fo rthcoming d iscussion some spe ci f i ca t ion of the relations (4.13),(4 .18) , (4 .22 ) is necessary .

Le t us introduce the covariant m omentum densitie s n*01, n'121 , n0*11 ,w h i c h are the classical analogs of the covariant de riva tive s (3 .33 ) . In t er m s ofthe canonica l m omentum densit ie s P (P "

a, pt ) and P (P

+, P ") they arev A A * -l\ OCA OCAdefined by the relations

( _ 2 n< P)>

50

rf.-in^.vtp;.vxP^v;pt + vtPi, (4.23a)(4.23b)

(4.23c)

(4.23d)

It is ev ident (see Eqs.(4.15), (3.39)) that the densities IT*0' and lfm genera te s(on the Poisson brackets) the Kac-Moody-like extensions of the S C K U ) and SO(8)group a lgebras re spe ct ive ly . The densities n1 ' 21 0 ' are associated with the cose tS O ( l , 9 y [ S O ( U ) x S O ( 8 ) ] .

The Poisson brackets (4.6) m ay bewr itten in the follow ing form

[P. 0 ] , = \d o (8F78P (a )r J m- Jrfo (SP/ofl^o) 80/ 8 ^(0 ) - &T/SKm(.a)

8G/8P (a)) + (4.24)

JJ da 3 * ^ 0 ) 0 - 5(rK1)(a)Ffor the functionals F and fl living in the phase space (4 .17 ) (com prise w ithEq.(2.55)). All the expressions (4.23) vanish in the week sense (4 .18) . Hence theprim ary harm onic c onstraints has the fo l low ing fo rm in the discussed approach

- 0 , (4.25a)

n I ? J K i ) - - o (4 .25b)51


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o , (4.25c)At last, the expr ession (4.22) for the canonical Ham iltonian m ay b e specified i

a s fo l low s

P n (o ) (4.26)

where L denote the Lagrangian density for the action (4.3).Lets resum e the results of this subsection w hich define the starting point for

the next one.Hence , the phase space o f the d iscussed system is pa ramete rized by the

variables (4.17) , Poisson brackets is defined by the standard relations (4.6) forthe ordinary variab les and by the relations (4.14), (4.15), (4.19) for the harm onicones T) . The canonical Ham iltonian is defined by the relation (4.26). And the setof pr im ary c onstraints include the relations (4 ,7a-d) and (4.25).

4.2. Irreducible f irst class constraintsfor the D=IO , N=2B Green-Sckwarz superstring

The first class c onstraints can be extracted by m eans of the w ell-know n Diracprocedure of checking the constraints conservation during evolution [50].

The evolution of the dynamical variables of system w ith constraints is definedby a gener alized Ham iltonian, which is the sum of the canonical Ham iltonian and thepr oducts of the prim ary constraints on the corr esponding Lagrange m ultipliers. Forthe discussed dynam ical system with die prim ary constraints (4.7a-d), (4.25) the

7 ) Of c ourse , the sim ple expr essions (4.23) and the initial Poisson bracketsdefinition m ay b e used for the calculations, because the Poisson br ackets were notchanged (see above ) .

52

gener alized Hamiltonian has the follow ing for m

tf = Jdff H'(t,o) ,

H'(t,o) - H0(x.o) + ' v; D > ) + ^ v (4.27)

aHere the canonical Ham iltonian H is defined by the general expr ession(4 .26 ) fo r any dynamica l system l iv ing on the phase space (4 .17 ) . For thediscussed superstring form ulation (4.3) it has the follow ing form

V /


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c a' - 0 and n i j 0 (4.25a,d) has no the nontrivial consequences .(Let's rem ind, that if* and TV generate 50(1,1) a nd S0 ( 8 ) transform ations onthe Poisson br ackets). How ever the conservation of the other 16 harmonicconstraints Tl [21: - 0 (4.25) ha s nontrivial consequences for the Lagrangem ultipliers

P % um ' * p " (4.32a,b,This m eans, that the gauge symmetry under the transformations from the coset spaceSO ( l , 9 ) / [SO ( l , l ) S0 (g ) ] are absent in the discussed form ulation. Such fact w asdiscussed in Rcfs.[22,23] in details.

