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On odd-dimensional modular tensor categories

Agustina Czenky, Julia Plavnik

Abstract

We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we prove that MNSD modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that MNSD tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.

On odd-dimensional modular tensor categories

Abstract

We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we prove that MNSD modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that MNSD tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.

Paper Structure

This paper contains 15 sections, 23 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fusion categories
  4. The universal grading
  5. Nilpotent, solvable and weakly group-theoretical fusion categories
  6. Modular tensor categories
  7. Centralizers in braided fusion categories
  8. Actions of $\mathcal{G(C)}$ on $\mathcal{O(C)}$ in MTCs.
  9. Results for classification of MTCs by dimension
  10. Non-pointed MTCs with irreducible degrees having a common factor $p$.
  11. MTCs with dimension a power of $p$.
  12. Weakly group-theoretical MTCs
  13. Perfect MTCs
  14. Results for classification of MTCs by rank
  15. Application: low rank odd-dimensional MTCs

Key Result

Lemma 2.3

Let $g$ be an invertible object in $\mathcal{C}_{\mathrm{ad}}$ such that $\theta_g=1$. Suppose $g\otimes X=X$ for all non-invertible simple $X\not\in \mathcal{C}_{\mathrm{ad}}$. Then the rows of the S-matrix corresponding to (the isomorphism classes of) $g$ and 1 are equal.

Theorems & Definitions (52)

  • Conjecture 1.1
  • Conjecture 1.2
  • Remark 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Proposition 3.1
  • proof
  • ...and 42 more