Superconformal algebra

Superconformal algebra in 3+1D

According to the superconformal algebra with ${\displaystyle \mathcal{N}}$ supersymmetries in 3+1 dimensions is given by the bosonic generators ${\displaystyle P_{\mu }}$, ${\displaystyle D}$, ${\displaystyle M_{\mu \nu }}$, ${\displaystyle K_{\mu }}$, the U(1) R-symmetry ${\displaystyle A}$, the SU(N) R-symmetry ${\displaystyle T_j^i}$ and the fermionic generators ${\displaystyle Q^{\alpha i}}$, ${\displaystyle {\overline{Q}}_i^{\dot{\alpha }}}$, ${\displaystyle S_i^{\alpha }}$ and ${\displaystyle {\overline{S}}^{\dot{\alpha }i}}$. Here, ${\displaystyle \mu ,\nu ,\rho ,\dots }$ denote spacetime indices; ${\displaystyle \alpha ,\beta ,\dots }$ left-handed Weyl spinor indices; ${\displaystyle \dot{\alpha },\dot{\beta },\dots }$ right-handed Weyl spinor indices; and ${\displaystyle i,j,\dots }$ the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

${\displaystyle [M_{\mu \nu },M_{\rho \sigma }]={\eta }_{\nu \rho }{M}_{\mu \sigma }-{\eta }_{\mu \rho }{M}_{\nu \sigma }+{\eta }_{\nu \sigma }{M}_{\rho \mu }-{\eta }_{\mu \sigma }{M}_{\rho \nu }}$

${\displaystyle [M_{\mu \nu },P_{\rho }]={\eta }_{\nu \rho }{P}_{\mu }-{\eta }_{\mu \rho }{P}_{\nu }}$

${\displaystyle [M_{\mu \nu },K_{\rho }]={\eta }_{\nu \rho }{K}_{\mu }-{\eta }_{\mu \rho }{K}_{\nu }}$

${\displaystyle [M_{\mu \nu },D]=0}$

${\displaystyle [D,P_{\rho }]=-P_{\rho }}$

${\displaystyle [D,K_{\rho }]=+K_{\rho }}$

${\displaystyle [P_{\mu },K_{\nu }]=-2M_{\mu \nu }+2{\eta }_{\mu \nu }D}$

${\displaystyle [K_n,K_m]=0}$

${\displaystyle [P_n,P_m]=0}$

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

${\displaystyle \left\{Q_{\alpha i},{\overline{Q}}_{\dot{\beta }}^j\right\}=2{\delta }_i^j{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }}$

${\displaystyle \left\{Q,Q\right\}=\left\{\overline{Q},\overline{Q}\right\}=0}$

${\displaystyle \left\{S_{\alpha }^i,{\overline{S}}_{\dot{\beta }j}\right\}=2{\delta }_j^i{\sigma }_{\alpha \dot{\beta }}^{\mu }{K}_{\mu }}$

${\displaystyle \left\{S,S\right\}=\left\{\overline{S},\overline{S}\right\}=0}$

${\displaystyle \left\{Q,S\right\}=}$

${\displaystyle \left\{Q,\overline{S}\right\}=\left\{\overline{Q},S\right\}=0}$

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

${\displaystyle [A,M]=[A,D]=[A,P]=[A,K]=0}$

${\displaystyle [T,M]=[T,D]=[T,P]=[T,K]=0}$

But the fermionic generators do carry R-charge:

${\displaystyle [A,Q]=-\frac{1}{2}Q}$

${\displaystyle [A,\overline{Q}]=\frac{1}{2}\overline{Q}}$

${\displaystyle [A,S]=\frac{1}{2}S}$

${\displaystyle [A,\overline{S}]=-\frac{1}{2}\overline{S}}$

${\displaystyle [T_j^i,Q_k]=-{\delta }_k^i{Q}_j}$

${\displaystyle [T_j^i,{\overline{Q}}^k]={\delta }_j^k{\overline{Q}}^i}$

${\displaystyle [T_j^i,S^k]={\delta }_j^k{S}^i}$

${\displaystyle [T_j^i,{\overline{S}}_k]=-{\delta }_k^i{\overline{S}}_j}$

Under bosonic conformal transformations, the fermionic generators transform as:

${\displaystyle [D,Q]=-\frac{1}{2}Q}$

${\displaystyle [D,\overline{Q}]=-\frac{1}{2}\overline{Q}}$

${\displaystyle [D,S]=\frac{1}{2}S}$

${\displaystyle [D,\overline{S}]=\frac{1}{2}\overline{S}}$

${\displaystyle [P,Q]=[P,\overline{Q}]=0}$

${\displaystyle [K,S]=[K,\overline{S}]=0}$

Superconformal algebra in 2D

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.