The consistency condition fo r Eqs.(4.32a) and (4.32b) leads to the relation

a' ewhich results in the vanishing o f the $0 ( 8 ) vector Lagrange m ultiplier

(4.33)in the case of a nondegenerate world-sheet metric (or, m ore precisely, nondegeneratewor ld-sheet m oving fr ame). Using Eq.(4.33) , w e ca n se e that Eqs.(4.32a,b) producethe follow ing secondary c onstraint

CO" H ( i ) - 0 ,C in (4.34)whi ch is the c - co m p o ne m of Eq.(3.44) ') .

Together with the Eqs.(4.30a,b), Eq.(4.34) give the possibility to decomp osethe com ponent of the Cartan form p ull-back co onto the basis o f two vectors mof the target space m oving fram e w hich, therefore, are tangent to the w orld-sheet onm ass shall (i.e. on the shell defined by the equations of m otion in the targetspace).

The requirem ent of the preservation of the Grassmannian spinor constraints ) ' ( a ) - 0 (4 .7b) g ives the expression for the Grassmannian Lagrange m ultipliers i^ 2and 4 i through the dynamical variables

-.01-21

t - i " O i l '

(4.35a)

(4.35b)

The rest of the Grassmannian Lagrange multipliers 5* 1 and % ' rem ain independent andp l a y ' the role of the parameters of the fermionic K-symmetry in the f ramework ofHamiltonian form alism. W e should stress that this sym m etry is present in the theoryonly for the definite choice of the numerical coeffic ient a' included in the Wess-Zumino term (4.3b) of the superstring action (4.3). If this coefficient is differentfrom i jj , , the conservation conditions for the Grassmannian constraints (4.7b )

Da(a)result in the relations

) It is an interesting to note that for the case of a degenerate world-sheet m etric, which c orresponds to the null-superstrings [52,53,42,43,19]Bqs.(4.32a) and (4.32b) becom e consistent without using Eq.(4.33) , and thesecondary constraints (4.34) are absent (see [42,43,19]).

54 55


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VOA Pm* ( - 1

(4.36)From Eq .(4 .36 ) w e m ay g e t the re la tions s imila r to Bqs. (4 .35a ,b ) no t on ly fo r d ie4 + 1 and t,\ l . but also for the rem aining Grassmannian Lagrange m ultipliers * ' and^ 2 . (The prim ary and secondary constraints (4.7a), (4.30) should be used in suchcom puta tions) . So the fe rm ion ic K sym metry is absent fo r the theory w ith thenum erical coeff icient in front of Wess-Zum ino term d *

If w e choos e this coeff icient a* to be equal to (- ~^) , instead of + ,then the re la t ion (4 .36 ) g ive s the expressions l ike Eqs. (4 .35a ,b ) , but fo r the

Lagrange mult ip l ie rs * ' and j ! . The Lagrangian m ultipliers S*2 a nd l ' , w h i c hA A A A

rem ain undeterm ined in this case , play the role of the parameters of the K-symm etrytransform ations.

The conservation of the constraints (4.7a)

H e r e Q^ , Q^*1101 . Q^ LXfl are fee coefficients of the pull-back of the SO(1,9)Cartan form s (3 .34 ) on the world sheet. They are related to the do differ ential andthe ir fo rm m ay b e de rived f rom Eqs.(3 .34 ) using the re la t ion

u^ "21The p ro je ct ions o f Eq .(4 .37 ) on Ihe comp osed ve cto rs u^ - , < r - , < r(3 .11 ) o f d ie moving f ram e system (3 .10 ) g ive us the fo l low ing expressions fo r theLagrange m ult ip l ie rs

flo p < + 0 p *- 21 a V 2 1 - 41 ( 4 . 3 8 a )

' Cp t t - 2 ]+ p Olft]

( 4 . 3 8 c )

y ie lds the fo l low ing re la t ion

At last, w e should ver ify the conservation of the secondary constraints (4.30a,b )and (4.34). They m ay b e presented as the pr ojections of one constraint

% " j (4 .39 )

(al+ 2 1 0 )

(4.37)56

!" 21 '* 21 u (l )onto the moving f rame v e cto rs !" 21 , u'*21 , u (l ) . The requirem ent of theconserv ation of this constraint leads to the equation sim ilar to (4.37). M oreover ,the pr ojections of this equation onto the m oving fr ame v ectors u[~ 2 ' and uj^ 21co incide w ith Eqs. (4 .39a ) and (4 ,39b ) , re spe ct ive ly . Howev e r , i ts p r o je ct ion onto

