Quantization of Integrable Systems and Four Dimensional Gauge Theories

Abstract

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 class ical integrable system underlying the moduli space of vacua of the ordinary four dim ensional N = 2 theory. The ? -parameter of the ?-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the super symmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimension al twisted superpotential which becomes the Yang-Yang function of the integrable syst em. We present the thermodynamic-Bethe-ansatz like formulae for these funct ions and for the spectra of commuting Hamiltonians following the direct computatio n in gauge theory. The general construction is illustrated at the examples of the m any-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser sys tem, 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