Classical elliptic integrable systems from the moduli space of instantons

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.