57


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the m oving f ram e ve cto rs u d i f fers f rom Eq .(4 .38c) and has the fo rm

(4.40)

^ ] +

This relation corresp onds to the requirement of the conservation of the secondaryconstraints (4.3 4) and thus is absent in the case of null-supentr ing (as well uthe constraint (4.34) oneself). Consequently the corr esponding "boost" sym m etry[20 ,21 ] , w hich charac terize the superparticle [18,20,21] and null superstring theory[19,4 2,43 ], is absent in the case of twistor-like superstring form ulation. This isbec ause the discussed superstring action (3.1), (4.3) contains spinor harmonicvariab le s o f bo th types: the v Q * harmonics a s w e l l a s the v j [ ones. The e ight"boost" sym m etrie s [20 ,21 ] consist in the shi f ting one o f these harm onics b yharm onics of another type

" C C A 8v It is clear that such sym m etry is present incontains only one of the types of the harm onics

the theories w hich form ulationv a * o r v ! . This p rope rty issatisfied not only in the twistor-like form ulation of m assless superp article ,

null superstrings and null super-p-branes [18-21,42,43], but also in the heteroticstr ing fonnulations o f the type o f ones d iscussed in [323 4 ,3 63 8 , 40 ] ) .

58

Eqs.(4 .38c ) and (4 .39 ) can be so lved w ith re spe ct to the Lagrange m ult ip l ie rsal+2W) an d a

lJ 0 } ? *(4.41a)

ai-!w> . l r - o v a i Q i J i H i> . 2 / 1 _ , + 1 1 \ a e a i v : +

(4 .41b)

Thus the verification of the conservation of the constraints under the evolution iscompleted and, hence, the complete set of the first class constraints is extracted(up to a transition to som e linear comb inations of them). They m a y b e defined as thevariations of the genera lized Hamiltonian H1 (4.27), (4.28) with respect to thegeneralized Lagrange multipliers, which se t m a y contain th e undetermined fieldparameters of the canonical Ham iltonian playing th e role of the Lagrange m ultipliersfo r di e secondary constraints (besides the original Lagrange multipliers). In thediscussed case w e m a y u se as a generalized Lagrange multipliers th e moving framedensity variables p 01*21 w hich is related with the original Lagrang e m ultipliers

) These form ulations m a y b e named half-twistor-like bec ause only one of theVirasoro constraints is "twistorized" (i.e. solved using twistor-likeprescription) in them. A nd just this fact explain th e presence of the (heteroticsuperstring) "boost" sym metry in them. Indeed, as it is easily to see fromthe discussed superstring form ulation (3.10 ), (4.3), th e inclusion of both typesof harmonic variable is necessary just for the "twistirization" of both Virasoroconstraints. And so , the form ulations in which only one of them is "twistirized"m ay b e constructed using only on e type of spinor harmonics and, consequently,may have the "boost" symmetry .

59


33/44

by Eqs. (4 .31a .b ) l * .After the substitution of the expressions for the dependent Lagrange

m ult ip l ie rs (4 .31 ) . (4 .3 3 ) , (4 .35 ) . (4 .38a ,b ) , (4 .40 ) in to the expr essions (417 )and taking into account (4.2S), the generalized Ham iltonian may b e wr itten asf o l l o w s

( 4 . 4 3 b )

\}.X-L- CO n [ - 2 1 T

I a" (4.42) 2567 2cb T (4.43c)The first class constraints has the following form

, [ - _ 2 atf 'A* _ r ^coT anau

" coT V (p )x " coT " o F(p)r (4 .43a )

+ 2 p t + J ! t p j p j i - 2 Pn u - o ,

[-:it - o

+

l0 ) It is im portant to note that the choice of the generalized Lagrang em ultip l ie rs is a ve ry de l ica te po int So , i f w e try to use the com ponents o f thewor ld-shee t repe re ( " ' a s the genera l ized Lagrange mult ip l ie rs ( instead of

0 *om ponents o f ve cto r densit ies (o/) l/2 ), then the extracting ofthe co rre sponding f i rst c la ss constraint be com es p rob lematic b e cause o f thenonlinear dependence of the resulting expr ession for the gener alized Ham iltonian one 01 7i. In the d iscussed case such p rob lem m ay be so lved by using the re la t ions(4.31) of the discussed repere var iables with the original Lagrange m ultipliers andrequ ir ing that new genera l ized Lagrange m ultip l ie r m ust be expressed b y l inearrelation through the original one.

