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 does not 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 ^[1][2] the $\mathcal{N}=1$ superconformal algebra in 3+1D 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 $Q^{\alpha i}$, $\overline{Q}^{\dot\alpha}_i$, $S^\alpha_i$ and ${\overline{S}}^{\dot\alpha i}$. Here, $\mu,\nu,\rho,\dots$ denote spacetime indices; $\alpha,\beta,\dots$ left-handed Weyl spinor indices; $\dot\alpha,\dot\beta,\dots$ right-handed Weyl spinor indices; and $i,j,\dots$ the internal R-symmetry indices.

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:

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

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

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

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

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

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

Superconformal algebra in 2D

There are two possible algebras; a Neveu–Schwarz algebra and a Ramond algebra.

See also

• Conformal symmetry
• Super Virasoro algebra
• Supersymmetry algebra

References

1. West, Peter C. (1997). "Introduction to rigid supersymmetric theories". arXiv:hep-th/9805055.
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.