New Exact Quantization Condition for Toric Calabi-Yau Geometries

TL;DR

This paper introduces a simplified, exact quantization condition for quantum systems from toric Calabi-Yau three-folds, capturing all non-perturbative energy contributions and linking to topological invariants.

Contribution

It presents a novel, more straightforward exact quantization condition that includes non-perturbative effects and relates to topological invariants of toric Calabi-Yau geometries.

Findings

The new quantization condition is consistent with previous results.

It reveals non-trivial relations among Gopakumar-Vafa invariants.

The approach simplifies the analysis of quantum spectra in these geometries.

Abstract

We propose a new exact quantization condition for a class of quantum mechanical systems derived from local toric Calabi-Yau three-folds. Our proposal includes all contributions to the energy spectrum which are non-perturbative in the Planck constant, and is much simpler than the available quantization condition in the literature. We check that our proposal is consistent with previous works and implies non-trivial relations among the topological Gopakumar-Vafa invariants of the toric Calabi-Yau geometries. Together with the recent developments, our proposal opens a new avenue in the long investigations at the interface of geometry, topology and quantum mechanics.