Non-perturbative Quantum Theories and Integrable Equations

TL;DR

This paper reviews how classical integrable systems serve as powerful tools for understanding non-perturbative phenomena in quantum string and gauge theories, highlighting parallels with topological string theories.

Contribution

It elucidates the connections between integrable systems and non-perturbative quantum theories, emphasizing spectral curves, deformations, and associativity equations with illustrative examples.

Findings

Spectral curves encode non-perturbative data.

Hidden parallels between 2d topological and 4d non-topological theories.

Explicit examples demonstrate the integrable approach.

Abstract

I review the appearance of classical integrable systems as an effective tool for the description of non-perturbative exact results in quantum string and gauge theories. Various aspects of this relation: spectral curves, action-angle variables, Whitham deformations and associativity equations are considered separately demonstrating hidden parallels between topological 2d string theories and naively non-topological 4d theories. The proofs are supplemented by explicit illustrative examples.