Modular invariants and NIM-reps

TL;DR

This paper introduces the encircling module in pivotal module categories over spherical fusion categories, proving its isomorphism to NIM-reps and linking modular invariants to categorical structures, generalizing previous results.

Contribution

It defines the encircling module, proves its isomorphism to NIM-reps, and connects modular invariants with categorical constructions, extending prior work in the field.

Findings

Encircling module is isomorphic to NIM-rep as a fusion algebra module.

Diagonal entries of modular invariants correspond to NIM-rep multiplicities.

Dimension conditions for modular invariance are automatically satisfied in indecomposable module categories.

Abstract

Given a pivotal module category over a spherical fusion category, we introduce the encircling module, a module over the fusion algebra defined using the pivotal structure, and prove that it is isomorphic to the NIM-rep as a fusion algebra module. When applied to the $\mathcal{TM}$ realisation of the modular invariant partition function (arXiv:1911.09024), this yields an identification of the diagonal entries of the modular invariant with the NIM-rep multiplicities, providing a categorical generalisation of B\"ockenhauer, Evans and Kawahigashi's results (arXiv:math/9907149). We also show that for indecomposable module categories the dimension condition on $\mathcal{TM}$ required for modular invariance is automatically satisfied, and that $\mathcal{TM}$ recovers the full centre construction of Fjelstad, Fuchs, Runkel and Schweigert (arXiv:hep-th/0612306, arXiv:0807.3356).