Quantization of integrable systems and four dimensional gauge theories

Nikita A. Nekrasov, Samson L. Shatashvili

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

210 Citations (Scopus)


We study four dimensional N = 2 supersymmetric gauge theory in the 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 εparameter is identi- fied 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 on which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-Ansatz like formulae for these functions and 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 L2-spectrum. We very briefly discuss the quantization of Hitchin system.

Original languageEnglish
Title of host publicationXVIth International Congress on Mathematical Physics
PublisherWorld Scientific Publishing Co.
Number of pages25
ISBN (Electronic)9789814304634
ISBN (Print)981430462X, 9789814304627
Publication statusPublished - 1 Jan 2010
Externally publishedYes


  • Bethe ansatz
  • Finite size corrections
  • Gauge theory
  • Instantons
  • Many-body systems
  • S-matrix


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