Citation: Reutter, David. "Uniqueness of unitary structure for unitarizable fusion categories." Communications in Mathematical Physics 397.1 (2023): 37-52.
Web: https://arxiv.org/abs/1906.09710
Tags: Mathematical, Monoidal-categories, Modular-tensor-categories
This paper proves a very nice result: unitarity is an inessential structure. It is a structure, but if it exists then it is unique up to equivalence. That is, if two unitary fusion categories are equivalent as categories then they must be equivalent as unitary fusion categories. In fact, the authors prove a stronger version of this: "Every monoidal equivalence between unitary fusion categories is monoidally naturally isomorphic to a unitary monoidal equivalence. In particular, any unitarizable fusion category admits a unique unitary structure (up to unitary monoidal equivalence)".
This is a fact that is treated as dead-obvious in the physics literature. It has been assumed a-priori to be true. Physicists often define unitary as a property, and not a structure. They call a fusion category unitary if the F-matrices are unitary. Of course, there are different ways of choosing the F-matrices though. The result in this context is interpreted as follows: if there are two different unitary choices of gauge, then those two choices of gauge can be transformed from one to another by a unitary change of gauge. Here, by "unitary" we mean that the maps from fusion spaces to themselves chosen in the change of gauge are unitary.
An important thing to note about this paper is that the result is actually nontrivial. It was conjectured explicitly in the literature years before it was proved:
> Henriques, André, and David Penneys. "Bicommutant categories from fusion categories." Selecta Mathematica 23.3 (2017): 1669-1708.
and there was another paper which proved a partial result:
> Galindo, César, Seung-Moon Hong, and Eric C. Rowell. "Generalized and quasi-localizations of braid group representations." International Mathematics Research Notices 2013.3 (2013): 693-731.
This paper is well-paired with an earlier work which showed that every braiding on a unitary fusion category is automatically unitary. Ribbon structures on unitary fusion categories are not automatically unitary, though. The result there is that there is a UNIQUE unitary ribbon structure. The idea is that different ribbon structures will differ by phases, and those phases will appear in the quantum dimensions making them negative:
> Galindo, César. "On braided and ribbon unitary fusion categories." Canadian Mathematical Bulletin 57.3 (2014): 506-510.
Of course, there are many non-unitarizable fusion categories. A cool fact is that this non-unitarizability can already appear at the level of fusion rings. There are fusion rings which have nice non-unitary categorifications (all the way up to ribbon, even) but cannot be made into unitary fusion categories with any choice of categorification:
> Rowell, Eric C. "On a family of non-unitarizable ribbon categories." Mathematische Zeitschrift 250 (2005): 745-774.