Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String

TL;DR

This paper establishes a precise relationship between two quantization schemes for mirror curves of toric Calabi-Yau threefolds, revealing deep connections with nonperturbative topological string theory and spectral theory.

Contribution

It generalizes the correspondence between NS and Grassi-Hatsuda-Marino quantization schemes to higher genus mirror curves and demonstrates their equivalence through spectral and theta function analysis.

Findings

Existence of at least g quantum Riemann theta functions for genus g

Spectra coincide with intersections of theta divisors from different schemes

Infinite constraints among Gopakumar-Vafa invariants are required for equivalence

Abstract

We establish the precise relation between the Nekrasov-Shatashvili (NS) quantization scheme and Grassi-Hatsuda-Marino conjecture for the mirror curve of arbitrary toric Calabi-Yau threefold. For a mirror curve of genus $g$, the NS quantization scheme leads to $g$ quantization conditions for the corresponding integrable system. The exact NS quantization conditions enjoy a self S-duality with respect to Planck constant $\hbar$ and can be derived from the Lockhart-Vafa partition function of nonperturbative topological string. Based on a recent observation on the correspondence between spectral theory and topological string, another quantization scheme was proposed by Grassi-Hatsuda-Marino, in which there is a single quantization condition and the spectra are encoded in the vanishing of a quantum Riemann theta function. We demonstrate that there actually exist at least $g$ nonequivalent quantum Riemann theta functions and the intersections of their theta divisors coincide with the spectra determined by the exact NS quantization conditions. This highly nontrivial coincidence between the two quantization schemes requires infinite constraints among the refined Gopakumar-Vafa invariants. The equivalence for mirror curves of genus one has been verified for some local del Pezzo surfaces. In this paper, we generalize the correspondence to higher genus, and analyze in detail the resolved $\mathbb{C}^3/\mathbb{Z}_5$ orbifold and several $SU(N)$ geometries. We also give a proof for some models at $\hbar=2\pi/k$.