Pivotal Module Categories, Factorization Homology and Modular Invariant Modified Traces

2025/12/22 by Jorge Becerra, Becerra, Jorge, Lukas Woike +1
Mathematics · #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2512.19669

openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

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 A, we prove that the factorization homology ∫_ΣA of a compact oriented surface Σ with n marked boundary intervals, at least one per connected component, comes with the structure of a pivotal module category over 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 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 A extends in a canonical way to the factorization homology of Σ. The resulting traces have the remarkable property of being modular invariant, i.e. fixed by the mapping class group action.

Citations

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Related

• Categorified Open Topological Field Theories
• Reflection Equivariance and the Heisenberg Picture for Spaces of Conformal Blocks
• Modular functors from conformal blocks of rational vertex operator algebras
• Hopf and Frobenius algebras in conformal field theory
• Frobenius Algebras, Factorization Homology and the Reshetikhin-Turaev Invariants