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Computing associators of endomorphism fusion categories

by Daniel Barter, Jacob C. Bridgeman, Ramona Wolf

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Submission summary

Authors (as registered SciPost users): Jacob C Bridgeman · Ramona Wolf
Submission information
Preprint Link: scipost_202204_00039v1  (pdf)
Code repository:
Date accepted: 2022-07-27
Date submitted: 2022-04-25 10:34
Submitted by: Bridgeman, Jacob C
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
Approaches: Theoretical, Computational


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)

Author comments upon resubmission

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.

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


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.

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Anonymous on 2022-05-03  [id 2440]

I thank you the authors for addressing my comments and questions. I am now satisfied with the manuscript to be published.