60

(4 .43d)

(4 .43e )

(4 .43 f )

( 4 . 4 3 g )

where the expressions D^ , L^ (I l^ ) are defined by the relations

c& T " 5(4.44)

(4.45)

M

The first class constraints (4.43a,b) generate the reparam eterization sym m etryw ith paramete rs p 01 * 11 on Poisson brackets. The first class constraints (4.4 3 c4 ),(4.43e), (4.43 f) generate the K-sym m etry transformations (w ith param eters % *1 and^ s ) , SO(1 ,1 ) symm etry (w ith param eters a"" ) , SO(1 ,1) sym m etiy (w ith paramete rs

61

' t &


34/44

a' J) and, finally, the sym m etry under the arbitrary shifting o f the reper e densityc o m p o n e n ts p 0 1 J ' (w ith paramete rs p 01 *21 ) . The la st sym m etry m eans the Lagrangem ultiplier nature of the variables p 01 * 21 ).

The connect ion o f the reparamete riza t ion sym metry genera tors (4 .43a^0 w iththe well know n V irasoro constraints should be discussed in the next section.

Thus the com plete set of the coven ant and irreducib le first class constraint 'for the D=10, N=H B superstring in the twistor-like Lorentz harmonic for m ulation(3 .1 ) , (4 .3 ) is de rived .

5 . ALGEBRA OF IRREDUCIBLE SYMMETRIESAND SECOND CLASS CONSTRAINT SYMPLECTIC STRUCTURE

FOR D=1 0 , N*EB GREEN-SCHWARZ SUPERSTRING

5.J. First class constraints and their algebra

To sim p l i f y the a lgebra o f the gauge sym m etries genera ted by the f i rst c la ssconstraints (4.43), let us redefine them, using som e linear transform ations insideof die first class constraints set T o form ulate the results of such r edefinition ina com pact fo rm , le t us in troduce the fo l low ing b oson ic and fe rmion ic b lo cks

L L

w h e r e .[!! pi -2 1(p)T ' (0

(5.1a)

(5 .1b )

(5.2a)

(5-2b)

(5 .3a )

(5 .3b )

and expressions D^ , L^ (1*1,2) are defined by the relations (4.44), (4.4S), or bythe expressions

62 63


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- o (5 .7c )

The algebraic structure associated w ith block s (5.4) , (5.3) is very simp le one

{ DlaS) J>fc


36/44

] p - 2i

Vjp) , ] p ] p ,

(5 .10b)

( 5 . 1 0 c ) ;

(5 .10d) '

as it is easy to see from Eq s.(5.6), (5.7)The f i rst c la ss constra ints Y ^ = (Y^ , Y ^) (4 .43a -d) , wh ich ge nentes K-sym metryand rep aram eterization transform ations, may b e redefined as follow s

Y > >

L ((o) - 2 pTl+21

L 2 (a) - 2 p 1 1 1 1 t [ + 2 1 . 2 - 4

The distinctions in Eqs.(5.11), (5.12) with respec t to (4.43a-d) layi) in the adding of the expressions

and2 _Xc a ' _ [ - 2

(5.11b)

(5 .12b)

w hich are pr oportional to the first class constraints constraints (5.11b ), (5.12b )(or, equiv alently, to (4.43b), (4.43d )), to the constraints (4.43a, c) resp ectivelyand

ii) in the m ultiplying of the resulting expressions on the overall f actors2 p a nd 2 p " r e s p ec t iv e ly .

The algeb ra of reparameterization and K-symm etry transform ations, assoc iatedw ith the same lig ht-like direction tangent to the world sheet, is realized in thefo rm o f the fo l low ing b ra cke t re lat ions

1 Y 1 i - r 1 K Y 1M Y N Jp " C M N Y N

- 4

- 4

r v 1 Y 1 1 = C 2 K Y 2 1 M ' N JP MN N '

[L 2(o), L 2(c0]p = 4EL J(o), B j ( o O ] p - 4 ( c a T 1 j (

- 4

,2(O) + L 2 (


37/44

where t l ^ V ) is the o-com ponent of the SO( U) Cartan form (3.32a); they transformas the connection (or gauge field) com ponent under the gauge SO (1,1)transformations.

