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 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 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 : 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 satisfyingthe 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.