Tambara-Yamagami, loop groups, bundles and KK-theory

TL;DR

This paper provides geometric and KK-theoretic interpretations of modular invariants in conformal field theories, especially for loop groups and Tambara-Yamagami categories, revealing new structural insights.

Contribution

It introduces a KK-theory framework for modular invariants of loop groups and describes Tambara-Yamagami categories as bundles over groupoids, extending previous reconstruction results.

Findings

KK-theory interpretation of all modular invariants for loop groups of tori

Description of Tambara-Yamagami categories as bundles over groupoids

Reconstruction of doubles of Tambara-Yamagami categories for even-order groups

Abstract

This paper is part of a sequence interpreting quantities of conformal field theories K-theoretically. Here we give geometric constructions of the associated module categories (modular invariants, nimreps, etc). In particular, we give a KK-theory interpretation of all modular invariants for the loop groups of tori, as well as most known modular invariants of loop groups. In addition, we find unexpectedly that the Tambara-Yamagami fusion category has an elegant description as bundles over a groupoid, and use that to interpret its module categories as KK-elements. We establish reconstruction for the doubles of all Tambara-Yamagami categories, generalizing work of Bischoff to even-order groups. We conclude by relating the modular group representations coming from finite groups and loop groups to the Chern character and to the Fourier-Mukai transform