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 Rp,q is SO(p+1,q+1) and its Lie algebra is so(p+1,q+1). The superconformal algebra is a Lie superalgebra containing the bosonic factor so(p+1,q+1) and whose odd generators transform in spinor representations of so(p+1,q+1). Given Kač's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of p and q. A (possibly incomplete) list is

osp∗(2N|2,2) in 3+0D thanks to usp(2,2)≃so(4,1);

osp(N|4) in 2+1D thanks to sp(4,R)≃so(3,2);

su∗(2N|4) in 4+0D thanks to su∗(4)≃so(5,1);

su(2,2|N) in 3+1D thanks to su(2,2)≃so(4,2);

sl(4|N) in 2+2D thanks to sl(4,R)≃so(3,3);

real forms of F(4) in five dimensions

osp(8∗|2N) in 5+1D, thanks to the fact that spinor and fundamental representations of so(8,C) are mapped to each other by outer automorphisms.

Superconformal algebra in 3+1D

According to the superconformal algebra with N supersymmetries in 3+1 dimensions is given by the bosonic generators Pμ, D, Mμν, Kμ, the U(1) R-symmetry A, the SU(N) R-symmetry Tji and the fermionic generators Qαi, Q¯iα˙, Siα and S¯α˙i. Here, μ,ν,ρ,… denote spacetime indices; α,β,… left-handed Weyl spinor indices; α˙,β˙,… right-handed Weyl spinor indices; and i,j,… the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

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.