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Pivotal Module Categories, Factorization Homology and Modular Invariant Modified Traces

Jorge Becerra, Lukas Woike

TL;DR

This work provides a topological origin for pivotal module categories by showing that, for a unimodular finite ribbon category $\mathcal{A}$, the factorization homology $\int_\Sigma \mathcal{A}$ forms a pivotal left $\mathcal{A}^{\boxtimes n}$-module for surfaces with $n$ marked boundary intervals. The construction yields symmetric Frobenius structures on internal skein algebras, enabling explicit open conformal field theory correlators through factorization homology and a canonical modular invariant extension of the modified trace to $\int_\Sigma \mathcal{A}$. The authors develop a gluing framework that propagates pivotal structures from disks to general surfaces, and demonstrate how trace functions arise from trivializations of Nakayama functors, with strong compatibility under mapping class group actions. Collectively, the results connect topological quantum field theory, non-semisimple modular data, and open CFTs, providing tools for computing correlators and invariants in non-semisimple settings.

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 $\int_Σ\mathcal{A}$ 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 $\mathcal{A}^{\boxtimes n}$. 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 $Σ$. The resulting traces have the remarkable property of being modular invariant, i.e. fixed by the mapping class group action.

Pivotal Module Categories, Factorization Homology and Modular Invariant Modified Traces

TL;DR

This work provides a topological origin for pivotal module categories by showing that, for a unimodular finite ribbon category , the factorization homology forms a pivotal left -module for surfaces with marked boundary intervals. The construction yields symmetric Frobenius structures on internal skein algebras, enabling explicit open conformal field theory correlators through factorization homology and a canonical modular invariant extension of the modified trace to . The authors develop a gluing framework that propagates pivotal structures from disks to general surfaces, and demonstrate how trace functions arise from trivializations of Nakayama functors, with strong compatibility under mapping class group actions. Collectively, the results connect topological quantum field theory, non-semisimple modular data, and open CFTs, providing tools for computing correlators and invariants in non-semisimple settings.

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 , we prove that the factorization homology of a compact oriented surface with marked boundary intervals, at least one per connected component, comes with the structure of a pivotal module category over . 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 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 extends in a canonical way to the factorization homology of . The resulting traces have the remarkable property of being modular invariant, i.e. fixed by the mapping class group action.
Paper Structure (20 sections, 15 theorems, 56 equations, 1 figure)

This paper contains 20 sections, 15 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Preliminaries
  3. Finite tensor categories
  4. Finite module categories and internal homs
  5. Balanced Deligne products
  6. The distinguished invertible element and unimodularity
  7. Nakayama functors
  8. Relative Serre functors
  9. Pivotal module structures
  10. A brief review of factorization homology
  11. Factorization as a module category
  12. Pivotal module structure on the factorization homology of a unimodular finite ribbon category
  13. The special case of disks
  14. Gluing pivotal structures
  15. Trace functions
  16. ...and 5 more sections

Key Result

Corollary 2.1

Let $\mathcal{A}$ be a finite tensor category, $\mathcal{M}$ a finite right $\mathcal{A}$-module category and $\mathcal{N}$ a finite left $\mathcal{A}$-module category. Then the projective objects of $\mathcal{M}\boxtimes_\mathcal{A}\mathcal{N}$ are direct summands of objects of the form $X\boxtimes

Figures (1)

  • Figure 1: A compact oriented surface $\Sigmait$ with three parametrized intervals (in blue) in its boundary.

Theorems & Definitions (28)

  • Corollary 2.1
  • proof
  • Lemma 2.2
  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 18 more