Chiral and real N = 2 Galilei superconformalalgebras

Zhanna Kuznetsova

CMCC - Federal University of ABCSanto Andre - Sao Paulo, Brazil

Supersymmetry in Integrable Systems - SIS’13December 29, 2013

Outline

Superconformal algebras

N-extended l-superconformal Galilei algebras

Outline

Superconformal algebras

N-extended l-superconformal Galilei algebras

Based on:

1. Z. K., M. Rojas and F. Toppan, Classification of irreps andinvariants of the N-extended supersymmetric quantummechanics, JHEP 0603 (2006) 043513.

2. Z. K., Irreducible representations of supersymmetricquentum mechanics, ROMP 61 (2008) 295.

3. Z. K. and F. Toppan, D-module representations of N=2,4,8superconformal algebras and their superconformalmechanics, J. Math. Phys. 53 (2012) 098.

4. S. Khodaee and F. Toppan, Critical scaling dimensions ofD-module representations of N=4,7,8 superconformalalgebras and constraints on superconformal mechanics, J.Math. Phys. 53 (2012) 103518.

5. N. Aizawa, Z. K. and F. Toppan, Chiral and Real N=2supersymmetric `-conformal Galilei algebra, J. Math.Phys. 54 (2013) 093506.

Superconformal algebras and their representationsA complete list of Lie simple conformal superalgebras wasobtained inD. Fattori and V. G. Kac, Classification of finite simple Lieconformal superalgebras, J. Algebra, 258(1) (2002) 23.

Here we will consider the D-module representations forN-extended superconformal mechanics. The superalgebrasbelow are listed in terms of the odd generators (number N ofsupersymmetries).

Representations space (modules): supermultiplets of the globalN-extended Supersymmetry algebra (the dynamical Liesuperalgebra of SUSY QM)

{Qi ,Qj} = 2δijH, [H,Qi ] = 0, i , j = 1, . . . ,N

H can be considered as a Hamiltonian and Qi are SUSYgenerators.

Root multiplet’s (d ,d) sizeWe call as a root multiplet a multiplet with d fields of massdimension λ and d fields of mass dimension λ+ 1/2. Thestructure of this multiplet can be represented as (d ,d).

N = 1 1 N = 9 16 N = 17 256 N = 25 4096N = 2 2 N = 10 32 N = 18 512 N = 26 8192N = 3 4 N = 11 64 N = 19 1024 N = 27 16384N = 4 4 N = 12 64 N = 20 1024 N = 28 16384N = 5 8 N = 13 128 N = 21 2048 N = 29 32768N = 6 8 N = 14 128 N = 22 2048 N = 30 32768N = 7 8 N = 15 128 N = 23 2048 N = 31 32768N = 8 8 N = 16 128 N = 24 2048 N = 32 32768

The matrix size in D-module representations is, therefore,2d × 2d .The modulo 8 property of the irreps of the N-extendedsupersymmetry is in consequence of modulo 8 (Bottperiodicity) property of Clifford algebras.

Other admissible irreps

include different type of fields, with mass dimensions

λ, λ+12, λ+ 1, ..., λ+ (m − 1)

12.

NotationsI (d1,d2, . . . ,dm) - the field content of the multipletI m is the length of the multiplet

For example, (1,3,3,1) length-4 N = 3 multiplet contains:1 field with mass dimension λ,3 fields with λ+ 1/2,3 fields with λ+ 1 and1 field with λ+ 3/2.

Admissible irreps

I For any N, all length-3 multiplets (d − k ,d , k) arerepresentations of N-extended supersymmetry.

