Pivotal Module Categories, Factorization Homology and Modular Invariant Modified Traces

Abstract

The algebraic notion of a pivotal module category was developed by Schaumann and Shimizu and is central to the description of boundary conditions in conformal field theory according to a proposal by Fuchs and Schweigert. In this paper, we present a large class of examples of pivotal module categories of topological origin: For a unimodular finite ribbon category $\mathcal{A}$, we prove that the factorization homology $\int_\Sigma \mathcal{A}$ of a compact oriented surface $\Sigma$ with $n$ marked boundary intervals, at least one per connected component, comes with the structure of a pivotal module category over $\mathcal{A}^{\boxtimes n}$. This endows the internal skein algebras of Ben-Zvi-Brochier-Jordan, in particular the elliptic double, with a symmetric Frobenius structure. As application, we obtain, for each choice of $\mathcal{A}$, a family of full open conformal field theories, each of which comes with correlation functions for all surfaces with marked boundary intervals that are explicitly computable using factorization homology. As a further application, we explain how modified traces can be 'integrated' over surfaces: We show that the modified trace for $\mathcal{A}$ extends in a canonical way to the factorization homology of $\Sigma$. The resulting traces have the remarkable property of being modular invariant, i.e. fixed by the mapping class group action.