Superconformal algebra

In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup).

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 the  superconformal algebra with $$\mathcal{N}$$ supersymmetries in 3+1 dimensions 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 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.