I length-4 irreps are present only for N = 3,5,6,7 andstarting from N ≥ 9, length-5 irrep appear starting fromN = 10.

length-2 length-3 length-4N=1 (1,1)N=2 (2,2) (1,2,1)N=3 (4,4) (3,4,1), (2,4,2), (1,4,3) (1,3,3,1)N=4 (4,4) (3,4,1), (2,4,2), (1,4,3)N=5 (8,8) (7,8,1), (6,8,2), (5,8,3), (4,8,4) (1,5,7,3), (3,7,5,1)

(3,8,5), (2,8,6), (1,8,7) (1,6,7,2), (2,7,6,1), (2,6,6,2)(1,7,7,1)

N=6 (8,8) (7,8,1), (6,8,2), . . ., (1,8,7) (1,6,7,2), (2,7,6,1), (2,6,6,2)(1,7,7,1)

N=7 (8,8) (7,8,1), (6,8,2), . . ., (1,8,7) (1,7,7,1)N=8 (8,8) (7,8,1), (6,8,2), . . ., (1,8,7)

The supersymmetry generators Qi which act on the multipletspresented in the table are matrices with 0, ±1 or differentialoperators entries.

Graphical presentation

(4,4)  

(1,4,3)  

(3,4,1)  

(2,4,2)  

Inhomogeneous supermultiplets

Linear multiplets of extended supersymmetry are usuallyformulated with homogeneous transformations for theircomponent fields.But in some cases (O. Lechtenfeld and F. Toppan) it is possileto extend the SUSY transformations by addition of aninhomogeneous term.

This possibility exists forI N = 2 irrep (0,2,2)

I N = 4 irreps (0,4,4) and (1,4,3)

I N = 8 irreps (0,8,8), (1,8,7), (2,8,6) and (3,8,5).These multiplets have an extra constant auxiliary field whichchanges the dynamics of the corresponding model.

Example.Inhomogeneous N = 2 (0,2,2) ≡ (0, ψi ,gj , cj) supermultiplet.The transformations can be expressed as

Q1ψj = gi , Q1gi = ψi ,

Q2ψi = εik gk , Q2gi = −εik ψk ,

where gk = gk + ck , ck ∈ R.Rotating in (c1, c2) plane one can define c1 = 0, c2 ≡ c > 0.

€

ψ1€

ψ1

€

ψ2

€

˙ ψ 2

€

˙ ψ 1

€

c2

€

g2

€

g1

€

c1

Conformal algebra

0 + 1 conformal algebra: so(1,2) ≈ sl(2)

The action of sl(2) generators on mass (or scaling) dimension scomponent fields isH = ∂t , D = −(t∂t + s) and K = −(t2∂t + 2st).(s = λ, λ+ 1

2 , . . .)

In the D-module representations these generators arerepresented as diagonal matrices whose entries in D and K aredifferent for the different types of component fields.

The scaling (or mass) dimension λ of a global multiplet is nowpromoted to be the conformal weight of the representation ofthe superconformal algebra.

Without supersymmetry

H = ∂t ,

D = −(t∂t + λ),

K = −(t2∂t + 2λt).

λ is the scaling dimension.

Other realizations: ”hyperbolic” and ”trigonometric”,G. Papadopoulos, Class. Quant. Grav. 30 (2013) 075018.

For any λ (non-criticality)

[D,H] = H,[D,K ] = −K ,[H,K ] = 2D.

(D is the dilatation operator).

Abstract finite conformal Lie superalgebras from theKac’s classification

N = 0 : sl(2),

N = 1 : osp(1,2),

N = 2 : sl(2|1),

N = 3 : osp(3|2),

N = 4 : A(1,1), D(2,1;α),

N = 5 : B(2,1),

N = 6 : A(2,1), B(1,2), D(3,1),

N = 7 : B(3,1), G(3),

N = 8 : D(4,1), D(2,2), A(3,1), F (4),

. . . : . . . ,

D-module representations in the classification ofsuperalgebras

1. N = 2: three inequivalent SUSY irreps (2,2), (1,2,1) and(0,2,2)inhom produce three inequivalent D-modulerepresentations of the superconformal algebra A(1,0).

2. N = 4:a) for λ = 0, all homogeneous (k ,4,4− k) and

inhomogeneous λ = −1 (1,4,3) multipletsproduce A(1,1) SCA;

b) for λ 6= 0 and k = 1,3,4 we have D(2,1;α)SCA with

(1,4,3) : α = λ; (3,4,1) : α = −λ;

(4,4) : α = −2λ

c) for λ 6= 0 and k = 2 we get sl(2|2)representation with u(1) central charge.

