Quantization of Integrable Systems and Four Dimensional Gauge Theories
Abstract
We study four dimensional {N} = 2 supersymmetric gauge theory in the Ωbackground with the two dimensional {N} = 2 superPoincare 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 ∊parameter 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 on Σ which becomes the YangYang function of the integrable system. We present the thermodynamicBetheAnsatz 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 manybody systems, such as the periodic Toda chain, the elliptic CalogeroMoser 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.
 Publication:

XVITH INTERNATIONAL CONGRESS ON MATHEMATICAL PHYSICS. Held 38 August 2009 in Prague
 Pub Date:
 March 2010
 DOI:
 10.1142/9789814304634_0015
 arXiv:
 arXiv:0908.4052
 Bibcode:
 2010maph.conf..265N
 Keywords:

 gauge theory;
 instantons;
 Bethe Ansatz;
 manybody systems;
 finite size corrections;
 Smatrix;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Analysis of PDEs;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 32 pp. 1 figure