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Superconformal algebra in dimension greater than 2

The conformal group of the ( p + q ) {\displaystyle (p+q)} -dimensional space R p , q {\displaystyle \mathbb {R} ^{p,q}} is S O ( p + 1 , q + 1 ) {\displaystyle SO(p+1,q+1)} and its Lie algebra is s o ( p + 1 , q + 1 ) {\displaystyle {\mathfrak {so}}(p+1,q+1)} . The superconformal algebra is a Lie superalgebra containing the bosonic factor s o ( p + 1 , q + 1 ) {\displaystyle {\mathfrak {so}}(p+1,q+1)} and whose odd generators transform in spinor representations of s o ( p + 1 , q + 1 ) {\displaystyle {\mathfrak {so}}(p+1,q+1)} . Given Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of p {\displaystyle p} and q {\displaystyle q} . A (possibly incomplete) list is

• o s p ∗ ( 2 N | 2 , 2 ) {\displaystyle {\mathfrak {osp}}^{*}(2N|2,2)} in 3+0D thanks to u s p ( 2 , 2 ) ≃ s o ( 4 , 1 ) {\displaystyle {\mathfrak {usp}}(2,2)\simeq {\mathfrak {so}}(4,1)} ;
• o s p ( N | 4 ) {\displaystyle {\mathfrak {osp}}(N|4)} in 2+1D thanks to s p ( 4 , R ) ≃ s o ( 3 , 2 ) {\displaystyle {\mathfrak {sp}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,2)} ;
• s u ∗ ( 2 N | 4 ) {\displaystyle {\mathfrak {su}}^{*}(2N|4)} in 4+0D thanks to s u ∗ ( 4 ) ≃ s o ( 5 , 1 ) {\displaystyle {\mathfrak {su}}^{*}(4)\simeq {\mathfrak {so}}(5,1)} ;
• s u ( 2 , 2 | N ) {\displaystyle {\mathfrak {su}}(2,2|N)} in 3+1D thanks to s u ( 2 , 2 ) ≃ s o ( 4 , 2 ) {\displaystyle {\mathfrak {su}}(2,2)\simeq {\mathfrak {so}}(4,2)} ;
• s l ( 4 | N ) {\displaystyle {\mathfrak {sl}}(4|N)} in 2+2D thanks to s l ( 4 , R ) ≃ s o ( 3 , 3 ) {\displaystyle {\mathfrak {sl}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,3)} ;
• real forms of F ( 4 ) {\displaystyle F(4)} in five dimensions
• o s p ( 8 ∗ | 2 N ) {\displaystyle {\mathfrak {osp}}(8^{*}|2N)} in 5+1D, thanks to the fact that spinor and fundamental representations of s o ( 8 , C ) {\displaystyle {\mathfrak {so}}(8,\mathbb {C} )} are mapped to each other by outer automorphisms.

Superconformal algebra in 3+1D

According to ¹² the superconformal algebra with N {\displaystyle {\mathcal {N}}} supersymmetries in 3+1 dimensions is given by the bosonic generators P μ {\displaystyle P_{\mu }} , D {\displaystyle D} , M μ ν {\displaystyle M_{\mu \nu }} , K μ {\displaystyle K_{\mu }} , the U(1) R-symmetry A {\displaystyle A} , the SU(N) R-symmetry T j i {\displaystyle T_{j}^{i}} and the fermionic generators Q α i {\displaystyle Q^{\alpha i}} , Q ¯ i α ˙ {\displaystyle {\overline {Q}}_{i}^{\dot {\alpha }}} , S i α {\displaystyle S_{i}^{\alpha }} and S ¯ α ˙ i {\displaystyle {\overline {S}}^{{\dot {\alpha }}i}} . Here, μ , ν , ρ , … {\displaystyle \mu ,\nu ,\rho ,\dots } denote spacetime indices; α , β , … {\displaystyle \alpha ,\beta ,\dots } left-handed Weyl spinor indices; α ˙ , β ˙ , … {\displaystyle {\dot {\alpha }},{\dot {\beta }},\dots } right-handed Weyl spinor indices; and i , j , … {\displaystyle i,j,\dots } the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

[ M μ ν , M ρ σ ] = η ν ρ M μ σ − η μ ρ M ν σ + η ν σ M ρ μ − η μ σ M ρ ν {\displaystyle [M_{\mu \nu },M_{\rho \sigma }]=\eta _{\nu \rho }M_{\mu \sigma }-\eta _{\mu \rho }M_{\nu \sigma }+\eta _{\nu \sigma }M_{\rho \mu }-\eta _{\mu \sigma }M_{\rho \nu }} [ M μ ν , P ρ ] = η ν ρ P μ − η μ ρ P ν {\displaystyle [M_{\mu \nu },P_{\rho }]=\eta _{\nu \rho }P_{\mu }-\eta _{\mu \rho }P_{\nu }} [ M μ ν , K ρ ] = η ν ρ K μ − η μ ρ K ν {\displaystyle [M_{\mu \nu },K_{\rho }]=\eta _{\nu \rho }K_{\mu }-\eta _{\mu \rho }K_{\nu }} [ M μ ν , D ] = 0 {\displaystyle [M_{\mu \nu },D]=0} [ D , P ρ ] = − P ρ {\displaystyle [D,P_{\rho }]=-P_{\rho }} [ D , K ρ ] = + K ρ {\displaystyle [D,K_{\rho }]=+K_{\rho }} [ P μ , K ν ] = − 2 M μ ν + 2 η μ ν D {\displaystyle [P_{\mu },K_{\nu }]=-2M_{\mu \nu }+2\eta _{\mu \nu }D} [ K n , K m ] = 0 {\displaystyle [K_{n},K_{m}]=0} [ P n , P m ] = 0 {\displaystyle [P_{n},P_{m}]=0}

where η is the Minkowski metric; while the ones for the fermionic generators are:

