Hochschild Cohomology, Modular Tensor Categories,
and Mapping Class Groups

• Preprint: Series "Hamburger Beiträge zur Mathematik": Nr. 825
• Preprint: Series "Center for Mathematical Physics": ZMP-HH/20-5
• Preprint: XXX preprint archive: math.QA/2003.06527
• Book: SpringerBriefs Math. Phys., Vol. 44, Springer, Berlin, 2023

Already early in the development of the theory, it was noted that such projective representations can be obtained even if the modular category is not semisimple. In this case, the arising functors are no longer exact, but they are still left exact, so that it is natural to study their derived functors. The main result of the present work is that the cohomology groups arising from these derived functors still carry a projective action of the mapping class group in such a way that the original action is recovered in degree zero. It is therefore appropriate to call these spaces derived block spaces.

This result can be applied to a special case: For the category, one can use the representation category of a factorizable ribbon Hopf algebra, and for the surface, one can use the torus. In this case, the mapping class group is isomorphic to the modular group, and the cohomology groups become the Hochschild cohomology groups of the Hopf algebra. In this way, we recover the main result of our previous article [LMSS1].

The approach used to reach our present result is inspired by the principle of 'propagation of vacua' (cf. [TUY, Par. 2.2, p. 476]). This technique introduces an additional boundary component on the surface that, if labeled with the monoidal unit of the category, leads to block spaces that are canonically isomorphic to the block spaces without the additional boundary component. In our construction, the new boundary component serves as the position where we insert a projective resolution of the unit object. By the functoriality of the block spaces, we then obtain a cochain complex on which the mapping class group of the surface with one additional boundary component acts. The main point of the argument will be that this additional boundary component can be closed again when passing to cohomology. To establish this, we proceed in two steps, using standard techniques from the theory of mapping class groups, namely the capping sequence in the first step and the Birman sequence in the second step.

The present work sets out the general theory needed to establish the result just described, but also prepares the ground for the computation of explicit examples of these mapping class group representations, which will be addressed in our forthcoming article [LMSS2]. A new aspect of our approach is that we consider an action of the mapping class group on the fundamental group by requiring that the base point of the fundamental group be kept fixed. In this way, it is possible to avoid the necessity to identify homomorphisms that are related by simultaneous conjugation that can be found in many of the other articles on topological or conformal field theory.

The material is organized as follows: Section 1 reviews surfaces, fundamental groups, and mapping class groups as well as the capping sequence and the Birman sequence. It also introduces the action of the mapping class group on the fundamental group just mentioned. Section 2 explains how representations of mapping class groups are assigned to certain tensor categories in topological field theory. For this, we use the framework created by V. Lyubashenko in his articles [L1] and [L2]; in particular, we use the approach to surfaces via nets and ribbon graphs described in his articles. In Section 3, we first state and prove our main result and then explain why this result generalizes our previous one from [LMSS1]. In fact, our present result was already mentioned in [LMSS1], and it was also described in [FS2]. Here, we are now supplying proofs for our claims.

Throughout the text, the word 'projective' is used frequently. It has two very different meanings: When speaking about projective modules and projective resolutions, the term is used in the sense of ring theory and homological algebra. When speaking about projective space and projective representations or actions, the term is used in the sense of projective geometry. In particular, for a vector space V, we denote the associated projective space, i.e., the set of its one-dimensional subspaces, by P(V), and the projectivity or homography induced by a bijective linear map f by P(f). The set of all projectivities from P(V) to itself forms the projective linear group PGL(V), which is isomorphic to the general linear group GL(V) modulo the scalar multiples of the identity transformation. By a projective representation or projective action, we mean a group homomorphism from a given group to the projective linear group.

We will assume throughout that our base field K is perfect, i.e., that all of its algebraic extensions are separable (cf. [J, Chap. IV, § 1, p. 146]). In particular, algebraically closed fields are perfect, as are fields of characteristic zero.

In general, we compose mappings and morphisms in a category from right to left, i.e., we have (g ○ f)(x) = g(f(x)), so that f is applied first. In contrast, we concatenate paths in the fundamental group from left to right, i.e., in the concatenation κ ɣ, the path κ is traced out first, while the path ɣ is traced out afterwards.

We use the symbol V for the category of finite-dimensional vector spaces and the symbol S_n for the symmetric group in n letters. Additional notation will be explained in the text.

While carrying out this research, the first and the third author were partially supported by the RTG 1670 'Mathematics inspired by String Theory and Quantum Field Theory' and by the 'Deutsche Forschungsgemeinschaft' under Germany's Excellence Strategy EXC 2121 'Quantum Universe' – 390833306, while the second and the fourth author were partially supported by NSERC grant RGPIN-2017-06543.