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"A short guide to anyons and modular functors", Simon Burton, 2023

Reviewed October 1, 2023

Citation: Burton, Simon. "A short guide to anyons and modular functors." arXiv preprint arXiv:1610.05384 (2016).

Web: https://arxiv.org/abs/1610.05384

Tags: Expository, Pedagogical, TQFTs, Modular-tensor-categories

This short expository article does exactly what it says it will: it details the theory of anyons and modular functors. The thesis is that the theory of MTCs is amazing and compact, and takes the two-dimensional theory of anyons and "compresses" it into 1 dimension. TQFTs and modular functors are the full 2-dimensional picture that is getting compressed. This article makes the connection clear, and shows plenty of pretty pictures along the way.

The TQFT axioms are layed out really well, and this a good place to learn about them. The hexagon and pentagon axioms are given beautiful little diagrams. Simon claims his target audience to be "working physicists", but this exposition seems suited to mathematicians as well.

One point to note is that Burton presents his paper as some sort of uncovering of the deeply held secret that MTCs encode (2+1)-dimensional information. This is a bit strange, because the points he discussed seem well-known to experts. Perhaps, though, this is a subtelty which isn't discussed enough in other MTC introductions.