# Computing associators of endomorphism fusion categories

### Submission summary

 Authors (as Contributors): Jacob C Bridgeman · Ramona Wolf
Submission information
Code repository: https://github.com/JCBridgeman/FSymbolsFromModules
Date accepted: 2022-07-27
Date submitted: 2022-04-25 10:34
Submitted by: Bridgeman, Jacob C
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Mathematical Physics
• Quantum Physics
Approaches: Theoretical, Computational

### Abstract

Many applications of fusion categories, particularly in physics, require the associators or $F$-symbols to be known explicitly. Finding these matrices typically involves solving vast systems of coupled polynomial equations in large numbers of variables. In this work, we present an algorithm that allows associator data for some category with unknown associator to be computed from a Morita equivalent category with known data. Given a module category over the latter, we utilize the representation theory of a module tube category, built from the known data, to compute this unknown associator data. When the input category is unitary, we discuss how to ensure the obtained data is also unitary. We provide several worked examples to illustrate this algorithm. In addition, we include several Mathematica files showing how the algorithm can be used to compute the data for the Haagerup category $\mathcal{H}_1$, whose data was previously unknown.

Published as SciPost Phys. 13, 029 (2022)

We thank the referees for their comments and recommendations. We have attached replies to the initial submission.

### List of changes

A collection of minor changes detailed in the response to referees.

### Submission & Refereeing History

Resubmission scipost_202204_00039v1 on 25 April 2022
Submission 2110.03644v2 on 19 October 2021

## Reports on this Submission

### Anonymous Report 1 on 2022-6-8 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202204_00039v1, delivered 2022-06-08, doi: 10.21468/SciPost.Report.5203

### Report

Dear Authors and Editors,

I would like to thank the authors for their various clarifications, both in the reply to my report, as well as in the resubmitted manuscript. With these changes/clarifications, I recommend publication of the paper in SciPost in the current version.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -