Analytic Expressions for Singular Vectors of the $N=2$ Superconformal Algebra

TL;DR

This paper derives explicit formulas for singular vectors in the N=2 superconformal algebra, revealing the structure of Verma modules and identifying conditions for linearly independent singular vectors at the same grade.

Contribution

It provides analytic expressions for singular vectors in the N=2 superconformal algebra and analyzes their linear independence within Verma modules.

Findings

Existence of two linearly independent neutral singular vectors at the same grade

Construction of a two-dimensional space of singular vectors

Identification of conditions for singular vectors with same charge and grade

Abstract

Using explicit expressions for a class of singular vectors of the $N=2$ (untwisted) algebra and following the approach of Malikov-Feigin-Fuchs and Kent, we show that the analytically extended Verma modules contain two linearly independent neutral singular vectors at the same grade. We construct this two dimensional space and we identify the singular vectors of the original Verma modules. We show that in some Verma modules these expressions lead to two linearly independent singular vectors which are at the same grade and have the same charge.