Classical elliptic integrable systems from the moduli space of instantons

TL;DR

This paper reviews how the moduli space of instantons relates to classical integrable systems, deriving key matrices and solutions, and exploring dualities and eigenvectors using advanced cohomology techniques.

Contribution

It introduces a novel derivation of Krichever's Lax matrix from instanton moduli space cohomology, extending to K-theoretic and elliptic cases, and connects instanton partition functions to integrable system solutions.

Findings

Derived Krichever's Lax matrix from instanton cohomology.

Provided a natural eigenvector and horizontal section in terms of instanton partition functions.

Progressed understanding of spectral duality and quantum-classical duality in integrable systems.

Abstract

This paper is intended to serve as a review of a series of papers with Nikita Nekrasov, where we achieved several important results concerning the relation between the moduli space of instantons and classical integrable systems. We derive I. Krichever's Lax matrix for the elliptic Calogero-Moser system from the equivariant cohomology of the moduli space of instantons. This result also has K-theoretic and elliptic cohomology counterparts. Our methods rely upon the so-called $\theta$-transform of the $qq$-characters vev's, defined as integrals of certain classes in these cohomology theories. The key step is the non-commutative Jacobi-like product formula for them. We also obtained a natural answer for the eigenvector of the Lax matrix and the horizontal section for the associated isomonodromic connection in terms of the partition function of folded instantons. As an application of our formula, we demonstrate some progress towards the spectral duality of the many-body systems in question, as well as give a new look at the quantum-classical duality between their trigonometric version and the corresponding spin chains.