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Family Floer program and non-archimedean SYZ mirror construction

Hang Yuan

TL;DR

This work develops a non-archimedean realization of the SYZ program by building a mirror Landau-Ginzburg model from a semipositive Lagrangian torus fibration. It introduces a categorical framework of A∞-algebras with topological labels and a divisor-axiom–respecting, unital ud-homotopy theory over the Novikov field $\Lambda$, enabling controlled wall-crossing and disk counting to produce a global, affinoid mirror. Local mirror charts arise from Maurer–Cartan formalisms on Lagrangian fibers; transition maps are glued via Fukaya’s trick and a new wall-crossing formula, yielding a coherent global non-archimedean space $X_0^ lat$ equipped with a superpotential $W_0^ lat$ and an affinoid torus fibration $\pi_0^ lat$. The construction generalizes Auroux’s conjectures to a non-archimedean setting, supplies explicit local-to-global glueing data, and has applications to SYZ reconstructions with singular fibers and to the spectral aspects of quantum cohomology through $W_0^ lat$’s critical values. Overall, the paper provides a rigorous, algebraically robust, non-archimedean framework for understanding quantum corrections and wall-crossing in mirror symmetry, with a precise cocycle condition ensuring the mirror is globally well-defined.

Abstract

Given a Lagrangian fibration, we provide a natural construction of a mirror Landau-Ginzburg model consisting of a rigid analytic space, a superpotential function, and a dual fibration based on Fukaya's family Floer theory. The mirror in the B-side is constructed by the counts of holomorphic disks in the A-side together with the non-archimedean analysis and the homological algebra of the $A_\infty$ structures. It fits well with the SYZ dual fibration picture and explains the quantum/instanton corrections and the wall crossing phenomenon. Instead of a special Lagrangian fibration, we only need to assume a weaker semipositive Lagrangian fibration to carry out the non-archimedean SYZ mirror reconstruction.

Family Floer program and non-archimedean SYZ mirror construction

TL;DR

This work develops a non-archimedean realization of the SYZ program by building a mirror Landau-Ginzburg model from a semipositive Lagrangian torus fibration. It introduces a categorical framework of A∞-algebras with topological labels and a divisor-axiom–respecting, unital ud-homotopy theory over the Novikov field , enabling controlled wall-crossing and disk counting to produce a global, affinoid mirror. Local mirror charts arise from Maurer–Cartan formalisms on Lagrangian fibers; transition maps are glued via Fukaya’s trick and a new wall-crossing formula, yielding a coherent global non-archimedean space equipped with a superpotential and an affinoid torus fibration . The construction generalizes Auroux’s conjectures to a non-archimedean setting, supplies explicit local-to-global glueing data, and has applications to SYZ reconstructions with singular fibers and to the spectral aspects of quantum cohomology through ’s critical values. Overall, the paper provides a rigorous, algebraically robust, non-archimedean framework for understanding quantum corrections and wall-crossing in mirror symmetry, with a precise cocycle condition ensuring the mirror is globally well-defined.

Abstract

Given a Lagrangian fibration, we provide a natural construction of a mirror Landau-Ginzburg model consisting of a rigid analytic space, a superpotential function, and a dual fibration based on Fukaya's family Floer theory. The mirror in the B-side is constructed by the counts of holomorphic disks in the A-side together with the non-archimedean analysis and the homological algebra of the structures. It fits well with the SYZ dual fibration picture and explains the quantum/instanton corrections and the wall crossing phenomenon. Instead of a special Lagrangian fibration, we only need to assume a weaker semipositive Lagrangian fibration to carry out the non-archimedean SYZ mirror reconstruction.

Paper Structure

This paper contains 59 sections, 80 theorems, 317 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Application
  4. Outline
  5. Motivating ideas
  6. Mirror local charts
  7. Mirror transition maps
  8. Mirror cocycle condition
  9. $A_\infty$ structures in Lagrangian Floer theory
  10. $A_\infty$ structures with topological labels
  11. Definition
  12. Pseudo-isotopies
  13. Unitality and divisor axiom
  14. Category $\mathscr{UD}$
  15. Homotopy theory with divisor axiom
  16. ...and 44 more sections

Key Result

Theorem 1.1

Given Assumption assumption-mu ge 0 and assumption_obstruction_ideal below, we can associate to the pair $(X,\pi_0)$ a triple $\mathbb X^\vee:=(X_0^\vee,W_0^\vee, \pi_0^\vee)$ consisting of a non-archimedean analytic space $X_0^\vee$ over $\Lambda$, a global analytic function $W_0^\vee$, and an affi

Figures (2)

  • Figure 1: The shadowed disk represents the quantum correction (taken from MSClayII).
  • Figure 2: Homological perturbation

Theorems & Definitions (176)

  • Theorem 1.1
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.9
  • ...and 166 more