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 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, G, where or defined by the following relations: If in these relations, this yields the N = 2 Ramond algebra; while if are half-integers, it gives the N = 2 Neveu–Schwarz algebra. The operators generate a Lie subalgebra isomorphic to the Virasoro algebra. Together with the operators, they generate a Lie superalgebra isomorphic to the super Virasoro algebra, giving the Ramond algebra if are integers and the Neveu–Schwarz algebra otherwise. 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:
Properties
The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism of :
In the N = 2 Ramond algebra, the zero mode operators,, and the constants form a five-dimensional Lie superalgebra. They satisfy the same relations as the fundamental operators in Kähler geometry, with corresponding to the Laplacian, the degree operator, and the and operators.
Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism, of period two, is given by
give a construction using two commuting real bosonic fields, and a complex fermionic field 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. The current operator is defined by the standard construction from fermions and the two supersymmetric operators by This yields an N = 2 Neveu–Schwarz algebra with c = 3.
gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of for the discrete series representations of the Virasoro and super Virasoro algebra. Given a representation of the affine Kac–Moody algebra of SU at level with basis 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 simultaneously annihilated by the 's for positive 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.