N = 3 case; B(1,1) = osp(3,2) from (1,3,3,1):

H = 18 · ∂t ,

D = −18 · t∂t − Λ,

K = −18 · t2∂t − 2tΛ,Q1 = (−E38 + E47 − E52 + E61)∂t + E16 − E25 + E74 − E83,

Q2 = (E28 − E46 − E53 + E71)∂t + E17 − E35 − E64 + E82,

Q3 = (−E27 + E36 − E54 + E81)∂t + E18 − E45 + E63 − E72,

S1 = (−E38 + E47 − E52 + E61)t∂t + (E16 − E25 + E74 − E83)t +

−E52(2λ+ 2) + (−E38 + E47)(2λ+ 1) + E612λ,S2 = (E28 − E46 − E53 + E71)t∂t + (E17 − E35 − E64 + E82)t +

−E53(2λ+ 2) + (E28 − E46)(2λ+ 1) + E712λ,S3 = (−E27 + E36 − E54 + E81)t∂t + (E18 − E45 + E63 − E72)t +

−E54(2λ+ 2) + (−E27 + E36)(2λ+ 1) + E812λ,W1 = E34 − E43 + E78 − E87,

W2 = −E24 + E42 − E68 + E86,

W3 = E23 − E32 + E67 − E76,

for Λ = diag(λ, λ+ 1, λ+ 1, λ+ 1, λ+ 32 , λ+ 1

2 , λ+ 12 , λ+ 1

2 ).

...

D-module representations of superconformal algebrasN = 2 and N = 4 exist without restrictions on the value of λ.

Restrictions on λ appear for N = 7,8 superconformal finitealgebras.

l-conformal Galilei algebras.The kinematical groups were classified in H. Bacry, J.-M.Levy-Leblond, JMP 9 (1968) 1605.

The groups are split in two families, relative time groups(containing Poincare and two de Sitter groups) and absolutetime groups (with the Galilei and two Newton-Hooke groups asthe most familiar ones).

It was proven in J. Negro, M. A. del Olmo and A.Rodriguez-Marco, JMP 38 (1997) 3786, that the only Galileiconformal algebras are the (10 + 6l)-dimensional Liealgebras, parametrized by positive half-integer number l(l = 0,1/2,1, . . .).

These algebras got the name l-conformal Galilei algebras.

l =12

: Schroedinger algebral = 1: conformal Galilei algebra

Supersymmetric extensionsI C. Duval and P. A. Horvathy, JMP 35 (1994) 2516: l =

12

,N-supersymmetric extensions of Schroedinger algebras.

I de Azcarraga and J. Lukierski, Phys. Lett. B 678 (2009)411: l ≥ 1, Inonu-Wigner contraction to su(2,2|N).

I A. Bagchi and I. Mandal, Phys. Rev. D 80 (2009) 086011:l = N = 1 on (3 + 1),

I M. Sakaguchi, JMP 51 (2010) 042301: l = 1, in (3 + 1) and(2 + 1) dimensional spacetime, superspace contraction,

I S. Fedoruk and J. Lukierski, Phys. Rev. D 84 (2011)065002: possible general structure for l = 1 super-GCA,N = 4, D(2,1|α) with α = 1

I I. Masterov, JMP 53 (2012) 072904: N = 2super-extension for any d and l

I N. Aizawa, J. Phys. A 45 (2012) 475203: mass and exoticcentral extensions for N = 2 super-GCA, presented inMasterov‘s paper

D-module supersymmetrization of l-conformal Galileialgebra

Supersymmetric realizations of l-conformal Galilei algebra areobtained by adding to N-extended SCA new generatorsP(n)

i (n = 1,2 . . . ,2l) and closing the algebra.

The representations are listed in connection with irreps ofN-extended supersymmetry: (2,2) and (1,2,1) for N = 2,(4,4), (1,4,3), (2,4,2) and (3,4,1) for N = 4, etc.

