Superconformal algebra

Superconformal algebra

In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. It generates the superconformal group in some cases (In two Euclidean dimensions, the Lie superalgebra doesn't generate any Lie supergroup.).

In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, there is a finite number of known examples of superconformal algebras.

Superconformal algebra in 3+1D

According to [Citation
last = West
first = Peter C.
author-link =
title = "Introduction to rigid supersymmetric theories"
year = 1997
url = http://arxiv.org/abs/hep-th/9805055
] [Citation
last = Gates
first = S. J.
author-link = Sylvester Gates
last2 = Grisaru
first2 = Marcus T.
author2-link = Marcus Grisaru
last3 = Rocek
first3 = M.
last4 = Siegel
first4 = W.
title = Superspace, or one thousand and one lessons in supersymmetry
journal = Front. Phys.
volume = 58
year = 1983
pages = 1-548
url = http://arxiv.org/abs/hep-th/0108200
] ,the mathcal{N}=1 superconformal algebra in 3+1D is given by the bosonic generators P_mu, D, M_{mu u}, K_mu, the U(1) R-symmetry A and the SU(N) R-symmetry T^i_j and the fermionic generators Q^{alpha i}, overline{Q}^{dotalpha}_i, S^alpha_i and overline{S}^{dotalpha i}. mu, u, ho,dots denote spacetime indices, alpha,eta,dots left-handed Weyl spinor indices and dotalpha,doteta,dots right-handed Weyl spinor indices, and i,j,dots the internal R-symmetry indices.

The Lie superbrackets are given by: [M_{mu u},M_{ hosigma}] =eta_{ u ho}M_{musigma}-eta_{mu ho}M_{ usigma}+eta_{ usigma}M_{ homu}-eta_{musigma}M_{ ho u}: [M_{mu u},P_ ho] =eta_{ u ho}P_mu-eta_{mu ho}P_ u: [M_{mu u},K_ ho] =eta_{ u ho}K_mu-eta_{mu ho}K_ u: [M_{mu u},D] =0: [D,P_ ho] =-P_ ho: [D,K_ ho] =+K_ ho: [P_mu,K_ u] =-2M_{mu u}+2eta_{mu u}D: [K_n,K_m] =0: [P_n,P_m] =0This is the bosonic conformal algebra. Here, η is the Minkowski metric.

: [A,M] = [A,D] = [A,P] = [A,K] =0: [T,M] = [T,D] = [T,P] = [T,K] =0The bosonic conformal generators do not carry any R-charges.

: [A,Q] =-frac{1}{2}Q: [A,overline{Q}] =frac{1}{2}overline{Q}: [A,S] =frac{1}{2}S: [A,overline{S}] =-frac{1}{2}overline{S}

: [T^i_j,Q_k] = - delta^i_k Q_j: [T^i_j,overline{Q}^k] = delta^k_j overline{Q}^i: [T^i_j,S^k] =delta^k_j S^i: [T^i_j,overline{S}_k] = - delta^i_k overline{S}_jBut the fermionic generators do.

: [D,Q] =-frac{1}{2}Q: [D,overline{Q}] =-frac{1}{2}overline{Q}: [D,S] =frac{1}{2}S: [D,overline{S}] =frac{1}{2}overline{S}: [P,Q] = [P,overline{Q}] =0: [K,S] = [K,overline{S}] =0

Tells us how the fermionic generators transform under bosonic conformal transformations.

:left{ Q_{alpha i}, overline{Q}_{dot{eta^j ight} = 2 delta^j_i sigma^{mu}_{alpha dot{etaP_mu:left{ Q, Q ight} = left{ overline{Q}, overline{Q} ight} = 0:left{ S_{alpha}^i, overline{S}_{dot{eta}j} ight} = 2 delta^i_j sigma^{mu}_{alpha dot{etaK_mu:left{ S, S ight} = left{ overline{S}, overline{S} ight} = 0:left{ Q, S ight} = :left{ Q, overline{S} ight} = left{ overline{Q}, S ight} = 0

Superconformal algebra in 2D

See super Virasoro algebra. There are two possible algebras; a Neveu-Schwarz algebra and a Ramond algebra.

References

See also

* conformal symmetry
* SUSY algebra
* Super Virasoro algebra


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