Citation: Gustafson, Paul P. "Finiteness for mapping class group representations from twisted Dijkgraaf-Witten theory." Journal of Knot Theory and Its Ramifications 27.06 (2018): 1850043.
Web: https://arxiv.org/abs/1610.06069
Tags: Kitaev-quantum-double, Modular-tensor-categories, Topological-quantum-field-theory, No-go
In this paper, it is proved that the mapping class group representations associated to twisted quantum doubles of finite groups all have finite image. This includes the braid group representations as a special case, which was already established earlier in
> P. Etingof, E. C. Rowell, and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008), no. 1, 33-42
The Property-F conjecture states that a MTC should have finite braid group representations if and only if it is weakly integral, and a generalization states that it should have finite braid group representations if and only if it has finite mapping class group representations on every surface. This papers verifies this result in the special case of group-theoretical topological order.
The results of this paper are very combinatorial and direct. The proof is quite simple, and one can perhaps imagine it as a blueprint for further progress.