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 is SO(p+1,q+1) and its Lie algebra is. The superconformal algebra is a Lie superalgebra containing the bosonic factor and whose odd generators transform in spinor representations of. 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

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, , S^\alpha_i and. Here, denote spacetime indices; left-handed Weyl spinor indices; 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 where η is the Minkowski metric; while the ones for the fermionic generators are: The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators: But the fermionic generators do carry R-charge: Under bosonic conformal transformations, the fermionic generators transform as:

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.