The b rockets of the reparam eterization and K-symm etry generators, associatedw ith the different light-like wor ld sheet directions, have m ore com plicatedstnicture. How ever they are com pletely defined by the relations (5.7c)

> ) , Y ' ( o ' ) ] p

= - 16 (coO (5.15a)

L V ) ]p -(5.15b)

(5.15c)(5.144)

where tlo+2 ) '. &Q 2] ' > " the comp onents of the pull-backs of the covar iant Cartanforms (4.32b,c ) and

' ; ! ' - 2( P ) x

- 0 , (5.16a)

m n ij - 0 . (5.16b)68

a r e the f i r s t c l a s s c o n s t r a i n t s ( 4 . 4 3 e J ) g e n e r a t i n g the S O ( 1 , 1 ) and S0(8) p u g esym m etries (on the Poisson brackets).

The fact of closure of the of the supenepanuneterization sym m etry algebr a( i . e . the algebra of rcp anuneterizations and K-symm etry transform ations) on theS0 ( 1,1) and S0(8 ) g auge sym m etry transform ations is the significant one. It m eansthat SO( U)x SO( 8 ) gaug e sym m etry connects different light-like directions, tangentto the world sheet

The b racket relations of the SO(1,1) and SCK8) sym m etry g enerators (5.16a, b)with another first class constraints are defined b y the SO


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2 S i | i- (5 .18d)So , the algebra of the first class constraints

' Y N ( a )

[Y A(o) , Yj.(oO ] p = Jda" C^ C o, o ' io") Y n(o")is com p le te ly spe ci f ied by Eqs. (5 .13)-(5.15 ) , (5 .17 ) , (5 .18) . ex cep t fo r thebr acket relations of them w ith the rest two first class constraints Z^ 2 ' - 0(4.43g). These brackets are all vanish because of the absence of the variablesp[?2]O -m ^ C X p r e s s i o n s f or the first class constraints

The symm etry genera ted by the constra ints P ? ' - 0 (4 .43g ) ind ica te theLagrang e m ultiplier nature of the zweinb ein densities p ' t I i a in the discussedform ulation.

70

52. Second class constraints ,their algebra an d symplectic structure

The rest of the constraints (4.7), (4.25) are the second class ones. Theya lso may be de com posed natura lly onto the two se ts

Sf - (S} . Sj ) - 0associated w ith different light-like direc tions tangent to the w orld-sheet

Sj - 0 :L l f f lJ ) l(o) = u" 0' [P^

n I + 2 K V ) - o ,

(5.21)

(5.22)

- 2. a ae1< J m e I] - 0 . (5.22a)

(5 .22b)

(5 .22c)

p |+2,T (5 .2M)

(5.22e)

S j - 0 :

v ;a(o ) ^( a ) - 0 ,r - 2i

(5 -23)

- 0 , (5.23a)

(5 .23b )

71


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n 1 2 1 S ) ( o ) - o ,

. 0 ,

( 5 . 2 3 c )

(5 .23d)

(5 .23e )

A nondegenera te symp le ct ic structure Q1'

(5.24)

of the set of constraints (S.21), (5.22) is the block-diagonal one and is definedby the relations ,

[ s ; , s; )p - o ; ( :

[L l

] p - -2(ccO"' (S*1 fi o- Q

- 2

- i - ^ L ^ C ) . L I(i>(o0 ] p

6(o-oO ,

(5.25)

(5.25a)(5 .25b)

(5 .25c) j(5.25d)(5 .25e );

) p 2i Ljo)

] p - 2 ( c a O '

] p =

i w -1-11 L^XoO ] p

P [ 2 1 tSAB

- 2

(5.25f)

(5.26)

(5.26a)(5 .26b)

(5 .26c)

> 1 2 1 X 6 ( a a O ,(5.26d)

(5 .26e )

(5.26f)

All other br ackets between the pairs of the constraints from the sam e set (either(5.20), or (5.21)) vanish in the strong sense. The brac kets betw een theconstraints from different se ts an all equal to zero in the week sense

[S 1 , S 2 ) p - 0 (5.27)All of these br acket relations, which are nonvanishing in the strong sense, involv ethe constraints (5.22c) , or (5.23c)

[S j . S j } p * 0 ( - 0 ) : (5 .28)

72 73


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- 0 .

