Tropical disk potential for almost toric manifolds

TL;DR

The paper presents a tropical formula for disk potentials of Lagrangian tori in almost toric four-manifolds, extending previous results and providing explicit computations for del Pezzo surfaces.

Contribution

It generalizes Mikhalkin's results to almost toric manifolds and computes explicit potentials for Lagrangian tori in del Pezzo surfaces.

Findings

Derived a tropical formula for disk potentials in almost toric manifolds.

Computed potentials explicitly for Lagrangian tori in del Pezzo surfaces.

Connected the formula to existing work in the monotone case and the Gross-Siebert program.

Abstract

Using our previous work we give a tropical formula for disk potentials for Lagrangian tori in almost toric four-manifolds, that is, fibrations by Lagrangian tori with only toric and focus-focus singularities, generalizing results of Mikhalkin for holomorphic spheres in the projective plane. As examples, we directly compute potentials for Lagrangian tori in del Pezzo surfaces equipped with monotone symplectic forms. These formulas were established in the monotone case by different methods in Pascaleff-Tonkonog, and investigated from the point of view of the Gross-Siebert program in Carl-Pumperla-Siebert, Bardwell-Evans--Cheung--Hong--Lin and also Lau-Lee-Lin.