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Quantization of Integrable Systems and Four Dimensional Gauge Theories

Nikita A. Nekrasov, Samson L. Shatashvili

TL;DR

The paper develops a four-dimensional ${\mathcal N}=2$ gauge theory in an ${\Omega}$-background as a quantization framework for classical integrable systems tied to the moduli space of vacua. It identifies the deformation parameter ${\varepsilon}$ with the Planck constant and shows that the two-dimensional twisted superpotential ${\mathcal W}(a;\varepsilon)$ plays the role of the Yang-Yang function, governing the quantum spectrum via Bethe equations. The authors provide a detailed account for periodic Toda, elliptic Calogero-Moser, and Hitchin-type systems, including both classical and quantum descriptions, and present Thermodynamic Bethe Ansatz-like expressions for the spectra and observables. This Bethe/gauge correspondence yields a robust, gauge-theoretic route to quantize classical integrable systems and to compute their spectra and Yang-Yang data, with broad implications for quantum cohomology, representation theory, and beyond.

Abstract

We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system.

Quantization of Integrable Systems and Four Dimensional Gauge Theories

TL;DR

The paper develops a four-dimensional gauge theory in an -background as a quantization framework for classical integrable systems tied to the moduli space of vacua. It identifies the deformation parameter with the Planck constant and shows that the two-dimensional twisted superpotential plays the role of the Yang-Yang function, governing the quantum spectrum via Bethe equations. The authors provide a detailed account for periodic Toda, elliptic Calogero-Moser, and Hitchin-type systems, including both classical and quantum descriptions, and present Thermodynamic Bethe Ansatz-like expressions for the spectra and observables. This Bethe/gauge correspondence yields a robust, gauge-theoretic route to quantize classical integrable systems and to compute their spectra and Yang-Yang data, with broad implications for quantum cohomology, representation theory, and beyond.

Abstract

We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system.

Paper Structure

This paper contains 38 sections, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the Bethe/gauge correspondence
  3. Twisted chiral ring and quantum integrability
  4. Topological field theory
  5. Yang-Yang function and quantum spectrum
  6. Quantization from four dimensions
  7. Four dimensional gauge theory
  8. The $\Omega$-background and twisted masses
  9. Four dimensional theory on ${\bf R}^2_{\varepsilon}$
  10. Calculation of the twisted superpotential
  11. Integrable systems
  12. The classical story
  13. The ${\bf t}$-deformation
  14. Quantization
  15. Quantization and $\bf t$-deformation
  16. ...and 23 more sections

Figures (1)

  • Figure 1: The contour ${\mathcal{C}}$