Jp 5(0-00 - 0 ,

0 ,

(5.28a)

(5.28b)

(5.28c)

S ; }p = flj - (5.29)is described com pletely by the relations (5.25). (5.26. (5.28).

S3. Reparameterization generators an d Virasoro conditions.

V ) lp = 81' i1"1*11 1^(5" (2p'-2|V [ L2(o) + A,

1 ftnrt - o5(o-(o)8(o-o> 0.


41/44

is the fu st class constraint by definition. Thus, there is the follow ing(dependent) first class constraint

Vn(o) i f i f t; 0 - 0 (5 .34 ) |in the d iscussed dynamica l system . The l inear comb ination o f Eq s. (5 .32 ) and (S .34 )

(5.36)

$) a o e a i B ^ ( o ) - om ay b e wr itten in the form

t > ) t"\a) + 2 (ca'p )' u"^(o ) a / " Dfc) - 0( i f the comp le teness condit ions

fo r the com posed m oving f ram e variab le s (1 .8 ) , (1 .9 ) is taken in to a ccount) .It is easy to s ee that the first class constraint (5.35) coinc ides w ith theV irasoro condition (5.3 0a) up to the Grassmannian and harm onic degrees of freedom

(w hich are ab sent in the standard bosonic string form ulation).

6. CONCLUSION

So the c la ssica l m echanics o f the tw isto r l ike Lorentz harmonicfo rm ula tion o f the D- 10 , N-Hf i supe rstr ing [22 ,23] in the f ram eworks o fLagrangian and Hairdltonian appr oaches has been built . The equations of m otionhave been derived (Eqs. (3 .40 ) , (3 .44 ) , (3 .47 ) , (3 .51 )) using the concep t o f theadmissib le va ria t ion (3 .31 ) fo r the harmonic va riab le s . The com p le te se ts o fLorentz cov enant *nd irreducible constraints of the first and second classeshave been p re sented ( in Eqs. (4 .43a -g) and Eqs.(5 .2S)-(5 .33 ) , re spe ct ive ly ) . Thea lgebra o f the gauge sym m etrie s (Eqs. (5 .13)-(5 .15 ), (5 .17)-(5 .20 )) and symp le ct icstructure associated w ith the set of sec ond class constraints (Eq s.(5.25),(5 .26 ) , (5 .28) , (5 .29 )) have been ca lcu la ted

Thus we hav e sufficient information for the next steps towar ds covariantquantiza t ion o f D -10 superstr ing , wh ich c onsist o f the p rov id ing o f the conve rsionE54-56J] of the second class constraints into the Abelian first class ones and theconstruct ion o f the c la ssica l BRST charge (see [42 ,43 ,19 ] fo r the case o f nu llsiiper-p-br anes in D-4 ), These steps are under investigation now .

We hope that the twistor-like character of the Lorentz harm onic superstringfo rmula tion [22 ,23] sha ll he lp to so lve som e p rob lem s appearing in thepr evious app roaches to superstring quantization [14-17]. First of all, itsim plifies the pr ojection operators used for the Gr assmannian constraintdivision onto the irreducible first and second class ones. The structurefunctions independence on the inve rse degrees of the m omentum variab le s (seeEq s.(5 .BM 5.15) , (5 .17M5.20)) is the p rope rty o f the gauge sym m etrie s algebra ,wh ich r e su lts f rom m is s imp l i f i ca tion and f rom the p re sence o f the zw e inbe indensities in the tw istor-like superstring formulation [22,23 ]. It is possib le , thatthe BRST charg e of the discussed superstring form ulation has the same proper tyand, resp ectiv ely, the nonlocalMes are absent in the ver texes related with it.

76 77

..


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A C K N O W L E D G E M E N T S R E F E R E N C E S

The authors are sincer ely gr ateful to D.V . Volkov , D.P. Sorokin, V.I. Tkachand V.G . Zitna for the interest to the work and stiniulatiiig disc ussions.

One of the authors (I.A.B.) would like to thank Professor Ab dus Salam, theInternational Atomic Energ y Ag ency and UNESCO for hosp itality at the InternationalCentre fo r Theore tica l Physics (Trie ste ) , where the work was c omp le ted .

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I.A. Bandos and A.A. Zheltukhin- D=10 Superstring: Lagrangian and Hamiltonian Mechanics in...

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