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## "Microscopic definitions of anyon data", Kyle Kawagoe, Michael Levin, 2019

*Reviewed March 23, 2024*

*Citation:* Kawagoe, Kyle, and Michael Levin. "Microscopic definitions of anyon data." Physical Review B 101.11 (2020): 115113.

*Web:* https://arxiv.org/abs/1910.11353

*Tags:* Physical, Modular-tensor-categories

In this paper, the authors give a microscopic definition
of the anyon data associated to an MTC. That is, if a phase
is well-behaved enough to have well-defined anyonic superselection which can be manipulated freely
and whose statistics are described by some UBFC, then one can define corresponding physically measurable F- and R-symbols which satisfy the pentagon
and hexagon equations. This is the first real proof that the MTC describing a phase
is actually an invariant of that phase.

A subtle point is to what extent this is actually unique. If I am given a phase,
is their experimental setup actually always going to give me the same answer? What
if two equivalent phases could give two different MTCs? The answer is that no,
this is impossible. By smoothly deforming from one setup to another, the resulting F- and R-symbols
must deform smoothly as well. By Ocneanu rigidity, this means that they cannot change at all.

This is a great reference to be aware of for people who are interested in first-principles
justifications of the category theoretic description of anyons.