Exact quantization conditions for cluster integrable systems

TL;DR

This paper introduces exact quantization conditions for Goncharov-Kenyon quantum integrable systems, leveraging enumerative geometry of toric Calabi-Yau manifolds, extending previous work on mirror curve quantization and relativistic Toda lattice.

Contribution

It proposes a new conjecture for exact quantization conditions based on enumerative geometry, generalizing prior quantization approaches for integrable systems.

Findings

Explicit tests confirm the conjecture for C^3/Z_5 and C^3/Z_6 orbifolds.

The approach unifies geometric and quantum integrable system frameworks.

Results suggest broader applicability to other toric Calabi-Yau related systems.

Abstract

We propose exact quantization conditions for the quantum integrable systems of Goncharov and Kenyon, based on the enumerative geometry of the corresponding toric Calabi-Yau manifolds. Our conjecture builds upon recent results on the quantization of mirror curves, and generalizes a previous proposal for the quantization of the relativistic Toda lattice. We present explicit tests of our conjecture for the integrable systems associated to the resolved C^3/Z_5 and C^3/Z_6 orbifolds.