D-module representations are denoted as G(2,2), G(1,2,1), etc.

Our construction has two parameters:d - the dimension of the associated space andl - which gives the maximum number of extra generators P(n)

i .

N = 2

G(1,2,1) algebra is a different (but equivalent) realization of thealgebra considered by Masterov and by Aizawa

G(2,2) is a new algebra.

The generators of l-Galilean SCA act on component fields as :

P(n)i = tn∂xi , Mij = xi∂xj − xj∂xi

H = ∂t , D = −t∂t − λ, K = −t2∂t − 2λt ,

where λ→ λ = λ+ ld∑

i=1

xi∂xi .

This construction, for G(1,2,1) - real and G(2,2) - chiral cases,includes:-so(d) subalgebra generators Mij ;-the abelian ideal of the bosonic CGA P(n)

i ;-sl(2) generators: H, D and K ;-bosonic R-symmetry generator R;

-the supercharges Qa and their superconformal partners Sa;

together with the odd generators X (n)a,i from the commutators

[Sa,P(n)i ] and

the even generators J(n)i from the set of anti-commutators

{Sa,X(n)b,i }.

The common part of G(1,2,1) and G(2,2) is

[D,H] = H, [D,K ] = −K , [H,K ] = 2D

[D,Qa] =12

Qa, [K ,Qa] = Sa,

[H,Sa] = Qa, [D,Sa] = −12

Sa

{Qa,Qb} = 2δabH, {Sa,Sa} = −2δabK ,{Qa,Sb} = −2δabD + εabR[R,Qa] = −εabQb, [R,Sa] = −εabSb

The other, specific for G(2,2), (anti)commutators are

[H,P(n)i ] = nP(n−1)

i , [D,P(n)i ] = −(n − l)P(n)

i ,

[K ,P(n)i ] = −(n − 2l)P(n+1)

i , [R,P(n)i ] = 2lJ(n)

i ,

[Qa,P(n)i ] = nX (n−1)

a,i , [Sa,P(n)i ] = (n − 2l)X (n)

a,i ,

[H,X (n)a,i ] = nX (n−1)

a,i , [D,X (n)a,i ] = −(n − l +

12

)X (n)a,i

[K ,X (n)a,i ] = −(n − 2l + 1)X (n+1)

a,i , [R,X (n)a,i ] = −(2l + 1)εabX (n)

b,i

{Qa,X(n)b,i } = δabP(n)

i − εabJ(n)i ,

{Sa,X(n)b,i } = δabP(n+1)

i − εabJ(n+1)i ,

[H, J(n)i ] = nJ(n−1)

i , [D, J(n)i ] = −(n − l)J(n)

i

[K , J(n)i ] = −(n − 2l)J(n+1)

i , [R, J(n)i ] = −2lP(n)

i ,

[Qa, J(n)i ] = −εabnX (n−1)

b,i , [Sa, J(n)i ] = −εab(n − 2l)X (n)

b,i

[Mij ,Mkl ] = δjkMil + δilMjk + δjlMki + δikMlj

[P(J,Xa)(n)k ,Mij ] = δikP(J,Xa)n

j − δjkP(J,Xa)(n)i .

The inequivalence of the two supersymmetrizations isrecovered from their structure constants and the different totalnumber of generators entering G(2,2) and G(1,2,1).

We have for both algebras:4 bosonic and 4 fermionic generators entering sl(2|1)subalgebrathe d(d − 1)/2 so(d) generators Mij ,d(2l + 1) generators P(n)

i ,(2d × 2l) X (n)

a,i ,

d × (2l + 1) J(n)i generators in G(2,2) and

d × (2l − 1) J(n)i for G(1,2,1).

Central extensions

Both G(1,2,1) and G(2,2) have two different types of centralextensions:

1. mass central extension M for any d and half-integer l

[P(m)i ,P(n)

j ] = δijδm+n,2l ImM,

{X (m)a,i ,X

(n)b,j } = δabδijδm+n,2l−1αmM

[J(m)i , J(n)

j ] = δijδm+nImβmM,

where Im = (−1)m+l+1/2(2l −m)!m!αm = (−1)m+l−1(2l − 1−m)!m! andβm = (−1)m+l+1/2(2l − 2−m)!m! for G(1,2,1);Im = c0(−1)m(2l −m)!m!

