Citation: Etingof, Pavel, and David Penneys. "Rigidity of non-negligible objects of moderate growth in braided categories." arXiv preprint arXiv:2412.17681 (2024).
Web: https://arxiv.org/abs/2412.17681
Tags: Mathematical, Monoidal-categories
In this article, the authors show some conditions in which objects satisfying some conditions similar to rigidity are actually rigid. Namely, in a category satisfying all of the axioms of a braided fusion category except for rigidity, you only need to impose that every object X has some Y such that X \otimes Y has a fusion channel to the identity. The snake diagrams are automatic by their results. The problem in the absence of a braiding is still open.
The setting is quite general in the sense that they do not assume semisimpicity or related niceness axioms.
Questions like these are relevant because in some situations it is hard to prove that a category is rigid, but it is easier to prove some results about the fusion ring. In particular, Bakalov-Kirilov are able to take the data of a modular functor and turn it into a modular fusion category, modulo the fact that they can't prove it is rigid! This paper closes the gap and proves that it is rigid.
This problem also shows up in the entanglement bootstrap program, when trying to prove rigidity. The reason that things are simpler in the entanglement bootstrap program is that there is naturally the structure of a traced monoidal category inherited from the partial trace operation on the level of states, and the trace structure can be used to prove the rigidity structure. It is good to be aware of the fact that the partial trace structure on the density matrices is being used in such a key way at this moment of the entanglement bootstrap program, and to be able to compare it to this work.
A similar story happens in the world of bimodules between subfactors. In this world, the existence of a conditional expectation/partial trace is used in a key way to show that non-negligible bimodules are dualizable, as in proposition 7.17 of
> Bartels, Arthur, Christopher L. Douglas, and André Henriques. "Dualizability and index of subfactors." Quantum topology 5.3 (2014): 289-345.