Higher genus mapping class group invariants from factorizable Hopf algebras

TL;DR

This paper constructs new invariants of mapping class groups for Riemann surfaces using factorizable ribbon Hopf algebras, advancing the understanding of non-semisimple conformal field theories.

Contribution

It extends Lyubashenko's construction to bimodule categories over finite-dimensional factorizable ribbon Hopf algebras, producing invariants for all genus and hole counts.

Findings

Established invariants for all g and n in the mapping class group

Generalized invariants to include ribbon automorphisms of H

Motivated by applications to logarithmic conformal field theories

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,omega), where omega is a ribbon automorphism of H. Our results are motivated by the quest to understand 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.