Lectures on Gauge theories and Many-Body systems

TL;DR

This paper explores two deep correspondences between gauge theories and integrable many-body systems, highlighting their classical and quantum connections across various dimensions and supersymmetry conditions.

Contribution

It introduces new links between gauge theory dynamics and Calogero--Moser systems via Hamiltonian reduction and instanton counting, expanding understanding of their quantum and classical relations.

Findings

Gauge-theoretic dynamics relate to Calogero--Moser systems through Hamiltonian reduction.

Supersymmetric gauge theories connect classical and quantum problems via instanton counting.

Non-local observables satisfy Dyson--Schwinger equations and lead to Schrödinger equations for many-body systems.

Abstract

These lectures study two correspondences between gauge theories and integrable many-body systems. The first arises from infinite-dimensional Hamiltonian reduction and relates gauge-theoretic dynamics directly to Calogero--Moser-type systems and their quantum counterparts. The second emerges in supersymmetric gauge theory through instanton counting and non-perturbative dualities, linking classical problems on one side to quantum problems on the other. A central motivation comes from the observation that conjugacy classes of holonomies in gauge theory can be interpreted as configurations of indistinguishable particles on a circle. In quantum theory these particle positions become random variables, and the correspondence may be either exact or approximate depending on spacetime dimension and supersymmetry. We focus on the Calogero--Moser--Sutherland family associated with root systems of type A and with SU(N) gauge theories in dimensions from one to six. In low dimensions the correspondence is direct and involves matrix quantum mechanics, two-dimensional Yang--Mills theory, and higher-dimensional Chern--Simons-type theories. In four, five, and six dimensions with eight supercharges, the correspondence takes a more indirect form through supersymmetric gauge theory and the Omega-deformation. We introduce non-local observables whose correlation functions satisfy non-perturbative Dyson--Schwinger equations and, in four dimensions, lead to Schrodinger equations for many-body systems. The notes are divided accordingly: the first part develops symplectic-reduction constructions of Calogero--Moser systems, while the second studies localization, partition measures, and order/disorder observables in supersymmetric gauge theory.