Hochschild cohomology, modular tensor categories and mapping class groups

Abstract

It is known that the mapping class group of a compact oriented surface of genus g and n boundary components acts projectively on certain Hom-spaces, called spaces of chiral conformal blocks, arising from a modular tensor category. If the tensor category is nonsemisimple, the Hom-functor is not exact and one can ask if there is a similar construction that gives a projective action of the mapping class group on the right derived functors. We will approach this question in two steps: In the first step, which is carried out in Chapter 2, we consider the case where the modular tensor category is given by the category of finite-dimensional modules over a factorizable ribbon Hopf algebra, and take the torus as the surface. In this case, the mapping class group is the modular group SL(2, Z), and one of the Hom-spaces mentioned above is the center of the factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group, and we show that it is indeed possible to extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra. The second step, which is carried out in Chapter 3, is to consider the general case. We construct from an arbitrary modular tensor category and for every compact oriented surface with finitely many labeled boundary components a cochain complex of finite-dimensional vector spaces. Its zeroth cohomology group is the vector space appearing in the original construction. We show that the mapping class group of the surface acts projectively on the cohomology groups of this cochain complex in such a way that it reduces to the original action in degree zero. As we explain, this generalizes the results in Chapter 2.