On Low-Rank Multiplicity-Free Fusion Categories

TL;DR

This thesis develops algorithms and software to explicitly compute fusion and braiding data for all multiplicity-free pivotal fusion categories up to rank 7, with applications in topological quantum computation.

Contribution

It introduces new algorithms and the Anyonica software for classifying and constructing fusion categories up to rank 7, including non-commutative and generalized fusion rings.

Findings

Explicit F, R, and pivotal coefficients for categories up to rank 7.

Classification of all multiplicity-free fusion rings up to rank 9.

Development of the Anyonica software for fusion system computations.

Abstract

This thesis explains the methods and algorithms we used to obtain explicit F symbols, R symbols, and pivotal coefficients of all multiplicity-free pivotal fusion categories up to rank 7. The thesis starts by introducing the concept of a unitary modular fusion system via two applications: modeling anyons for topological quantum computation and calculating braid group representations. Next, the notions of a pivotal, spherical, braided, ribbon, and modular fusion system are introduced. Unitarity and its implications on the pivotal structure are discussed as well. The next part of the thesis is devoted to algorithms for finding fusion systems and compatible structures. First, an algorithm to find low-rank fusion rings is explained, and its results are given. Special attention is given to the structure of non-commutative fusion rings and the construction of songs, which are generalizations of the Tambara-Yamagami and Haagerup Izumi fusion rings, is given. Then, the algorithms used to find fusion systems are discussed. Particular attention is given to how the individual steps for solving the consistency equations are done with Anyonica, a software package we developed for working with fusion systems. Gauge and automorphism equivalence are reviewed, and algorithms that put solutions in a unitary gauge and remove redundant solutions are given. Some results on the categorification of all multiplicity-free pivotal fusion rings up to rank 7 are presented. The final part of the thesis is devoted to building models of anyons on graphs and how their behavior differs from those in the plane. The appendices contain a minimal mathematical exposition on fusion categories with their relation to fusion systems, a list of all multiplicity-free fusion rings up to rank 9 with information on categorifiability, a list of all multiplicity-free fusion categories up to rank 7, and data on some graph-braid models.