Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries

TL;DR

This paper explores quantum integrable systems from dimer models and Calabi-Yau geometries, computing spectra and quantum periods using perturbative and Bohr-Sommerfeld methods, and connecting to topological string theory.

Contribution

It provides new exact analytic results for quantum and classical periods of higher genus Calabi-Yau geometries and determines quantum period differential operators.

Findings

Spectra computed via perturbation and Bohr-Sommerfeld methods

Exact analytic results for quantum and classical periods

Agreement with topological string free energy calculations

Abstract

We study a class of quantum integrable systems derived from dimer graphs and also described by local toric Calabi-Yau geometries with higher genus mirror curves, generalizing some previous works on genus one mirror curves. We compute the spectra of the quantum systems both by standard perturbation method and by Bohr-Sommerfeld method with quantum periods as the phase volumes. In this way, we obtain some exact analytic results for the classical and quantum periods of the Calabi-Yau geometries. We also determine the differential operators of the quantum periods and compute the topological string free energy in Nekrasov-Shatashvili (NS) limit. The results agree with calculations from other methods such as the topological vertex.