Citation: Hagge, Tobias, and Matthew Titsworth. "Geometric invariants for fusion categories." arXiv preprint arXiv:1509.03275 (2015).
Web: https://arxiv.org/abs/1509.03275
Tags: Mathematical, Monoidal-categories
In this paper, Hagge and Titsworth discuss the problem of determining whether two fusion categories are gauge equivalent. Their approach is quite mathematical, based on Mumford's geometric invariant theory:
> Mumford, David, John Fogarty, and Frances Kirwan. Geometric invariant theory. Vol. 34. Springer Science & Business Media, 1994.
Wikipedia describes geometric invariant theory nicely: "Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces". This is really perfect, because that's exactly the situation we find ourselves in when discussion fusion categories. Fusion categories are described by their F-symbols. These F-symbols are only described up to change of gauge, with is the action of a gauge group. The space of fusion categories is a moduli space of F-symbols under the action of the gauge group (and under the action of the symmetry group of the fusion ring which fits together with the gauge group as a semidirect product).
The theorems in geometric invariant theory, of course, require assumptions. In particular, you need the group action to be sufficiently nice. For one need the orbits of G to be closed. The proof of this requires showing that the stabilizer of the action of G is independent of the point being acted on - there is a globally defined set of "trivial" gauge transformations. This is a result proved by Zhenghan and others:
> Davidovich, Orit, Tobias Hagge, and Zhenghan Wang. "On arithmetic modular categories." arXiv preprint arXiv:1305.2229 (2013).
Additionally, you need to show that the orbits of the G-action are irreducible as algebraic sets, which uses Ocneanu rigidity.
The conclusion of geometric invariant theory is that there is always a finite set of gauge-invariant polynomials which distinguish the space of gauge equivalence classes of F-symbols, and that the number of polynomials in this distinguishing set is bounded by the number of gauge equivalence classes of F-symbols. Additionally, geometric invariant theory can also be used to construct polynomials which are invariant under the action of symmetries of the fusion ring which distinguish full equivalence classes of fusion categories, and the number of such polynomials is equal to the number of fusion categories with the given fusion ring.
Naturally, one might expect that the results of geometric invariant theory would be non-constructive and thus not practically useful. One of the most surprising features of this paper is that, in the case of multiplicity free fusion categories, the geometric invariant theory can be made algorithmic. That is, the authors construct an algorithm (and implement it in mathematica) to find the space of gauge-equivalence classes of fusion categories with given small fusion rings.
This paper forms the core of Titsworth's PhD thesis:
> Titsworth, Matthew Kelly. Arithmetic data for anyonic systems and topological phases. The University of Texas at Dallas, 2016.