Determinant Formula for the Topological N=2 Superconformal Algebra

TL;DR

This paper derives the Kac determinant for the Topological N=2 superconformal algebra, analyzes singular vectors and submodules, and extends findings to the Ramond N=2 algebra, clarifying previous misconceptions.

Contribution

It provides the first detailed determinant formula and classification of submodules for the Topological N=2 superconformal algebra, including new insights into chiral and Ramond modules.

Findings

Derived the Kac determinant for the algebra

Identified four types of submodules in Verma modules

Extended results to Ramond N=2 algebra

Abstract

The Kac determinant for the Topological N=2 superconformal algebra is presented as well as a detailed analysis of the singular vectors detected by the roots of the determinants. In addition we identify the standard Verma modules containing `no-label' singular vectors (which are not detected directly by the roots of the determinants). We show that in standard Verma modules there are (at least) four different types of submodules, regarding size and shape. We also review the chiral determinant formula, for chiral Verma modules, adding new insights. Finally we transfer the results obtained to the Verma modules and singular vectors of the Ramond N=2 algebra, which have been very poorly studied so far. This work clarifies several misconceptions and confusing claims appeared in the literature about the singular vectors, Verma modules and submodules of the Topological N=2 superconformal algebra.