{ Q α i , Q ¯ β ˙ j } = 2 δ i j σ α β ˙ μ P μ {\displaystyle \left\{Q_{\alpha i},{\overline {Q}}_{\dot {\beta }}^{j}\right\}=2\delta _{i}^{j}\sigma _{\alpha {\dot {\beta }}}^{\mu }P_{\mu }} { Q , Q } = { Q ¯ , Q ¯ } = 0 {\displaystyle \left\{Q,Q\right\}=\left\{{\overline {Q}},{\overline {Q}}\right\}=0} { S α i , S ¯ β ˙ j } = 2 δ j i σ α β ˙ μ K μ {\displaystyle \left\{S_{\alpha }^{i},{\overline {S}}_{{\dot {\beta }}j}\right\}=2\delta _{j}^{i}\sigma _{\alpha {\dot {\beta }}}^{\mu }K_{\mu }} { S , S } = { S ¯ , S ¯ } = 0 {\displaystyle \left\{S,S\right\}=\left\{{\overline {S}},{\overline {S}}\right\}=0} { Q , S } = {\displaystyle \left\{Q,S\right\}=} { Q , S ¯ } = { Q ¯ , S } = 0 {\displaystyle \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 {\displaystyle [A,M]=[A,D]=[A,P]=[A,K]=0} [ T , M ] = [ T , D ] = [ T , P ] = [ T , K ] = 0 {\displaystyle [T,M]=[T,D]=[T,P]=[T,K]=0}

But the fermionic generators do carry R-charge:

[ A , Q ] = − 1 2 Q {\displaystyle [A,Q]=-{\frac {1}{2}}Q} [ A , Q ¯ ] = 1 2 Q ¯ {\displaystyle [A,{\overline {Q}}]={\frac {1}{2}}{\overline {Q}}} [ A , S ] = 1 2 S {\displaystyle [A,S]={\frac {1}{2}}S} [ A , S ¯ ] = − 1 2 S ¯ {\displaystyle [A,{\overline {S}}]=-{\frac {1}{2}}{\overline {S}}} [ T j i , Q k ] = − δ k i Q j {\displaystyle [T_{j}^{i},Q_{k}]=-\delta _{k}^{i}Q_{j}} [ T j i , Q ¯ k ] = δ j k Q ¯ i {\displaystyle [T_{j}^{i},{\overline {Q}}^{k}]=\delta _{j}^{k}{\overline {Q}}^{i}} [ T j i , S k ] = δ j k S i {\displaystyle [T_{j}^{i},S^{k}]=\delta _{j}^{k}S^{i}} [ T j i , S ¯ k ] = − δ k i S ¯ j {\displaystyle [T_{j}^{i},{\overline {S}}_{k}]=-\delta _{k}^{i}{\overline {S}}_{j}}

Under bosonic conformal transformations, the fermionic generators transform as:

[ D , Q ] = − 1 2 Q {\displaystyle [D,Q]=-{\frac {1}{2}}Q} [ D , Q ¯ ] = − 1 2 Q ¯ {\displaystyle [D,{\overline {Q}}]=-{\frac {1}{2}}{\overline {Q}}} [ D , S ] = 1 2 S {\displaystyle [D,S]={\frac {1}{2}}S} [ D , S ¯ ] = 1 2 S ¯ {\displaystyle [D,{\overline {S}}]={\frac {1}{2}}{\overline {S}}} [ P , Q ] = [ P , Q ¯ ] = 0 {\displaystyle [P,Q]=[P,{\overline {Q}}]=0} [ K , S ] = [ K , S ¯ ] = 0 {\displaystyle [K,S]=[K,{\overline {S}}]=0}

Superconformal algebra in 2D

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

References

1. West, P. C. (2002). "Introduction to Rigid Supersymmetric Theories". Confinement, Duality, and Non-Perturbative Aspects of QCD. NATO Science Series: B. Vol. 368. pp. 453–476. arXiv:hep-th/9805055. doi:10.1007/0-306-47056-X_17. ISBN 0-306-45826-8. S2CID 119413468. 0-306-45826-8 ↩

2. Gates, S. J.; Grisaru, Marcus T.; Rocek, M.; Siegel, W. (1983). "Superspace, or one thousand and one lessons in supersymmetry". Frontiers in Physics. 58: 1–548. arXiv:hep-th/0108200. Bibcode:2001hep.th....8200G. /wiki/Martin_Rocek ↩