Abstract
We study the implications of the anyon fusion equation a×b=c on global properties of 2+1D topological quantum field theories (TQFTs). Here a and b are anyons that fuse together to give a unique anyon, c. As is well known, when at least one of a and b is abelian, such equations describe aspects of the one-form symmetry of the theory. When a and b are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and a and b lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian a and b can also be an indication of zero-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, M24), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.
Highlights
Topological quantum field theories (TQFTs) in 2 + 1 dimensions and their anyonic excitations lie at the heart of important physical [1], mathematical [2], and computational [3] systems and constructions
In order to take the shortest route to answering some of the questions posed in the introduction and in order to establish the existence of fusion rules of the form (2) in prime TQFTs, we will start with an analysis of Wilson lines
We have seen that the existence of fusions of non-abelian anyons having a unique outcome is intimately connected with the global structure of the corresponding TQFT
Summary
Topological quantum field theories (TQFTs) in 2 + 1 dimensions and their anyonic excitations lie at the heart of important physical [1], mathematical [2], and computational [3] systems and constructions. More complicated cosets do sometimes have such fixed points, and we will construct an explicit example of such a prime TQFT that has fusion rules of the form (8) and (2) To summarize, this discussion leads us to the following questions we will answer in subsequent sections: 1. In order to take the shortest route to answering some of the questions posed in the introduction and in order to establish the existence of fusion rules of the form (2) in prime TQFTs, we will start with an analysis of Wilson lines These objects form a closed fusion subcategory that is particular easy to analyze..
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