Hochschild Cohomology, Modular Tensor Categories,
and Mapping Class Groups II

• Preprint: Series "Hamburger Beiträge zur Mathematik": Nr. 1012
• Preprint: Series "Center for Mathematical Physics": ZMP-HH/26-25
• Preprint: XXX preprint archive: mathQA/2608.11193
• Book:

The present text constitutes a direct continuation of our previous work [LMSS2], which comprises the first three sections of the complete text, whereas the present second part begins with the fourth section. References to the first three sections are therefore found in the first part.

In this first part, we have generalized the action of mapping class groups of surfaces on the spaces of chiral conformal blocks, or briefly the block spaces, to the so-called derived block spaces. For this purpose, we have used a framework created by V. Lyubashenko, which we have reviewed in Paragraph 2.5 and which describes surfaces via nets and ribbon graphs. The key point of our argument was described in Paragraph 3.2: We considered a surface with an additional boundary component, which could be labeled with a projective resolution of the unit object, so that the block spaces of this modified surface formed a cochain complex. We then showed that the action of the mapping class group of the modified surface descended to an action of the mapping class group of the original surface in cohomology, i.e., on the derived block spaces.

It is natural to ask whether these mapping class group representations on the derived block spaces are really new in the sense that they are not isomorphic to the representations on the original block spaces. In Paragraph 7.4 below, we answer this question affirmatively: The representations that appear in higher cohomological degree are in general not isomorphic to those in degree zero. Furthermore, we show in Paragraph 7.6 that these new representations can also not always be obtained from those in degree zero by taking Yoneda products. In other words, the submodule that is generated from the degree zero component under the Yoneda product may not contain these new representations.

To reach these goals, we consider a special modular category, namely the category of representations of the Drinfel'd double of a finite group, a category that is nonsemisimple if the characteristic of the base field divides the order of the finite group under consideration. For this category, we are able to show that the mapping class group representations constructed by V. Lyubashenko are isomorphic to those arising from the so-called representation variety, i.e., the set of group homomorphisms from the fundamental group of the surface to the given finite group. We give this isomorphism as Theorem 4.28 in Paragraph 4.10. Another key fact that we use is a correspondence between certain cohomology groups of the Drinfel'd double of our finite group and certain cohomology groups of that finite group itself, a relation that we establish as Proposition 5.3 in Paragraph 5.2. Taken together, these facts show that the derived block spaces can be computed as the cohomology groups of our finite group with coefficients in a linearized version of the representation variety, a result that we state as Corollary 5.4 directly afterwards.

Let us now review our contents in greater detail and in a more linear fashion. Like any quasitriangular Hopf algebra, also the Drinfel'd double of a finite group comes with two natural R-matrices, denoted by R and R̃ in Paragraph 4.1. Interchanging these two R-matrices corresponds graphically to interchanging overcrossings and undercrossings. While none of the two R-matrices is preferred over the other, one of the two can be expressed more easily in terms of the standard basis, at least for the model of the Drinfel'd double that we are using.

In general, the mapping class group representations considered here depend on the R-matrix used. However, we show in Section 4, more precisely in Theorem 4.36 of Paragraph 4.12, that both R and R̃ lead to isomorphic mapping class group representations in the cases that are relevant for us. As the formulas arising from R̃ are considerably simpler, at least for the conventions that we are using, we work with R̃ in the sequel. Another key result of Section 4 is the description of the relevant mapping class group actions via the representation variety in Theorem 4.28 that was already mentioned above.

In Section 5, we reduce the computation of the relevant cohomology groups to group cohomology, as also already mentioned above. Moreover, the appearing cohomology groups are modules over the cohomology ring of the unit object of the category, which is an algebra with respect to the Yoneda product. This module structure plays an important role in Section 7. To define this product, we pass to Yoneda's description of the cohomology groups, which is not based on projective resolutions, in contrast to the description that we gave in Paragraph 3.1.

The module structure over the Yoneda algebra is compatible with the passage to group cohomology just discussed. In group cohomology, however, it is customary not to use the Yoneda product itself, but rather the so-called cup product, which in a certain sense can be seen as a generalization of the Yoneda product. Although, strictly speaking, the cup product is not necessary for our treatment, we explain in the appendix not only the relation between the two products, but also discuss there the cup product in the setting of general tensor categories, which, compared to the case of groups or Hopf algebras, requires the modification of some arguments.

As stated above, we have seen in Section 4 that the relevant mapping class group representations can be realized on the representation variety. The given finite group acts on the representation variety by conjugation, and this action commutes with the action of the mapping class group. For the subsequent computations, it would be helpful if the representation variety could be decomposed into a Cartesian product in which the mapping class group acts on one factor and the given finite group acts on the other. While this is not possible in general, we show in Section 6 that it is possible to come rather close to such a decomposition. The corresponding variant of this decomposition, which is stated as Proposition 6.1 in Paragraph 6.1, descends to cohomology by a particularly simple case of the universal coefficient theorem, as we show in Theorem 6.2 directly afterwards.

