N = 2 superconformal algebra
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  as a gauge algebra of the U(1) fermionic string. Definition

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, G, where  (for the Ramond basis) or  (for the Neveu–Schwarz basis) defined by the following relations:

c is in the center

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 :    with inverse:   
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    In terms of Kähler operators,  corresponds to conjugating the complex structure. Since , the automorphisms  and  generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group . Twisted operators  were introduced by  and satisfy:  so that these operators satisfy the Virasoro relation with central charge 0. The constant  still appears in the relations for  and the modified relations

Constructions

Free field construction
 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. SU(2) supersymmetric coset construction
 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(2) 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 
 (Neveu–Schwarz)
 (Ramond)
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. Kazama–Suzuki supersymmetric coset construction
 generalized the SU(2) coset construction to any pair consisting of a simple compact Lie group  and a closed subgroup  of maximal rank, i.e. containing a maximal torus  of , with the additional condition that the dimension of the centre of  is non-zero. In this case the compact Hermitian symmetric space  is a Kähler manifold, for example when . 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 . See also
Virasoro algebra
Super Virasoro algebra
Coset construction
Type IIB string theory

Notes

References

String theory
Conformal field theory
Lie algebras
Representation theory
Supersymmetry