In mathematical physics, the 2D N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional conformal field theory.
It has important applications in mirror symmetry.
It was introduced by M. Ademollo, L. Brink, and A.
D'Adda et al. (1976) as a gauge algebra of the U(1) fermionic string.
There are two slightly different ways to describe the N = 2 superconformal algebra, called the N = 2 Ramond algebra and the N = 2 Neveu–Schwarz algebra, which are isomorphic (see below) but differ in the choice of standard basis.
The N = 2 superconformal algebra is the Lie superalgebra with basis of even elements c, Ln, Jn, for n an integer, and odd elements G+r, G−r, where
in these relations, this yields the N = 2 Ramond algebra; while if
are half-integers, it gives the N = 2 Neveu–Schwarz algebra.
generate a Lie subalgebra isomorphic to the Virasoro algebra.
, they generate a Lie superalgebra isomorphic to the super Virasoro algebra, giving the Ramond algebra if
When represented as operators on a complex inner product space,
is taken to act as multiplication by a real scalar, denoted by the same letter and called the central charge, and the adjoint structure is as follows: Green, Schwarz, and Witten (1988a, 1988b) give a construction using two commuting real bosonic fields
is defined to the sum of the Virasoro operators naturally associated with each of the three systems where normal ordering has been used for bosons and fermions.
is defined by the standard construction from fermions and the two supersymmetric operators
Di Vecchia et al. (1986) gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of Goddard, Kent & Olive (1986) for the discrete series representations of the Virasoro and super Virasoro algebra.
Given a representation of the affine Kac–Moody algebra of SU(2) at level
satisfying the supersymmetric generators are defined by This yields the N=2 superconformal algebra with The algebra commutes with the bosonic operators The space of physical states consists of eigenvectors of
and the supercharge operator The supercharge operator commutes with the action of the affine Weyl group and the physical states lie in a single orbit of this group, a fact which implies the Weyl-Kac character formula.
[2] Kazama & Suzuki (1989) generalized the SU(2) coset construction to any pair consisting of a simple compact Lie group
, with the additional condition that the dimension of the centre of
In this case the compact Hermitian symmetric space
The physical states lie in a single orbit of the affine Weyl group, which again implies the Weyl–Kac character formula for the affine Kac–Moody algebra of