Twisted Affine Lie Algebras, Fusion Algebras, and Congruence Subgroups

TL;DR

This paper explores the structure of twisted affine Lie algebra characters, their relation to congruence subgroups, and the construction of fusion algebras with new positivity conjectures and explicit group actions.

Contribution

It introduces a fusion algebra structure for twisted affine Lie algebra characters, studies their quotients, and proposes positivity conjectures for structure constants.

Findings

Constructed a fusion algebra with negative structure constants.

Derived formulas for congruence subgroup actions on character spaces.

Proved positivity conjectures in specific cases.

Abstract

The space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of $SL(2,\mathbb{Z})$. By embedding the characters in the space spanned by theta functions, we study an $SL(2,\mathbb{Z})$-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram automorphisms on conformal blocks, have negative structure constants with respect to the (usual) basis indexed by the dominant integral weights of the Lie algebra. We give positivity conjectures for the new structure constants and prove them in some illuminating cases. We then compute formulas for the action of congruence subgroups on these character spaces and give explicit descriptions of the quotients using the affine Weyl group.