Exact quantization conditions for the relativistic Toda lattice

TL;DR

This paper proposes exact quantization conditions for the relativistic Toda lattice using spectral theory and topological string theory, incorporating non-perturbative effects related by S-duality, and tests these conditions for N=3.

Contribution

It introduces a novel set of exact quantization conditions involving Nekrasov-Shatashvili free energy and non-perturbative contributions, extending to general toric Calabi-Yau manifolds.

Findings

Quantization conditions match explicit spectral calculations for N=3

Conditions involve S-duality related non-perturbative terms

Framework potentially applicable to broader quantum integrable systems

Abstract

Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy, which resums the perturbative WKB expansion, but they require in addition a non-perturbative contribution, which is related to the perturbative result by an S-duality transformation of the Planck constant. We test the quantization conditions against explicit calculations of the spectrum for N=3. Our proposal can be generalized to arbitrary toric Calabi-Yau manifolds and might solve the corresponding quantum integrable system of Goncharov and Kenyon.