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.

Superconformal algebra in dimension greater than 2

The conformal group of the -dimensional space is 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 Kač's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of and. A list is
According to the superconformal algebra with supersymmetries in 3+1 dimensions is given by the bosonic generators,,,, the U R-symmetry, the SU R-symmetry and the fermionic generators,, and. Here, denote spacetime indices; left-handed Weyl spinor indices; right-handed Weyl spinor indices; and 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.