Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra

TL;DR

This paper analyzes the characters of irreducible N=2 superconformal algebra representations, deriving resolutions, examining automorphism effects, and expressing characters through theta functions and spectral flow transformations.

Contribution

It introduces new resolutions for N=2 representations and establishes identities linking N=2 and affine sl(2) characters, enhancing understanding of their structure and automorphisms.

Findings

Derived BGG and new resolutions for N=2 representations

Expressed characters in terms of theta functions and spectral flow

Established identities relating N=2 and affine sl(2) characters

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 $\hat{sl}(2)$ 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 $\hat{sl}(2)$ characters.