From symplectic cohomology to Lagrangian enumerative geometry

TL;DR

This paper establishes a connection between Floer theory and Lagrangian enumerative geometry by linking symplectic cohomology elements to mirror Landau-Ginzburg potentials, including higher Maslov index cases, with applications to Liouville domain geometry.

Contribution

It introduces a novel framework connecting symplectic cohomology with holomorphic disk enumeration and mirror symmetry, extending to higher Maslov index potentials.

Findings

Identifies elements in symplectic cohomology mirror to Landau-Ginzburg potentials.

Discovers relations between higher disk potentials and symplectic cohomology rings.

Explores applications to the geometry of Liouville domains.

Abstract

We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains.