The wall-crossing formula and Lagrangian mutations

TL;DR

This paper establishes a general wall-crossing formula linking disk potentials of monotone Lagrangian submanifolds with their Floer theory, introduces mutations of Lagrangian tori, and explores their implications in del Pezzo surfaces and higher-dimensional toric Fano varieties.

Contribution

It introduces geometric mutations of Lagrangian tori and proves a wall-crossing formula connecting their disk potentials, advancing understanding of mirror symmetry and Floer theory.

Findings

Derived a general wall-crossing formula for disk potentials.

Defined and studied mutations of Lagrangian tori in various geometries.

Identified new Lagrangian tori corresponding to different mirror chambers.

Abstract

We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behavior away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one. In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds. In higher dimension, this provides new Lagrangian tori in toric Fanos corresponding to different chambers of the mirror variety, including ones which are conjecturally separated by infinitely many walls from the chamber containing the standard toric fibre.