Quantized mirror curves and resummed WKB

TL;DR

This paper develops a first principles approach to quantize mirror curves in toric Calabi-Yau threefolds, providing explicit formulas for eigenvalues and eigenfunctions, and demonstrating their accuracy through numerical checks.

Contribution

It introduces a novel ansatz based on modular duality to derive closed-form quantization conditions and eigenfunctions for difference equations from mirror symmetry.

Findings

Explicit quantization conditions for rational ar/2

Eigenfunctions expressed in closed form

Successful numerical validation of theoretical results

Abstract

Based on previous insights, we present an ansatz to obtain quantization conditions and eigenfunctions for a family of difference equations which arise from quantized mirror curves in the context of local mirror symmetry of toric Calabi-Yau threefolds. It is a first principles construction, which yields closed expressions for the quantization conditions and the eigenfunctions when $\hbar/2\pi \in \mathbb Q$. The key ingredient is the modular duality structure of the underlying quantum integrable system. We use our ansatz to write down explicit results in some examples, which are successfully checked against purely numerical results for both the spectrum and the eigenfunctions. Concerning the quantization conditions, we also provide evidence that, in the rational case, this method yields a resummation of conjectured quantization conditions involving enumerative invariants of the underlying toric Calabi-Yau threefold.