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Splitting the diagonal for broken maps

Sushmita Venugopalan, Chris Woodward

TL;DR

This work develops the split Fukaya algebra by degenerating the edge matching conditions of broken maps to split edges, and formalizes deformed maps that interpolate between broken and split configurations. It proves that the broken and split Fukaya algebras are $A_\infty$-homotopy equivalent, enabling a robust framework to study unobstructedness through tropical and combinatorial data. The approach recovers and extends unobstructedness results for toric Lagrangians, and provides an alternate proof of Fukaya–Oh–Ohta–Ono in toric settings, with applications to almost toric four-manifolds. By connecting diagonal splitting in toric geometry, SFT degenerations, and tropical geometry, the paper offers a versatile toolkit for understanding Floer theory in degenerate or toric-type geometries, including explicit disk potential comparisons via the Batyrev–Givental potential. The results illuminate how tropical and cone-analytic data govern obstructions and potentials, with potential implications for mirror symmetry constructions in degenerating families.

Abstract

In previous work, we introduced a version of the Fukaya algebra associated to a degeneration of a symplectic manifold, whose structure maps count collections of maps in the components of the degeneration satisfying matching conditions. In this paper, we introduce a further degeneration of the matching conditions (similar in spirit to Bourgeois' version of symplectic field theory) which results in a "split Fukaya algebra" whose structure maps are, in good cases, sums of products over vertices of tropical graphs. In the case of toric Lagrangians contained in a toric component of the degeneration, an invariance argument implies the existence of projective Maurer-Cartan solutions, which gives an alternate proof of the unobstructedness result of Fukaya-Oh-Ohta-Ono for toric manifolds. Our result also proves unobstructedness in more general cases, such as for toric Lagrangians in almost toric four-manifolds.

Splitting the diagonal for broken maps

TL;DR

This work develops the split Fukaya algebra by degenerating the edge matching conditions of broken maps to split edges, and formalizes deformed maps that interpolate between broken and split configurations. It proves that the broken and split Fukaya algebras are -homotopy equivalent, enabling a robust framework to study unobstructedness through tropical and combinatorial data. The approach recovers and extends unobstructedness results for toric Lagrangians, and provides an alternate proof of Fukaya–Oh–Ohta–Ono in toric settings, with applications to almost toric four-manifolds. By connecting diagonal splitting in toric geometry, SFT degenerations, and tropical geometry, the paper offers a versatile toolkit for understanding Floer theory in degenerate or toric-type geometries, including explicit disk potential comparisons via the Batyrev–Givental potential. The results illuminate how tropical and cone-analytic data govern obstructions and potentials, with potential implications for mirror symmetry constructions in degenerating families.

Abstract

In previous work, we introduced a version of the Fukaya algebra associated to a degeneration of a symplectic manifold, whose structure maps count collections of maps in the components of the degeneration satisfying matching conditions. In this paper, we introduce a further degeneration of the matching conditions (similar in spirit to Bourgeois' version of symplectic field theory) which results in a "split Fukaya algebra" whose structure maps are, in good cases, sums of products over vertices of tropical graphs. In the case of toric Lagrangians contained in a toric component of the degeneration, an invariance argument implies the existence of projective Maurer-Cartan solutions, which gives an alternate proof of the unobstructedness result of Fukaya-Oh-Ohta-Ono for toric manifolds. Our result also proves unobstructedness in more general cases, such as for toric Lagrangians in almost toric four-manifolds.

Paper Structure

This paper contains 23 sections, 23 theorems, 169 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Broken manifold
  4. Broken maps
  5. Tropical symmetry
  6. Broken treed holomorphic disks
  7. Convergence of broken maps
  8. Boundary strata
  9. Broken Fukaya algebras
  10. Deformed maps
  11. Split tropical graphs
  12. The idea of a split map
  13. Splitting the diagonal
  14. Preliminaries on cones and genericity
  15. Split tropical graphs
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let ${\eta_{\operatorname{gen}}} \in \mathfrak{t}$ be a generic direction of deformation. The broken Fukaya algebra $CF_{{\operatorname{brok}}}(\mathfrak{X},L)$ is $A_\infty$-homotopy equivalent to the split Fukaya algebra $CF_{{\operatorname{split}}}(\mathfrak{X}, L, {\eta_{\operatorname{gen}}})$.

Figures (13)

  • Figure 1: A polyhedral decomposition $\mathcal{P}$, its dual complex $B^\vee$ and a cutting datum.
  • Figure 2: The tropical graph $\Gamma_1$ is rigid, but $\Gamma_2$ is not rigid since the vertices inside the square can be moved to the dotted positions.
  • Figure 3: Split disk corresponding to a disk in the Hirzebruch surface $X=F_1$ of homology class $\delta_{D_2} +\delta_E$ passing through a fixed point $p$.
  • Figure 4: Quasi-split tropical graphs $({\tilde{\Gamma}}_1,\Gamma_1)$, $({\tilde{\Gamma}}_2,\Gamma_2)$. In both cases $e$ is the only split edge, and the direction condition is not satisfied for $e$ in ${\tilde{\Gamma}}_i$.
  • Figure 5: Split graphs $({\tilde{\Gamma}}_1,\Gamma)$, $({\tilde{\Gamma}}_2,\Gamma)$ with split edge $e$ and a cone direction ${\eta_{\operatorname{gen}}}$ satisfying $\langle {{\eta_{\operatorname{gen}}},(1,-1)} \rangle > 0$.
  • ...and 8 more figures

Theorems & Definitions (93)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • ...and 83 more