Staggered and affine Kac modules over $A_1^{(1)}$

TL;DR

This paper explores the representation theory of the affine Lie algebra $A_1^{(1)}$ at fractional level, introducing affine Kac modules, conjecturing staggered modules, and extending the coset construction and character formulas to deepen understanding of their structure and relation to Virasoro algebra.

Contribution

It introduces affine Kac modules, conjectures staggered modules, and extends the coset construction and character formulas for $A_1^{(1)}$ at fractional level.

Findings

Explicit formulas for irreducible $A_1^{(1)}$-characters

Evidence supporting the existence of staggered modules

Extension of the Fuchs-Astashkevich theorem to $A_1^{(1)}$

Abstract

This work concerns the representation theory of the affine Lie algebra $A_1^{(1)}$ at fractional level and its links to the representation theory of the Virasoro algebra. We introduce affine Kac modules as certain finitely generated submodules of Wakimoto modules. We conjecture the existence of several classes of staggered $A_1^{(1)}$-modules and provide evidence in the form of detailed examples. We extend the applicability of the Goddard-Kent-Olive coset construction to include the affine Kac and staggered modules. We introduce an exact functor between the associated category of $A_1^{(1)}$-modules and the corresponding category of Virasoro modules. At the level of characters, its action generalises the Mukhi-Panda residue formula. We also obtain explicit expressions for all irreducible $A_1^{(1)}$-characters appearing in the decomposition of Verma modules, re-examine the construction of Malikov-Feigin-Fuchs vectors, and extend the Fuchs-Astashkevich theorem from the Virasoro algebra to $A_1^{(1)}$.