Singular dimensions of the N=2 superconformal algebras II: the twisted N=2 algebra

TL;DR

This paper develops an adapted ordering method for the twisted N=2 superconformal algebra, identifying singular vectors, degenerate cases, and subsingular vectors, thereby advancing the understanding of its representation theory and embedding structures.

Contribution

It introduces a new adapted ordering for the twisted N=2 algebra, classifies singular vectors, and uncovers subsingular vectors, enhancing the algebra's representation analysis.

Findings

Complete Verma modules can have two-dimensional singular vector spaces.

G-closed Verma modules have at most one-dimensional singular vector spaces.

Explicit singular and subsingular vectors are provided at levels 1/2, 1, and 3/2.

Abstract

We introduce a suitable adapted ordering for the twisted N=2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels 1/2, 1, and 3/2 for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N=2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels 1/2, 1, and 3/2. Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N=2 embedding diagrams.