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Lectures on Gauge theories and Many-Body systems

Igor Chaban, Nikita Nekrasov

TL;DR

The work surveys how gauge theories and integrable many-body systems are interwoven through two principal routes: a Hamiltonian-reduction viewpoint that ties CMS-type Hamiltonians to gauge-theory phase spaces (e.g., $H_k=\frac{1}{k}\mathrm{Tr} L^k$) and a web of dualities that map classical problems to quantum ones via geometric data such as spectral curves and qq-characters. It then extends the CMS framework to elliptic and complexified settings, connects to Nekrasov’s instanton counting on partitions, and develops a rich structure of order and disorder observables, surface defects, and their pushforwards, all encoded by Lax operators and spectral curves. The resulting picture yields concrete computational tools for Lax representations, spectral curves, and defect-based constructions, enabling cross-fertilization between gauge-theory localization and integrable-system techniques. Overall, the paper builds a cohesive framework linking symplectic reduction, complex gauge data, and combinatorial/Geometric representation theory to advance the gauge–integrable-systems correspondence across dimensions and deformations.

Abstract

These lectures discuss two correspondences between gauge theories and integrable many-body systems. The first correspondence goes back to the work of many mathematicians and physicists in the 1980-1990's. It is realized by an infinite dimensional Hamiltonian reduction and its quantum counterpart. In this approach the quantization parameters of the gauge theory coincide with the quantization parameters of the many-body system. The second correspondence emerged in the mid-1990's, it involves non-trivial dualities, relating classical problems on one side to quantum on another and vice versa. This duality has various reincarnations: Fourier and Legendre transforms, Langlands duality, etc. The quantization parameters are mapped to geometric parameters. Simple questions on one side solve complicated ones on the other and vice versa.

Lectures on Gauge theories and Many-Body systems

TL;DR

The work surveys how gauge theories and integrable many-body systems are interwoven through two principal routes: a Hamiltonian-reduction viewpoint that ties CMS-type Hamiltonians to gauge-theory phase spaces (e.g., ) and a web of dualities that map classical problems to quantum ones via geometric data such as spectral curves and qq-characters. It then extends the CMS framework to elliptic and complexified settings, connects to Nekrasov’s instanton counting on partitions, and develops a rich structure of order and disorder observables, surface defects, and their pushforwards, all encoded by Lax operators and spectral curves. The resulting picture yields concrete computational tools for Lax representations, spectral curves, and defect-based constructions, enabling cross-fertilization between gauge-theory localization and integrable-system techniques. Overall, the paper builds a cohesive framework linking symplectic reduction, complex gauge data, and combinatorial/Geometric representation theory to advance the gauge–integrable-systems correspondence across dimensions and deformations.

Abstract

These lectures discuss two correspondences between gauge theories and integrable many-body systems. The first correspondence goes back to the work of many mathematicians and physicists in the 1980-1990's. It is realized by an infinite dimensional Hamiltonian reduction and its quantum counterpart. In this approach the quantization parameters of the gauge theory coincide with the quantization parameters of the many-body system. The second correspondence emerged in the mid-1990's, it involves non-trivial dualities, relating classical problems on one side to quantum on another and vice versa. This duality has various reincarnations: Fourier and Legendre transforms, Langlands duality, etc. The quantization parameters are mapped to geometric parameters. Simple questions on one side solve complicated ones on the other and vice versa.
Paper Structure (56 sections, 11 theorems, 273 equations, 2 figures, 1 table)

This paper contains 56 sections, 11 theorems, 273 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Calogero--Moser many-body system
  3. Definition of the system
  4. Integrals of motions
  5. A simple system
  6. Hamiltonian action
  7. Symplectic reduction
  8. Hamiltonians of the reduction
  9. Gauge theory counterpart
  10. Lagrangian description
  11. $2d$ Yang--Mills theory
  12. Hamiltonian description
  13. Calogero--Moser--Sutherland system
  14. Elliptic Calogero--Moser system
  15. Compactification of the phase space
  16. ...and 41 more sections

Key Result

Proposition 2.1

The natural $U(N)-$action on $\mathcal{P}_{L}$

Figures (2)

  • Figure 1: A Young diagram with boxes in $\partial_{+}\lambda$ and $\partial_{-}\lambda$ colored green and pink respectively.
  • Figure 2: A pair of transposed Young diagrams

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.2
  • Proposition 2.1
  • Remark 2.3
  • Theorem 2.1
  • Definition 2.5
  • ...and 39 more