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Modular invariants and NIM-reps

Alastair King, Leonard Hardiman

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 Böckenhauer, 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).

Modular invariants and NIM-reps

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 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 Böckenhauer, Evans and Kawahigashi's results (arXiv:math/9907149). We also show that for indecomposable module categories the dimension condition on required for modular invariance is automatically satisfied, and that recovers the full centre construction of Fjelstad, Fuchs, Runkel and Schweigert (arXiv:hep-th/0612306, arXiv:0807.3356).
Paper Structure (8 sections, 11 theorems, 62 equations)

This paper contains 8 sections, 11 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Modules over the Fusion Algebra
  3. Preliminaries on the tube category
  4. $\mathop{\mathrm{Spec}}\nolimits F$ and the tube category
  5. Introducing $\mathcal{TM}$
  6. The dimension of $\mathcal{TM}$
  7. $\mathcal{TM}$ and the full centre
  8. Comparing pivotal structures on a multifusion category

Key Result

Lemma 2.5

There exists a set of scalars $\{\lambda_{i}\}_{i \in \mathop{\mathrm{Irr}}\nolimits(\mathcal{B})}$ such that Proof. Consider the clique complex $\Delta$ of the unoriented graph defined by $\mathcal{J}$. Then, by it:ii and it:ij, $\mu$ defines a $1$-cochain on $\Delta$ with values in $\mathbb{K}^{*}$. By it:ijjk, $\mu$ is closed, i.e. $\mathop{\mathrm{\text{ d \!\!\!}}}\nolimits\mu=0$ and the des

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 2.7
  • Proposition 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 15 more