Representation theory of the affine Lie superalgebra sl(2|1) at fractional level

TL;DR

This paper analyzes the representation theory of the affine Lie superalgebra sl(2|1) at fractional levels, deriving embedding diagrams of singular vectors and revealing connections to N=2 superconformal algebra representations.

Contribution

It provides an analytical derivation of embedding diagrams for sl(2|1) at fractional levels, including a generalized MFF construction accounting for fermionic nilpotency.

Findings

Embedding diagrams of singular vectors are derived analytically.

The structure of these diagrams closely resembles N=2 superconformal algebra representations.

A generalized MFF construction is introduced for fractional levels.

Abstract

N=2 noncritical strings are closely related to the $\Slr/\Slr$ Wess-Zumino- Novikov-Witten model, and there is much hope to further probe the former by using the algebraic apparatus provided by the latter. An important ingredient is the precise knowledge of the $\hslc$ representation theory at fractional level. In this paper, the embedding diagrams of singular vectors appearing in $\hslc$ Verma modules for fractional values of the level ($k=p/q-1$, p and q coprime) are derived analytically. The nilpotency of the fermionic generators in $\hslc$ requires the introduction of a nontrivial generalisation of the MFF construction to relate singular vectors among themselves. The diagrams reveal a striking similarity with the degenerate representations of the $N=2$ superconformal algebra.