Morse superpotentials and blow-ups of surfaces

TL;DR

This paper investigates the Landau-Ginzburg mirror of blowups of toric surfaces, using tropical geometry to locate critical points of the superpotential and establish their correspondence with the surface's cohomology, confirming mirror symmetry.

Contribution

It introduces a tropical geometry-based method to identify critical points of superpotentials in mirror symmetry for blown-up toric surfaces, including non-Fano cases.

Findings

Critical points are non-degenerate for generic parameters.

Number of critical points equals the surface's cohomology rank.

Supports closed-string mirror symmetry for these surfaces.

Abstract

We study the Landau-Ginzburg mirror of toric/non-toric blowups of (possibly non-Fano) toric surfaces arising from SYZ mirror symmetry. Through the framework of tropical geometry, we provide an effective method for identifying the precise locations of critical points of the superpotential, and further show their non-degeneracy for generic parameters. Moreover, we prove that the number of geometric critical points equals the rank of cohomology of the surface, which leads to its closed-string mirror symmetry due to Bayer's earlier result.