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 M. Ademollo, L. Brink, and A. D'Adda et al. (1976) as a gauge algebra of the U(1) fermionic string. 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 r ∈ Z {displaystyle rin {mathbb {Z} }} (for the Ramond basis) or r ∈ 1 2 + Z {displaystyle rin {1 over 2}+{mathbb {Z} }} (for the Neveu–Schwarz basis) defined by the following relations: If r , s ∈ Z {displaystyle r,sin {mathbb {Z} }} in these relations, this yields theN = 2 Ramond algebra; while if r , s ∈ 1 2 + Z {displaystyle r,sin {1 over 2}+{mathbb {Z} }} are half-integers, it gives the N = 2 Neveu–Schwarz algebra. The operators L n {displaystyle L_{n}} generate a Lie subalgebra isomorphic to the Virasoro algebra. Together with the operators G r = G r + + G r − {displaystyle G_{r}=G_{r}^{+}+G_{r}^{-}} , they generate a Lie superalgebra isomorphic to the super Virasoro algebra,giving the Ramond algebra if r , s {displaystyle r,s} are integers and the Neveu–Schwarz algebra otherwise. When represented as operators on a complex inner product space, c {displaystyle c} 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 & Witten (1988) give a construction using two commuting real bosonic fields ( a n ) {displaystyle (a_{n})} , ( b n ) {displaystyle (b_{n})} and a complex fermionic field ( e r ) {displaystyle (e_{r})} L n {displaystyle L_{n}} 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 J n {displaystyle J_{n}} is defined by the standard construction from fermions and the two supersymmetric operators G r ± {displaystyle G_{r}^{pm }} by