The wall-crossing formula and Lagrangian mutations

Abstract

We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behaviour 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.