Families of Singular and Subsingular Vectors of the Topological N=2 Superconformal Algebra

TL;DR

This paper classifies and analyzes the structure of singular vectors in the Topological N=2 Superconformal algebra, revealing various types, family patterns, and subsingular vectors, with detailed examples and conditions for their occurrence.

Contribution

It provides a comprehensive classification of singular vectors, their family structures, and subsingular vectors in the Topological N=2 Superconformal algebra, including explicit examples and conditions.

Findings

Identified four types of singular vectors in chiral Verma modules.

Discovered twenty-nine types of singular vectors in complete Verma modules.

Presented a 38-member family of singular vectors at multiple levels.

Abstract

We analyze several issues concerning the singular vectors of the Topological N=2 Superconformal algebra. First we investigate which types of singular vectors exist, regarding the relative U(1) charge and the BRST-invariance properties, finding four different types in chiral Verma modules and twenty-nine different types in complete Verma modules. Then we study the family structure of the singular vectors, every member of a family being mapped to any other member by a chain of simple transformations involving the spectral flows. The families of singular vectors in chiral Verma modules follow a unique pattern (four vectors) and contain subsingular vectors. We write down these families until level 3, identifying the subsingular vectors. The families of singular vectors in complete Verma modules follow infinitely many different patterns, grouped roughly in five main kinds. We present a particularly interesting thirty-eight-member family at levels 3, 4, 5, and 6, as well as the complete set of singular vectors at level 1 (twenty-eight different types). Finally we analyze the D\"orrzapf conditions leading to two linearly independent singular vectors of the same type, at the same level in the same Verma module, and we write down four examples of those pairs of singular vectors, which belong to the same thirty-eight-member family.