Citation: Simon, Steven, Slingerland, Joost, "Straightening Out the Frobenius-Schur Indicator", 2022
Web: https://arxiv.org/abs/2208.14500.pdf
Tags: Modular-tensor-categories, Monoidal-categories
This paper addresses a very interesting question: what is going on with Frobenius-Schur indicators, really? The main answer is that the need for non-identity Frobenius-Schur indicators comes from the fact that to get knot invariants from a TQFT one needs a framing. A self-dual particle is its own time reversal. However, time reversal inverts framing so a self-dual particle is not really its own dual. This framing discrepancy is quantified in the FS indicator.
Whenever the Frobenius-Schur indicator induces a Z2 grading on the category, there is a nice scheme for getting isotopy invariance. In this scheme, a negative Frobenius-Schur indicator is replaced with a negative quantum dimension. In more general theories, one needs to resort to decorating the braids with extra data ("flags") which keep track of your framing. The non-local nature of these conventions is made necessary by the non-locality of the condition that there exist a consistent global framing.
Overall, this paper is a fantastic place to go for people trying to clear their heads about the Frobenius-Schur indicator, and the ability (or lack thereof) of TQFTs to admit isotopy invariant descriptions.