Microscopic Models for Fusion Categories

TL;DR

This thesis explores the connection between subfactors, fusion categories, and conformal field theories, focusing on the Haagerup subfactor and developing new techniques to identify corresponding CFTs through lattice models.

Contribution

It introduces methods to construct CFTs from subfactors, especially the Haagerup subfactor, by analyzing fusion categories and their extensions to modular tensor categories.

Findings

No direct evidence of CFTs from the fusion categories of the Haagerup subfactor

Highlights the need to study unitary modular tensor categories for CFT correspondence

Provides techniques for calculating F-symbols for fusion categories

Abstract

This is a PhD Thesis on the connection between subfactors (more precisely, their corresponding fusion categories) and Conformal Field Theory (CFT). Besides being a mathematically interesting topic on its own, subfactors have also attracted the attention of physicists, since there is a conjectured correspondence between these and CFTs. Although there is quite a persuasive body of evidence for this conjecture, there are some gaps: there exists a set of exceptional subfactors with no known counterpart CFT. Hence, it is necessary to develop new techniques for building a CFT from a subfactor. Here, it is useful to study the underlying mathematical structure in more detail: The even parts of every subfactor give rise to two Unitary Fusion Categories (UFCs), and it is a promising direction to study quantum spin systems constructed from these categories to find a connection to CFTs. The simplest example that requires new techniques for building a CFT is the Haagerup subfactor, since it is the smallest subfactor with index larger than 4. In this thesis, we investigate the question whether there is a CFT corresponding to the Haagerup subfactor via lattice models in one and two dimensions. The first task here is to find the F-symbols of the fusion category since these are crucial ingredients for the construction of a physical model in all of the models we consider in this thesis. We then investigate microscopic models such as the golden chain model and the Levin-Wen model in order to find evidence for a corresponding CFT. We find that there is no evidence for a corresponding CFT from the investigation of the UFCs directly and it is necessary to expand these studies to the corresponding unitary modular tensor category, which can, for instance, be obtained via the excitations of the Levin-Wen model.