Dirac Families for Loop Groups as Matrix Factorizations

TL;DR

This paper establishes an equivalence between the category of integrable lowest-weight representations of loop groups and a category of twisted matrix factorizations, using Dirac operators and super-potentials.

Contribution

It introduces a categorical equivalence that lifts previous K-group isomorphisms to an isomorphism of entire categories, utilizing Dirac families and super-potentials.

Findings

Categorical equivalence between loop group representations and matrix factorizations.

Use of Dirac operators to construct the equivalence.

Extension of K-group isomorphisms to full category equivalences.

Abstract

We identify the category of integrable lowest-weight representations of the loop group LG of a compact Lie group G with the linear category of twisted, conjugation-equivariant curved Fredholm complexes on the group G: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LG. This lifts the isomorphism of K-groups of [FHT1,2, 3] to an equivalence of categories. The construction uses families of Dirac operators.