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Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov

Abstract

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

Abstract

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.

Paper Structure

This paper contains 28 sections, 15 theorems, 99 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation and comparison with other approaches
  3. Motivation: the mirror of the pair of pants
  4. Beyond the pair of pants
  5. The A-model: Lagrangian Floer theory in trivalent configurations
  6. Objects and morphisms in ${\mathcal{F}}(M)$
  7. $A_\infty$-operations: propagating discs
  8. The $A_\infty$-relations
  9. Propagating discs with one concave corner
  10. Proof of Lemma \ref{['l:tot_mult_invariance']}
  11. Bifurcated propagating discs
  12. Infinite Hamiltonian perturbations
  13. Continuation $A_\infty$-homomorphisms
  14. The B-model: generalized Tate curves from combinatorial data
  15. Generalized Tate curve in terms of formal schemes
  16. ...and 13 more sections

Key Result

Theorem 1.3

Given combinatorial data as above, and setting $q_{e}=T^{A_{e}},$ the Fukaya category ${\mathcal{F}}(M)$ is equivalent to $\operatorname{Perf}(X_K)$.

Figures (6)

  • Figure 1: Wrapped Floer homology in the mirror of the pair of pants, $M=\bigcup_{i=1}^3 (\mathbb{C},0)$
  • Figure 2: Left: $\mathbf{L}_0=\cup \ell_0\in {\mathcal{F}}(\mathbb{C}^3,-xyz)$ and the thimble $\mathcal{T}(L_p)$. Right: the tropical Lagrangian pair of pants $\Lambda_0\simeq \cap\mathbf{L}_0\subset (\mathbb{C}^*)^2,$ and $\Lambda_p=\cap\mathcal{T}(L_p)$.
  • Figure 3: A propagating disc contributing to the Floer product $\mu^2$
  • Figure 4: A one-dimensional family of propagating discs with a concave corner
  • Figure 5: Extending a slit through a constant output component
  • ...and 1 more figures

Theorems & Definitions (34)

  • Example 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1: AA
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 24 more