Discrete Symmetries of the Superpotential and Calculation of Disk Invariants

TL;DR

This paper verifies the integrality of disk invariants in mirror symmetry by leveraging discrete symmetries of the superpotential to compute quantum corrections, with detailed analysis of specific local geometries.

Contribution

It introduces a method to determine quantum corrections using superpotential symmetries, ensuring integrality of disk invariants in mirror Landau-Ginzburg models.

Findings

Confirmed integrality of Ooguri-Vafa disk invariants

Demonstrated the role of superpotential symmetries in quantum corrections

Analyzed local P^2 and F_2 geometries in detail

Abstract

The integrality of Ooguri-Vafa disk invariants is verified using discrete symmetries of the superpotential of the mirror Landau-Ginzburg theory to calculate quantum corrections to the boundary variables. We show that these quantum corrections are completely determined if we assume that the discrete symmetry of the superpotential also holds in terms of the quantum corrected variables. We discuss the case of local P^2 blown up at three points and local F_2 blown up at two points in detail.