Pivotal Module Categories, Factorization Homology and Modular Invariant Modified Traces

TL;DR

This paper develops a framework connecting pivotal module categories, factorization homology, and modular invariants, providing new examples and applications in topological quantum field theories and conformal field theories.

Contribution

It constructs a large class of pivotal module categories from unimodular finite ribbon categories using factorization homology, and demonstrates their applications in conformal field theory and modular invariants.

Findings

Factorization homology of surfaces yields pivotal module categories.

Internal skein algebras acquire symmetric Frobenius structures.

Modified traces extend to factorization homology, exhibiting modular invariance.

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.