On principal realization of modules for the affine Lie algebra $A_1 ^{(1)}$ at the critical level

TL;DR

This paper provides a comprehensive realization of irreducible modules for the affine Lie algebra $A_1^{(1)}$ at the critical level using vertex algebra techniques, twisted modules, and Z-algebra approaches, including explicit bases.

Contribution

It introduces a complete construction of irreducible modules at the critical level with explicit bases, combining vertex algebra and Z-algebra methods.

Findings

Realization of all irreducible highest weight modules on a specific vector space.

Use of vertex algebraic techniques and twisted modules for construction.

Explicit combinatorial bases for the modules are provided.

Abstract

We present complete realization of irreducible $A_1 ^{(1)}$-modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight $A_1 ^{(1)}$-modules at the critical level are realized on the vector space $M_{\tfrac{1}{2} + \Bbb Z} (1) ^{\otimes 2}$ where $M_{\tfrac{1}{2} + \Bbb Z} (1) $ is the polynomial ring ${\Bbb C}[\alpha(-1/2), \alpha(-3/2), ...]$. Explicit combinatorial bases for these modules are also given.