We extend Schaumanns theory of pivotal structures on fusion categories matched to a module category and of module traces developed in arXiv:1206.5716 to the case of non-semisimple tensor categories, and use it to study eigenvalues of the squared anti
pode $S^2$ in weak Hopf algebras. In particular, we diagonalize $S^2$ for semisimple weak Hopf algebras in characteristic zero, generalizing the result of Nikshych in the pseudounitary case. We show that the answer depends only on the Grothendieck group data of the pivotalizations of the categories involved and the global dimension of the fusion category (thus, all eigenvalues belong to the corresponding number field). On the other hand, we study the eigenvalues of $S^2$ on the non-semisimple weak Hopf algebras attached to dynamical quantum groups at roots of $1$ defined by D. Nikshych and the author in arXiv:math/0003221, and show that they depend nontrivially on the continuous parameters of the corresponding module category. We then compute these eigenvalues as rational functions of these parameters. The paper also contains an appendix by G. Schaumann discussing the connection between our generalization of module traces and the notion of an inner product module category introduced in arXiv:1405.5667.
We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQ
FTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of generalised orbifolds for any $n$-dimensional defect TQFT: Given a defect TQFT $mathcal{Z}$, one obtains a new TQFT $mathcal{Z}_{mathcal{A}}$ by decorating the Poincare duals of triangulated bordisms with certain algebraic data $mathcal{A}$ and then evaluating with $mathcal{Z}$. The orbifold datum $mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.
We initiate a systematic study of 3-dimensional `defect topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is
the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.
We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It
is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Witten topological field theories.
This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categ
ories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal structure. This is work is motivated by defects in topological field theories.
