Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The cyclic open--closed map and variations of Hodge structures

Sheel Ganatra, Nick Sheridan

TL;DR

This work builds a precise bridge between the A-model and Fukaya category perspectives by constructing and analyzing the cyclic open--closed map in the setting of the big relative Fukaya category. It proves that this map intertwines the Getzler–Gauss–Manin and Dubrovin–Givental connections, and preserves pairings, thereby establishing a polarized variation of semi-infinite Hodge structures (VSHS) isomorphism under suitable hypotheses. The authors provide an isomorphism criterion (e.g., via homological smoothness or HMS in Calabi–Yau cases) that allows extraction of rational Gromov–Witten invariants from the Fukaya category and hence deduces genus-zero enumerative mirror symmetry from homological mirror symmetry for Calabi–Yau mirror pairs. The framework relies on the big relative Fukaya category, its bounding-cochain formalisms, and the established algebraic apparatus for cyclic/homological invariants, rendering the Frobenius manifold structure on quantum cohomology accessible from the Fukaya side. Applications include verification for Greene–Plesser and Batyrev-type mirrors, providing a conceptual route from HMS to enumerative predictions via VSHS correspondences.

Abstract

We construct the cyclic open--closed map for the big (i.e., bulk-deformed) relative Fukaya category, in the semipositive case, and show that it is a morphism of `polarized variations of semi-infinite Hodge structures'. We also give a natural criterion for the map to be an isomorphism, which is verified for example in the context of Batyrev mirror pairs. We conclude in such Calabi-Yau cases that the rational Gromov--Witten invariants can be extracted from the relative Fukaya category, and hence that enumerative mirror symmetry is a consequence of homological mirror symmetry for Calabi--Yau mirror pairs.

The cyclic open--closed map and variations of Hodge structures

TL;DR

This work builds a precise bridge between the A-model and Fukaya category perspectives by constructing and analyzing the cyclic open--closed map in the setting of the big relative Fukaya category. It proves that this map intertwines the Getzler–Gauss–Manin and Dubrovin–Givental connections, and preserves pairings, thereby establishing a polarized variation of semi-infinite Hodge structures (VSHS) isomorphism under suitable hypotheses. The authors provide an isomorphism criterion (e.g., via homological smoothness or HMS in Calabi–Yau cases) that allows extraction of rational Gromov–Witten invariants from the Fukaya category and hence deduces genus-zero enumerative mirror symmetry from homological mirror symmetry for Calabi–Yau mirror pairs. The framework relies on the big relative Fukaya category, its bounding-cochain formalisms, and the established algebraic apparatus for cyclic/homological invariants, rendering the Frobenius manifold structure on quantum cohomology accessible from the Fukaya side. Applications include verification for Greene–Plesser and Batyrev-type mirrors, providing a conceptual route from HMS to enumerative predictions via VSHS correspondences.

Abstract

We construct the cyclic open--closed map for the big (i.e., bulk-deformed) relative Fukaya category, in the semipositive case, and show that it is a morphism of `polarized variations of semi-infinite Hodge structures'. We also give a natural criterion for the map to be an isomorphism, which is verified for example in the context of Batyrev mirror pairs. We conclude in such Calabi-Yau cases that the rational Gromov--Witten invariants can be extracted from the relative Fukaya category, and hence that enumerative mirror symmetry is a consequence of homological mirror symmetry for Calabi--Yau mirror pairs.

Paper Structure

This paper contains 26 sections, 38 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Big relative Fukaya category
  3. Bounding cochains on the small category
  4. 'Small' bounding cochains on the big category
  5. Main results
  6. Implications for mirror symmetry of compact Calabi--Yau varieties
  7. Acknowledgments:
  8. Algebraic preliminaries
  9. Small bounding cochains
  10. Weak proper Calabi--Yau structures
  11. Pairings
  12. Connections
  13. Getzler--Gauss--Manin connection
  14. Recollection of the formalism for defining chain-level Floer theoretic operations
  15. Systems of families of domains
  16. ...and 11 more sections

Key Result

Theorem 1.6

There is a unital graded $\Lambda^{\operatorname{big}}$-algebra homomorphism called the big closed--open map. Quotienting by $\mathfrak{m}^{\operatorname{bulk}}_\Lambda$, one gets a unital graded $\Lambda^{\operatorname{sm}}$-algebra homomorphism called the small closed--open map. The small closed--open map coincides with the first-order deformation class of $\mathcal{F}^{\operatorname{big}}(X,D

Theorems & Definitions (87)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.5
  • Theorem 1.6: Theorem 1.10 of relfukii
  • proof
  • Corollary 1.7
  • proof
  • Theorem 1.8: Theorem 1.11 of relfukii
  • proof
  • ...and 77 more