Higher genus mapping class group invariants from factorizable Hopf algebras

Abstract

Lyubashenko's construction associates representations of mapping class groups Map g:n of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H. For any such Hopf algebra we find an invariant of Map g:n for all values of g and n. More generally, we obtain such invariants for any pair (H, ω), where ω is a ribbon automorphism of H. Our results are motivated by the quest to understand higher genus correlation functions of bulk fields in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple, so-called logarithmic conformal field theories.

Some remarks on the unrolled quantum group of sl(2)

“…Understanding a deeper connection between the representation theory of this paper and CFT deserves some attention. For example, it would be interesting to compare the CFT representations of SL(2, Z) (see [17]) and more generally mapping class group representations (see [19]) with those obtained from U H q sl (2) in the TQFT of [4].…”

Renormalized Hennings Invariants and 2 + 1-TQFTs

“…As explained in [21], the analysis of non-rational CFTs is less understood. However, recently there has been several results proven in this area, see for example [7,8,12,17,19,21,22,23,24,27,25]. We can ask: does the non-semisimple TQFT of this paper play a similar role in non-rational CFTs as does the modular TQFT in rational CFTs?…”

Full Logarithmic Conformal Field theory — an Attempt at a Status Report

“…For the theorem above to be of relevance, we have to make sure that modular Frobenius algebras exist. This has indeed been established for the case that the modular tensor category is the category of finite‐dimensional modules over a finite‐dimensional factorizable ribbon Hopf algebra: Theorem Let H be a finite‐dimensional factorizable ribbon Hopf algebra over an algebraically closed field and be a ribbon automorphism. Then is a modular Frobenius algebra in the modular category .…”