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A new criterion for integral modular categorification

Jingcheng Dong, Sebastien Palcoux

TL;DR

This work addresses the classification of integral modular fusion categories by deriving a general, highly effective necessary criterion for integral modular categorification. The central tool, Theorem ENOcrit2, leverages Galois-theoretic properties of cyclotomic integers to relate the FP dimensions of adjoint-subcategory objects to those with coprime dimensions, yielding a concrete inequality that rules out vast classes of candidates. The authors implement the criterion in SageMath and demonstrate dramatic progress: a complete classification up to rank 14 and, in the odd-dimensional case, rank 25, plus substantial reductions at rank 15. The results substantially constrain the landscape of integral modular data and furnish a practical computational framework for ongoing classification efforts with potential implications for topological quantum computation and related representation-theoretic structures.

Abstract

A generalization of an argument due to Etingof-Nikshych-Ostrik yields a highly efficient necessary criterion for integral modular categorification. This criterion allows us to complete the classification of categorifiable integral modular data up to rank 14, and up to rank 25 in the odd-dimensional case.

A new criterion for integral modular categorification

TL;DR

This work addresses the classification of integral modular fusion categories by deriving a general, highly effective necessary criterion for integral modular categorification. The central tool, Theorem ENOcrit2, leverages Galois-theoretic properties of cyclotomic integers to relate the FP dimensions of adjoint-subcategory objects to those with coprime dimensions, yielding a concrete inequality that rules out vast classes of candidates. The authors implement the criterion in SageMath and demonstrate dramatic progress: a complete classification up to rank 14 and, in the odd-dimensional case, rank 25, plus substantial reductions at rank 15. The results substantially constrain the landscape of integral modular data and furnish a practical computational framework for ongoing classification efforts with potential implications for topological quantum computation and related representation-theoretic structures.

Abstract

A generalization of an argument due to Etingof-Nikshych-Ostrik yields a highly efficient necessary criterion for integral modular categorification. This criterion allows us to complete the classification of categorifiable integral modular data up to rank 14, and up to rank 25 in the odd-dimensional case.

Paper Structure

This paper contains 8 sections, 13 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Galois theory
  3. Proof of Theorem \ref{['thm:ENOcrit2']}
  4. Applications
  5. Proof of Corollary \ref{['cor:r14']}
  6. Proof of Corollary \ref{['cor:Oddr25']}
  7. Discussion about rank 15
  8. SageMath codes

Key Result

Theorem 1.1

Let $\mathcal{C}$ be an integral modular fusion category. Assume there exists a non-invertible simple object $X \in \mathcal{C}_{ad}$ and a prime divisor $p$ of $\mathrm{FPdim}(X)$ that is coprime to $\mathrm{FPdim}(\mathcal{C}_{pt})$. Then there exists a non-invertible simple object $Y \in \mathcal

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 19 more