[Paper Review] On the reconstruction problem in mirror symmetry

Junwu Tu|arXiv (Cornell University)|Aug 29, 2012

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper constructs a rigid analytic space $M^{ aisebox{1pt}{\scriptscriptstyle\vee}}_0$ over the Novikov field as a deformation of the semi-flat complex structure on the dual torus fibration over the smooth locus $B_0$ of a Lagrangian torus fibration. Using $A_\infty$ homomorphisms that encode wall-crossing from holomorphic disk moduli, the construction realizes a non-Archimedean mirror symmetry framework under Maslov index zero and unobstructedness assumptions, providing a geometric realization of Fukaya's proposed gluing of Maurer-Cartan moduli spaces.

ABSTRACT

Let π: M a B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M_0^\chk over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B_0 of π. Transition functions of M_0^\chk are obtained via A-\infty homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.

Motivation & Objective

• To realize Fukaya's conceptual approach to mirror symmetry reconstruction via gluing of Maurer-Cartan moduli spaces of $A_\infty$ algebras.
• To construct a mirror space $M^{\scriptscriptstyle\vee}_0$ as a deformation of the semi-flat complex structure on the dual torus fibration over the smooth locus $B_0$.
• To incorporate instanton corrections from symplectic geometry into the transition functions of the mirror space using $A_\infty$ homomorphisms.
• To establish a global valuation map $\mathsf{val}: M^{\scriptscriptstyle\vee}_0 \to B_0$, generalizing the torus fibration structure in non-Archimedean geometry.
• To provide a geometric framework for homological mirror symmetry by ensuring unobstructedness and Maslov index zero conditions are satisfied.

Proposed method

• Constructs local affinoid domains $\mathscr{U}_i = \operatorname{Spec} \mathscr{O}_i$ using canonical models of $A_\infty$ algebras over the Novikov field $\Lambda$.
• Defines transition maps $\Psi_{ij}$ between local charts via $A_\infty$ homomorphisms that encode wall-crossing phenomena in holomorphic disk moduli spaces.
• Uses the valuation map $\mathsf{val}$ to ensure compatibility of local charts and to define a global valuation $\mathsf{val}: M^{\scriptscriptstyle\vee}_0 \to B_0$.
• Applies homotopy invariance of the Maurer-Cartan functor via a two-parameter family of $\omega$-tamed almost complex structures to verify the cocycle condition $\Psi_{jk}\Psi_{ij} = \Psi_{ik}$.
• Employs the non-Archimedean framework of rigid analytic geometry to handle convergence issues in Floer theory, replacing complex manifolds with spaces over the Novikov field.
• Derives explicit gluing formulas (4.12) that reduce to Kontsevich-Soibelman's construction when instanton corrections vanish.

Experimental results

Research questions

• RQ1 How can the reconstruction problem in mirror symmetry be realized via $A_\infty$-theoretic gluing of Maurer-Cartan moduli spaces?
• RQ2 What is the role of instanton corrections from holomorphic disks in deforming the semi-flat mirror structure?
• RQ3 How can a rigid analytic mirror space $M^{\scriptscriptstyle\vee}_0$ be constructed over the Novikov field that captures symplectic data from the original Lagrangian fibration?
• RQ4 In what way do $A_\infty$ homomorphisms encode wall-crossing behavior in the moduli of holomorphic disks?
• RQ5 Can a global valuation map $\mathsf{val}: M^{\scriptscriptstyle\vee}_0 \to B_0$ be defined that generalizes the torus fibration structure in non-Archimedean geometry?

Key findings

• The construction yields a rigid analytic space $M^{\scriptscriptstyle\vee}_0$ over the Novikov field $\Lambda$, which is a deformation of the semi-flat complex structure on the dual torus fibration over $B_0$.
• Transition functions $\Psi_{ij}$ are explicitly given by $A_\infty$ homomorphisms that incorporate instanton corrections from symplectic geometry, generalizing Kontsevich-Soibelman's construction.
• The cocycle condition $\Psi_{jk}\Psi_{ij} = \Psi_{ik}$ is verified using a two-parameter family of almost complex structures and homotopy invariance of the Maurer-Cartan functor.
• The valuation map $\mathsf{val}: M^{\scriptscriptstyle\vee}_0 \to B_0$ is globally well-defined and compatible with the gluing maps, generalizing the fibration structure in non-Archimedean geometry.
• The construction does not require polarization data on $M$, though such data may be needed for compactification of $M^{\scriptscriptstyle\vee}_0$.
• The Maslov index zero and unobstructedness assumptions are essential: the former is automatic for special Lagrangians in Calabi-Yau manifolds, and the latter is necessary for homological mirror symmetry to hold.