Resolutions and characters of irreducible representations of the N = 2 superconformal algebra

Abstract

We evaluate characters of irreducible representations of the N = 2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG resolution of the admissible N = 2 representations and also a new “3, 5, 7, …” resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every sl ̂ (2) character as a linear combination of spectral-flow transformed N = 2 characters; this identity involves a finite number of N = 2 characters in the case of unitary representations. Conversely, we find an integral representation for the admissible N = 2 characters as contour integrals of admissible sl ̂ (2) characters.