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Open Gromov-Witten invariants from the Fukaya category

Kai Hugtenburg

Abstract

This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L \subset X$. We associate to an object in an $A_\infty$-category an extension of the negative cyclic homology, called \emph{relative cyclic homology}. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian $L$ in the Fukaya category of a symplectic manifold $X$ to the $S^1$-equivariant relative quantum homology of $(X,L)$. Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane.

Open Gromov-Witten invariants from the Fukaya category

Abstract

This paper proposes a framework to show that the Fukaya category of a symplectic manifold determines the open Gromov-Witten invariants of Lagrangians . We associate to an object in an -category an extension of the negative cyclic homology, called \emph{relative cyclic homology}. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian in the Fukaya category of a symplectic manifold to the -equivariant relative quantum homology of . Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane.
Paper Structure (23 sections, 45 theorems, 182 equations)

This paper contains 23 sections, 45 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Relative cyclic homology
  5. The relative Getzler-Gauss-Manin connection
  6. Euler-gradings
  7. The canonical relative connection
  8. Technical setup
  9. Regularity assumptions
  10. The $\mathfrak{q}$-operations
  11. The $\mathfrak{q}_{-1}$-operations
  12. Bulk-deformations and bounding cochains
  13. Relative cyclic homology of the Fukaya $A_\infty$-algebra
  14. The relative quantum TE-structure
  15. The T-structure
  16. ...and 8 more sections

Key Result

Theorem 1.9

Let $L\subset X$ be an, oriented, relatively-spin Lagrangian submanifold equipped with a $U(\Lambda)$-local system. Suppose there exists a complex structure $J$ such that $(L,J)$ satisfy Assumptions assumptions. Furthermore, suppose $L$ is equipped with a bounding pair $(b,\gamma)$ with curvature $c which is a morphism of TE-structures over $R$. When $L$ is null-homologous, it fits into the commut

Theorems & Definitions (117)

  • Definition 1.1: Formal TEP-structure, see Her
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Conjecture 1.5
  • Definition 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Theorem 1.9: Theorem \ref{['thm: relative OC is a morphism of TE-structures']}
  • Theorem 1.10
  • ...and 107 more