Representations of N=2 superconformal vertex algebra

TL;DR

This paper classifies all irreducible modules of the N=2 superconformal vertex algebra at specific central charges, revealing their structure and parameterization, especially for integer and rational admissible cases.

Contribution

It provides a complete classification of irreducible modules for the N=2 superconformal vertex algebra at certain central charges, including unitary and rational cases.

Findings

All irreducible modules for $L_{c_m}$ are classified.

Unitary representations account for all irreducible modules when $m$ is an integer.

Irreducible modules for rational $m$ are parameterized by finite sets and rational curves.

Abstract

Let $L_c$ be simple vertex operator superalgebra(SVOA) associated to the vacuum representation of N=2 superconformal algebra with the central charge $c$. Let $c_m = {3m}/{m+2}$. We classify all irreducible modules for the SVOA $L_{c_m}$. When $m$ is an integer we prove that the set of all unitary representations of N=2 superconformal algebra with the central charge $c_m$ provides all irreducible $L_{c_m}$-modules. When $m \notin {\N} $ and $m$ is an admissible rational number we show that irreducible $L_{c_m}$-modules are parameterized with the union of one finite set and union of finitely many rational curves.