Superconformal algebra explained

In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup).

Superconformal algebra in dimension greater than 2

The conformal group of the

(p + q) -dimensional space R ^{p,q} is SO(p +1 ,q +1 ) and its Lie algebra is ak{so}(p+1,q+1) . The superconformal algebra is a Lie superalgebra containing the bosonic factor ak{so}(p+1,q+1) and whose odd generators transform in spinor representations of ak{so}(p+1,q+1) . Given Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of p and q . A (possibly incomplete) list isak{osp} ^{*(2N|2,2)} in 3+0D thanks to ak{usp}(2,2)\simeq ak{so}(4,1) ;ak{osp}(N|4) in 2+1D thanks to ak{sp}(4, R )\simeq ak{so}(3,2) ;ak{su} ^{*(2N|4)} in 4+0D thanks to ak{su} ^{*(4)\simeqak{so}(5,1)} ;ak{su}(2,2|N) in 3+1D thanks to ak{su}(2,2)\simeq ak{so}(4,2) ;ak{sl}(4|N) in 2+2D thanks to ak{sl}(4, R )\simeq ak{so}(3,3) ;

• real forms of

F( 4 ) in five dimensionsak{osp}(8 ^{*|2N)} in 5+1D, thanks to the fact that spinor and fundamental representations of ak{so}(8, C ) are mapped to each other by outer automorphisms.

Superconformal algebra in 3+1D

According to ^{[1] [2]} the superconformal algebra with

l{N} supersymmetries in 3+1 dimensions is given by the bosonic generators P _{\mu} , D , M _{\mu\nu} , K _{\mu} , the U(1) R-symmetry A , the SU(N) R-symmetry T ^{i} _{j} and the fermionic generators Q ^{\alpha} , \overline{Q} ^{• \alpha} _{i} , S ^{\alpha} _{i} and {\overline{S}} ^{• \alpha i} . Here, \mu,\nu,\rho, ... denote spacetime indices; \alpha,\beta, ... left-handed Weyl spinor indices; • • \alpha, \beta, ... right-handed Weyl spinor indices; and i,j, ... the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

[M _{\mu\nu} ,M _{\rho\sigma} ] =η _{\nu\rho} M _{\mu\sigma} -η _{\mu\rho} M _{\nu\sigma} +η _{\nu\sigma} M _{\rho\mu} -η _{\mu\sigma} M _{\rho\nu} [M _{\mu\nu} ,P _{\rho]=η} _{\nu\rho} P _{\mu-η} _{\mu\rho} P _{\nu} [M _{\mu\nu} ,K _{\rho]=η} _{\nu\rho} K _{\mu-η} _{\mu\rho} K _{\nu} [M _{\mu\nu} ,D] =0 [D,P _{\rho]=-P} _{\rho} [D,K _{\rho]=+K} _{\rho} [P _{\mu,K} _{\nu]=-2M} _{\mu\nu} +2η _{\mu\nu} D [K _{n,K} _{m]=0} [P _{n,P} _{m]=0} where η is the Minkowski metric; while the ones for the fermionic generators are:\left\{ Q _{\alpha} , \overline{Q} ^{j} _{• \beta} \right\} = 2 \delta ^{j} _{i} \sigma ^{\mu} _{\alpha • \beta} P _{\mu} \left\{ Q, Q \right\} = \left\{ \overline{Q}, \overline{Q} \right\} = 0 \left\{ S ^{i,} _{\alpha} \overline{S} _{• \beta j} \right\} = 2 \delta ^{i} _{j} \sigma ^{\mu} _{\alpha • \beta} K _{\mu} \left\{ S, S \right\} = \left\{ \overline{S}, \overline{S} \right\} = 0 \left\{ Q, S \right\} = \left\{ Q, \overline{S} \right\} = \left\{ \overline{Q}, S \right\} = 0

The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:

[A,M] = [A,D] = [A,P] = [A,K] =0 [T,M] = [T,D] = [T,P] = [T,K] =0

But the fermionic generators do carry R-charge:

[A,Q] =- (1)/(2) Q [A,\overline{Q}] = (1)/(2) \overline{Q} [A,S] = (1)/(2) S [A,\overline{S}] =- (1)/(2) \overline{S} [T ^{i} _{j,Q} _{k]=} - \delta ^{i} _{k} Q _{j} [T ^{k] =} _{j,{\overline{Q}}} \delta ^{k} _{j} {\overline{Q}} ^{i} [T ^{k] = \delta} _{j,S} ^{k} ^{j} S ^{i} [T ^{i} _{j,\overline{S}} _{k]=} - \delta ^{i} _{k} \overline{S} _{j}

Under bosonic conformal transformations, the fermionic generators transform as:

[D,Q] =- (1)/(2) Q [D,\overline{Q}] =- (1)/(2) \overline{Q} [D,S] = (1)/(2) S [D,\overline{S}] = (1)/(2) \overline{S} [P,Q] = [P,\overline{Q}] =0 [K,S] = [K,\overline{S}] =0

Superconformal algebra in 2D

See main article: super Virasoro algebra. There are two possible algebras with minimal supersymmetry in two dimensions; a Neveu–Schwarz algebra and a Ramond algebra. Additional supersymmetry is possible, for instance the N = 2 superconformal algebra.

See also

• Conformal symmetry
• Super Virasoro algebra
• Supersymmetry algebra

Notes and References

1. Book: 10.1007/0-306-47056-X_17. Introduction to Rigid Supersymmetric Theories . Confinement, Duality, and Non-Perturbative Aspects of QCD . NATO Science Series: B . 2002 . West . P. C. . 368 . 453–476 . hep-th/9805055 . 0-306-45826-8 . 119413468 .
2. Gates . S. J. . Grisaru . Marcus T. . Rocek . M. . Martin Rocek . Siegel . W. . Warren Siegel . 1983 . Superspace, or one thousand and one lessons in supersymmetry . Frontiers in Physics . 58 . 1–548 . hep-th/0108200 . 2001hep.th....8200G .