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Open 2D TFTs admit initial open-closed extensions

Shaul Barkan, Jan Steinebrunner, Adela YiYu Zhang

TL;DR

This work proves that any open 2D TFT valued in a symmetric monoidal ∞-category with suitable colimits extends canonically to an open-closed TFT, with the circle value given by the Hochschild homology of the disk value. It develops a space-level refinement of prior results by embedding calculus and arc-complex methods, yielding a dense Open subcategory inside the Open–Closed bordism category and enabling a canonical left Kan extension that preserves monoidal structure. The circle-encoded operations align with the cyclic/bar constructions, and the construction provides a natural action of surface moduli on Hochschild homology for $E_1$-Calabi–Yau algebras. These results connect to Lurie’s non-compact cobordism hypothesis and offer explicit models via arc complexes, paracyclic categories, and factorization-homology, with broad applications to Calabi–Yau and topological field theory contexts.

Abstract

We show that any open 2-dimensional topological field theory valued in a symmetric monoidal $\infty$-category (with suitable colimits) extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of its value at the disk. As a corollary, we obtain an action of the moduli spaces of surfaces on the Hochschild homology object of $E_1$-Calabi-Yau algebras. This provides a space level refinement of previous work of Costello over $\mathbb{Q}$ and Wahl-Westerland and Wahl over $\mathbb{Z}$, and serves as a crucial ingredient to Lurie's "non-compact cobordism hypothesis" in dimension 2. As part of the proof we also give a description of slice categories of the d-dimensional bordism category with boundary, which may be of independent interest.

Open 2D TFTs admit initial open-closed extensions

TL;DR

This work proves that any open 2D TFT valued in a symmetric monoidal ∞-category with suitable colimits extends canonically to an open-closed TFT, with the circle value given by the Hochschild homology of the disk value. It develops a space-level refinement of prior results by embedding calculus and arc-complex methods, yielding a dense Open subcategory inside the Open–Closed bordism category and enabling a canonical left Kan extension that preserves monoidal structure. The circle-encoded operations align with the cyclic/bar constructions, and the construction provides a natural action of surface moduli on Hochschild homology for -Calabi–Yau algebras. These results connect to Lurie’s non-compact cobordism hypothesis and offer explicit models via arc complexes, paracyclic categories, and factorization-homology, with broad applications to Calabi–Yau and topological field theory contexts.

Abstract

We show that any open 2-dimensional topological field theory valued in a symmetric monoidal -category (with suitable colimits) extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of its value at the disk. As a corollary, we obtain an action of the moduli spaces of surfaces on the Hochschild homology object of -Calabi-Yau algebras. This provides a space level refinement of previous work of Costello over and Wahl-Westerland and Wahl over , and serves as a crucial ingredient to Lurie's "non-compact cobordism hypothesis" in dimension 2. As part of the proof we also give a description of slice categories of the d-dimensional bordism category with boundary, which may be of independent interest.

Paper Structure

This paper contains 33 sections, 47 theorems, 101 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Bordism categories and field theories
  3. An embedding calculus perspective
  4. Related work
  5. Acknowledgments
  6. The bordism category and its slice
  7. Constructing the bordism category
  8. Slices of the bordism category
  9. Preliminaries and outline of the proof
  10. Factorization homology of $\mathrm{E}_1$-algebras
  11. Outline of the proof
  12. Formal operations on Hochschild homology
  13. Proposition A: embedding calculus in dimension 1
  14. The E1-algebra structure on D1
  15. Proof of \ref{['proposition A']}
  16. ...and 18 more sections

Key Result

Theorem 1.1

Suppose that the symmetric monoidal product in $\pazocal{V}$ preserves geometric realization in each variable. Let $A$ be an $\mathrm{E}_1$-Calabi--Yau algebra in $\pazocal{V}$. There are maps of spaces for any $i>0$ and $j\geq 0$, which assemble into a symmetric monoidal $\infty$-functor from the positive-boundary surface bordism category that sends $S^1$ to $\int_{S^1} A$. In particular, $\int

Figures (12)

  • Figure 1: The "whistle" bordism $D^1 \to S^1$ in $\mathrm{Bord}_2^\partial$.
  • Figure 2: The first isomorphism shows that the "whistle" bordism $W\colon D^1 \to S^1$ from \ref{['fig:bordex']} is a coalgebra map, and the second isomorphism is the "Cardy condition" expressing $W^\vee \circ W\colon D^1 \to D^1$ purely in terms of the multiplication and comultiplication on $D^1$.
  • Figure 3: A morphism in $\mathrm{Emb}_{S^1}^\square(W, V)$.
  • Figure 4: Non-example and example of morphism in $\pazocal{O}\pazocal{C}$.
  • Figure 5: Algebra multiplication and unit map of $D^1\in \mathrm{Bord}_2^\partial$.
  • ...and 7 more figures

Theorems & Definitions (110)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Definition 1.4
  • Theorem 1.5: cyclic
  • Theorem 1.6: Main theorem
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 100 more