We also show in Section 6 that, if our finite group is abelian, the direct sum of the derived block spaces in all degrees is free as a right module over the cohomology ring of the base field under the Yoneda product. More precisely, we show in Paragraph 6.2 that the representation variety can be used as a basis for this free module, so that the arising isomorphism is equivariant with respect to the action of the mapping class group. This means that the key features that we want to exhibit cannot be realized by using only abelian groups: In the abelian case, the mapping class group representations that appear for the derived block spaces in higher degree are the same as the mapping class group representations on the ordinary block spaces in degree zero.

In Section 7, we reach the explicit construction of examples that exhibit the features just mentioned. As it is necessary to work with a nonabelian group, we consider the smallest such group, the symmetric group on three letters. In order to keep the examples as simple as possible, we choose g=1 for the genus of our surface, so that the mapping class group is the modular group SL(2,ℤ). To be in a nonsemisimple situation, we need to consider base fields of characteristic 2 or 3. If the characteristic is 2, we find in Paragraph 7.2 that all mapping class group representations that appear in higher cohomological degree are still already present in degree zero. However, the direct sum of the derived block spaces is no longer free over the cohomology ring of the base field. The situation changes if we pass to characteristic 3, as we show in Paragraph 7.4: Here, new representations appear when the cohomological degree is congruent to 1 or 2 modulo 4. These new representations do also not appear in the submodule that is generated under the Yoneda product by the degree zero component, i.e., by the ordinary block spaces, as follows from Theorem 7.13 in Paragraph 7.6.

We conclude with an appendix that first reviews in Paragraph A.1 the periodic resolution for cyclic groups, which is one of our most important computational tools. The remaining parts of the appendix discuss the cup product in the context of general tensor categories, as already mentioned above.

We continue to use the notations and conventions introduced in the first part. In particular, we work over an algebraically closed base field denoted by K. For a set X, K[X] denotes the free vector space with X as a basis (cf. [Gr1, Chap. I, Sec. 1.7, p. 13]). For a group G and an element a ∈ G, we denote its centralizer by C(a). If this group acts on a set X, we denote the fixed point set by X^G. The notation U ≤ G indicates that the subset U of G is in fact a subgroup. The symbol 0 is used to denote not only the zero vector in a vector space, but also the zero vector space itself and more generally the zero object in an abelian category.

As explained in Paragraph 2.1, the left dual of an object X in a left rigid category is denoted by X^*, and the left dual of a morphism f is denoted by f^*. This applies in particular to the category of finite-dimensional vector spaces, where X^* and f^* then denote the dual vector space and the dual map, i.e., the transpose of f. We will use the same notation also in the infinite-dimensional setting. Generalizing this special case, we use the notation f^* also for the map given by precomposing f, and accordingly the notation f_* for the map given by postcomposing f. In addition, we also use f_* for the map induced by postcomposition in cohomology. In contrast, the transpose of a matrix M is denoted by M^T.

We also use the Kronecker symbol δ_a,b, which is equal to 1 if a=b and equal to 0 if a ≠ b. It needs to be distinguished from the dual basis elements in the dual group ring introduced in Paragraph 4.2, for which we use the notation δ_a, so that δ_a(b) = δ_a,b.

We note that our treatment of the Yoneda product in Paragraph 5.1 leads into territory that is no longer covered by the standard axioms of set theory put forward by E. Zermelo and A. Fraenkel, together with the axiom of choice, as it requires the formation of so-called big groups. To a lesser extent, this also concerns other aspects of category theory discussed here. Various ways to deal with these difficulties have been proposed. One of them is to adopt an additional set-theoretical axiom, namely the axiom of the existence of a universe, and this is the approach taken here. This approach is discussed in greater detail in [ML2, Chap. I, § 6, p. 21ff]. In order to form the big groups just mentioned, we need to assume in addition that the category under consideration is small with respect to the universe in the sense of [ML2, loc. cit., p. 22].

The authors would like to thank Sarah Witherspoon for interesting discussions on the present material and Marc Hoyois for pointing out reference [Bg].

While carrying out this research, the first and the third author were partially supported by the 'Deutsche Forschungsgemeinschaft' under Germany's Excellence Strategy EXC 2121 'Quantum Universe' - 390833306 and the Collaborative Research Center SFB 1624 'Higher Structures, Moduli Spaces and Integrability' - 506632645, while the second and the fourth author were partially supported by NSERC grant RGPIN-2017-06543.