Quantization condition from exact WKB for difference equations

TL;DR

This paper connects the non-perturbative quantization condition of topological string partition functions to the exact WKB analysis of difference equations, providing a new perspective on quantum mirror curves.

Contribution

It demonstrates that the quantization condition arises from single-valuedness and smoothness requirements, derived through exact WKB analysis of the difference equations.

Findings

Quantization condition linked to monodromy and smoothness.

Exact WKB analysis applied to difference equations.

Supports conjecture on quantum mirror curve eigenvalues.

Abstract

A well-motivated conjecture states that the open topological string partition function on toric geometries in the Nekrasov-Shatashvili limit is annihilated by a difference operator called the quantum mirror curve. Recently, the complex structure variables parameterizing the curve, which play the role of eigenvalues for related operators, were conjectured to satisfy a quantization condition non-perturbative in the NS parameter $\hbar$. Here, we argue that this quantization condition arises from requiring single-valuedness of the partition function, combined with the requirement of smoothness in the parameter $\hbar$. To determine the monodromy of the partition function, we study the underlying difference equation in the framework of exact WKB.