Superconformal algebra

Superconformal algebra in dimension greater than 2

The conformal group of the (p+q)-dimensional space \mathbb{R}^{p,q} is SO(p+1,q+1) and its Lie algebra is \mathfrak{so}(p+1,q+1). The superconformal algebra is a Lie superalgebra containing the bosonic factor \mathfrak{so}(p+1,q+1) and whose odd generators transform in spinor representations of \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 and q. A (possibly incomplete) list is

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

Superconformal algebra in 3+1D

According to | author3-link = Martin Rocek | author4-link = Warren Siegel

The Lie superbrackets of the bosonic conformal algebra are given by :[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_{\mu\nu},P_\rho]=\eta_{\nu\rho}P_\mu-\eta_{\mu\rho}P_\nu :[M_{\mu\nu},K_\rho]=\eta_{\nu\rho}K_\mu-\eta_{\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}+2\eta_{\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 i}, \overline{Q}{\dot{\beta}}^j \right} = 2 \delta^j_i \sigma^{\mu}{\alpha \dot{\beta}}P_\mu :\left{ Q, Q \right} = \left{ \overline{Q}, \overline{Q} \right} = 0 :\left{ S_{\alpha}^i, \overline{S}{\dot{\beta}j} \right} = 2 \delta^i_j \sigma^{\mu}{\alpha \dot{\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]=-\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}_j

Under bosonic conformal transformations, the fermionic generators transform as: :[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 :[P_\mu,S_{\alpha}^i]=\sigma_\mu^{\alpha\dot{\beta}} \overline{Q}{\dot{\beta}} :[P\mu,\overline{S}{\dot{\alpha}}i]= Q :[K\mu,Q{\alpha i}]=\overline{S} :[K_\mu,\overline{Q}_{\dot{\alpha}}^i]=S

:[M_{\mu\nu}, Q^\alpha] = ??? (\overline{\sigma}\mu\sigma\nu - \overline{\sigma}\nu \sigma\mu) Q :[M_{\mu\nu}, S^\alpha] = ??? (\overline{\sigma}\mu\sigma\nu - \overline{\sigma}\nu \sigma\mu) S

:[Q_\alpha,D]={1\over2}Q_\alpha :[S_\alpha,D]=-{1\over2}S_\alpha :[Q_\alpha,K_\nu]=-(\gamma_\nu)\alpha^\beta S\beta :[S_\alpha,P_n]=(\gamma n)\alpha^\beta Q_\beta

:[Q_\alpha^i,T_r]=\big( \delta_\alpha^\beta(\tau_{r_1})^i_j +(\gamma_5)\alpha^ \beta(\tau{r_2})^i_j\big) Q_\beta^j :[S_\alpha^i,T_r]=\big(\delta_\alpha^\beta(\tau_{r_1})^i_j-(\gamma_5)\alpha^ \beta(\tau{r_2})^i_j\big)Q_\beta^j :[Q_\alpha^i,A]=-i\frac{3}{4}(\gamma_5)\alpha^\beta Q\beta^i :[S_\alpha^i,A]={4-N\over4N}i(\gamma_5)\alpha^\beta S\beta^i\cr {Q_\alpha^i,S_\beta^j}=-2(C^{-1}{\alpha\beta}) D\delta^{ij}+(\gamma^{mn}C^{- 1}){\alpha\beta}J_{mn}\delta^{ij}+4i(\gamma_5C^{-1}{\alpha\beta}) A\delta^{ij} : &\quad-2(\tau{r_1})^{ij}(C^{-1}){\alpha\beta}+\big((\tau{r_2})^{ij}(\gamma_ 5C^{-1})_{\alpha\beta}\big)T_r&(2.41)}

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.

References

References

1. (2002). "Confinement, Duality, and Non-Perturbative Aspects of QCD".