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On the Morita invariance of Categorical Enumerative Invariants

Lino Amorim, Junwu Tu

TL;DR

This work develops CEI for Calabi–Yau $A_ infty$-categories with a nc-Hodge filtration splitting and proves their Morita invariance. It introduces unital cyclic CY models via a unital Darboux-type result, and builds a robust TCFT/DGLA framework to define genus-$g$ CEI $F^{\mathcal{C},s}_{g,n}$ that are independent of cyclic models and invariant under Morita equivalence. The results yield new invariants for smooth, proper CY3s and connect to classical enumerative theories (GW and BCOV), with concrete computations in matrix factorization settings and Frobenius algebras, and conjectural links to open–closed maps in the A-model. Overall, the paper provides a principled Morita-invariant mechanism to extract enumerative invariants from categories, with broad implications for homological mirror symmetry and topological string theory.

Abstract

Categorical Enumerative Invariants (CEI) are invariants associated with a unital, cyclic, smooth $A_\infty$-category and a splitting of its non-commutative Hodge filtration. In this paper, we extend the definition of CEI to Calabi-Yau $A_\infty$-categories with a splitting. Moreover, we formulate and prove the Morita invariance of CEI. As part of our proof, we develop tools to construct unital and cyclic models for Calabi-Yau categories. In particular, we prove a unital version of Kontsevich-Soibelman's Darboux theorem. As an application, we compute CEI in some new examples. Also, when applied to derived categories of coherent sheaves, our results yield new invariants of smooth, proper Calabi-Yau 3-folds.

On the Morita invariance of Categorical Enumerative Invariants

TL;DR

This work develops CEI for Calabi–Yau -categories with a nc-Hodge filtration splitting and proves their Morita invariance. It introduces unital cyclic CY models via a unital Darboux-type result, and builds a robust TCFT/DGLA framework to define genus- CEI that are independent of cyclic models and invariant under Morita equivalence. The results yield new invariants for smooth, proper CY3s and connect to classical enumerative theories (GW and BCOV), with concrete computations in matrix factorization settings and Frobenius algebras, and conjectural links to open–closed maps in the A-model. Overall, the paper provides a principled Morita-invariant mechanism to extract enumerative invariants from categories, with broad implications for homological mirror symmetry and topological string theory.

Abstract

Categorical Enumerative Invariants (CEI) are invariants associated with a unital, cyclic, smooth -category and a splitting of its non-commutative Hodge filtration. In this paper, we extend the definition of CEI to Calabi-Yau -categories with a splitting. Moreover, we formulate and prove the Morita invariance of CEI. As part of our proof, we develop tools to construct unital and cyclic models for Calabi-Yau categories. In particular, we prove a unital version of Kontsevich-Soibelman's Darboux theorem. As an application, we compute CEI in some new examples. Also, when applied to derived categories of coherent sheaves, our results yield new invariants of smooth, proper Calabi-Yau 3-folds.
Paper Structure (34 sections, 56 theorems, 270 equations)

This paper contains 34 sections, 56 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of the proof
  4. Organization of the paper
  5. Acknowledgments
  6. Conventions and Notations.
  7. Calabi--Yau $A_\infty$-categories
  8. $A_\infty$-bimodules and Hochschild invariants
  9. From Calabi--Yau to homotopy cyclic
  10. From homotopy cyclic to cyclic
  11. Unital Calabi--Yau structures
  12. Unital cyclic homology
  13. Unital Darboux theorem
  14. Splittings and unitality
  15. Definition of categorical enumerative invariants
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}$ and $\mathcal{D}$ be two $\mathbb{Z}/2\mathbb{Z}$-graded $A_\infty$-category over a field $\mathbb{K}$ of characteristic zero. Assume that they are both small categories and satisfy Condition ($\dagger\dagger$). Assume $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent through a intertwines CEI, i.e. we have

Theorems & Definitions (141)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 131 more