αm =Im

2l −mfor G(2,2).

2. Exotic extension Θ existing only for d = 2 and integer l

[P(m)i ,P(n)

j ] = εijδm+n,2l ImΘ

{X (m)a,i ,X

(n)(b,j)} = δabεijδm+n,2l−1αmΘ

[J(m)i , J(n)

j ] = εijδm+n,2l ImβmΘ,

where Im = (−1)m(2l −m)!m!,αm = (−1)m+1(2l − 1−m)!m! andβm = (−1)mm!(2l − 2−m)! for G(1,2,1);

while the structure constants Im and αm are the same as inthe mass extension for G(2,2) algebra.

Realization in terms of super-Heisenberg subalgebraoperators

For both l-Galilei SCA algebras with central extensions thegenerators of sl(2|1)⊕ so(d) can be expressed in terms ofP(n)

i ,X (n)a,i , J

(n)i and M (or Θ).

We will use the vector notations for P(n), X(m)a and J(n)

P(m) =(

P(m)1 ,P(m)

2 , . . . ,P(m)d

), X(m)

a =(

X (m)a,1 ,X

(m)a,2 , . . . ,X

(m)a,d

)and introduce two types of product

P(m)J(n) =d∑

i=1

P(m)i J(n)

i , P(m) × J(n) =2∑

i,j=1

εijP(m)i J(n)

j

(the last ”vector product” is introduced for the exotic extensiononly).

In these notations the algebra G(2,2) with the exotic extension can be presented in the following way

D =1

2θ

∑A=P,J

2l∑m=0

l − m

ImA(2l−m) × A(m) +

∑a=1,2

2l−1∑m=0

l − 12 − m

αmX(2l−1−m)

a × X(m)a

,H =

1

2θ

∑A=P,J

2l∑m=0

m

ImA(2l−m) × A(m−1) +

∑a=1,2

2l−1∑m=0

m

αmX(2l−1−m)

a × X(m−1)a

,K = −

1

2θ

∑A=P,J

2l∑m=0

m

ImA(2l+1−m) × A(m) +

∑a=1,2

2l−1∑m=0

m

αmX(2l−m)

a × X(m)a

,M12 =

1

2θ

∑A=P,J

2l∑m=0

1

ImA(2l−m)A(m) +

∑a=1,2

2l−1∑m=0

1

αmX(2l−1−m)

a X(m)a

,R =

1

θ

2l∑m=0

m

Im

(P(2l−m) × J(m) + P(m) × J(2l−m)

)

+1

θ

2l−1∑m=0

m + 1

αm

(X(m)

1 × X(2l−1−m)2 − X(2l−1−m)

1 × X(m)2

),

Qa =1

θ

2l∑m=0

m

Im

(P(2l−m) × X(m−1)

a − εabJ(2l−m) × X(m−1)b

),

Sa =1

θ

2l∑m=0

m

Im

(P(2l+1−m) × X(m−1)

a − εabJ(2l+1−m) × X(m−1)b

).

Here we denote by θ the eigenvalue of the central extension Θ.

Conclusions and open questions

I The construction of super-l-Galilean algebras can becontinued for higher values of N.For N = 4 the resulting algebras are infinite-dimensionalfor all multiplets; nevetheless, they close on a finite numberof generators as nonlinear W -algebras (at most quadratic).

I It is interesting to combine our method with theconstruction presented by Sakaguchi-Yoshida on thesuperconformal part considering the subalgebras ofpsu(2,2|4), osp(8|4) and osp(8 ∗ |4).

I Pseudo-supersymmetric algebras and their properties.I Investigation of their structure: singular vectors, Verma

modules and lowest/highest weight representations forthese superalgebras.

I Invariant actions, dynamics and physical applications.

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