Citation: Valera, Sachin J. "Fusion structure from exchange symmetry in (2+ 1)-dimensions." Annals of Physics 429 (2021): 168471.
Web: https://arxiv.org/abs/2004.06282
Tags: MTC-reconstruction, Modular-tensor-categories, Monoidal-categories
This paper seeks to help ground the axioms of MTCs. The idea is that he starts with a minimal set of axioms and then deduces that the system has quasiparticles and that those quasiparticles are described by MTCs. The highlights: He doesn't really prove that the system has quasiparticles. He just assumes it. He doesn't really prove that quasiparticles are described by MTCs - he misses the modularity axiom.
The pessimistic way of looking at this paper is that its doing all the easy parts and skipping all the hard parts. He doesn't start with weak axioms about the ground state like they do in entanglement bootstrap - he starts with axioms with already necessitate anyons. What he does is show that once you've assumed the ground state is describe by a fusion category in a physical way, you can extract the fusion category.
This paper probably won't be all that interesting to most experts - it seems like stuff that we all already knew. However, I imagine it could be useful to someone familiar with the algebraic language who wants to learn a bit more how these things actually relate to ground states of gapped Hamiltonians in a general way.
Another way of stating this gripe would be to state that none of his conclusions are particularly suprirsing. He assumes there are finitely many superselection sectors and that the fusion spaces are finite dimensional. He assumes that every anyon has an antiparticle. So, of course the theory is described by a (unitary) fusion category. He assumes that ribbon operators exist, so of course there is a braided structure. Every unitary braided fusion category is ribbon, so of course it is ribbon. He is not able to derive the modularity axiom, which would be the only surprising contribution.
All this being said, this is Sachin Valera's first paper so it makes sense that it wouldn't be as groundbreaking as other works in this field.