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Topological conformal field theories and Calabi-Yau categories

Kevin J. Costello

TL;DR

This work constructs a purely algebraic realization of Gromov-Witten-type invariants through topological conformal field theories (TCFTs). It proves that open TCFTs are equivalent to unital extended Calabi–Yau $A_\infty$ categories, with a homotopy-universal open-closed TCFT whose closed sector is the Hochschild homology of the associated $A_\infty$ category, and shows that moduli-space homology acts on these Hochschild groups. A detailed cellular model for open-closed TCFTs yields explicit generators, relations, and differentials, enabling a concrete bridge between algebraic structures (CY $A_\infty$ categories) and geometric moduli spaces. Under suitable assumptions, the constructed closed TCFT recovers the Gromov–Witten invariants of a symplectic manifold and aligns with the B-model Hochschild (co)homology, providing evidence for homological mirror symmetry via Hochschild-to-homology maps. Overall, the paper furnishes a rigorous algebraic backbone for both A- and B-models, clarifying higher-genus structures and their interactions with mirror symmetry.

Abstract

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A-infinity category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A-infinity version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that a good theory of open-closed Gromov-Witten invariants exists for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.

Topological conformal field theories and Calabi-Yau categories

TL;DR

This work constructs a purely algebraic realization of Gromov-Witten-type invariants through topological conformal field theories (TCFTs). It proves that open TCFTs are equivalent to unital extended Calabi–Yau categories, with a homotopy-universal open-closed TCFT whose closed sector is the Hochschild homology of the associated category, and shows that moduli-space homology acts on these Hochschild groups. A detailed cellular model for open-closed TCFTs yields explicit generators, relations, and differentials, enabling a concrete bridge between algebraic structures (CY categories) and geometric moduli spaces. Under suitable assumptions, the constructed closed TCFT recovers the Gromov–Witten invariants of a symplectic manifold and aligns with the B-model Hochschild (co)homology, providing evidence for homological mirror symmetry via Hochschild-to-homology maps. Overall, the paper furnishes a rigorous algebraic backbone for both A- and B-models, clarifying higher-genus structures and their interactions with mirror symmetry.

Abstract

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A-infinity category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A-infinity version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that a good theory of open-closed Gromov-Witten invariants exists for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.

Paper Structure

This paper contains 33 sections, 25 theorems, 116 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Topological conformal field theories
  3. Open and open-closed TCFTs
  4. Statement of the main results
  5. Relation with Deligne's conjecture
  6. Relations to the work of Moore-Segal and Lazariou
  7. The non-unital version of the result
  8. Outline of the proof of theorem A
  9. Examples and applications
  10. String topology
  11. The B model
  12. Gromov-Witten invariants and the Fukaya category
  13. The TCFT associated to Gromov-Witten invariants
  14. Comparing the TCFT associated to Gromov-Witten theory with that from the Fukaya category
  15. Acknowledgements
  16. ...and 18 more sections

Key Result

Corollary 1

The homology of moduli spaces acts on the Hochschild homology of any Calabi-Yau $A_\infty$ category $\mathcal{D}$. That is there are operations

Figures (11)

  • Figure 1: A Riemann surface with open-closed boundary. The open boundaries can be either incoming or outgoing boundaries, but this is not illustrated.
  • Figure 2: $A,B$ are D-branes, labelling free boundaries. $o$ is an incoming open boundary with $s(o) = A$, $t(o) = B$.
  • Figure 3: Open gluing, corresponding to composition. $o_1$ on $\Sigma_1$ is incoming, $o_2$ on $\Sigma_2$ is outgoing, and $s(o_1) = s(o_2) = A$, $t(o_1) = t(o_2) = B$. Note incoming and outgoing boundaries are parameterised in the opposite sense.
  • Figure 4: A surface in $\overline {\mathcal{N}} (l)$. The dots represent incoming or outgoing open boundaries. The boundaries with no dots are closed and outgoing.
  • Figure 5: A surface in $\mathcal{G}$, with $7$ open boundaries and one closed boundary. The inside of the annulus is the outgoing closed boundary component, the open boundaries may be incoming or outgoing.
  • ...and 6 more figures

Theorems & Definitions (57)

  • Definition
  • Definition
  • Corollary
  • Conjecture 1
  • Corollary
  • Conjecture 2
  • Definition 3.0.1
  • Definition 3.0.2
  • Definition 4.2.1
  • Definition 4.3.1
  • ...and 47 more