Higher genus mapping class group invariants from factorizable Hopf algebras

Abstract

Lyubashenkoʼs construction associates representations of mapping class groups Mapg: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 Mapg: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.