Quantization of Integrable Systems and Four Dimensional Gauge Theories

Abstract

Abstract. We study four dimensional N = 2 supersymmetric gauge theory in the Ω-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 ε-parameter of the Ω-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.

From strings in 6d to strings in 5d

“…Now we claim that the elliptic genus of the strings in 5d is just given by setting 1 = 0 in (3.5) which corresponds to the Nekrasov-Shatashvili limit [19]. In particular it is defined by the operation…”

Bethe/gauge correspondence on curved spaces

“…In [4][5][6] the correspondence, called the Bethe/gauge correspondence, between the two dimensional N = 2 supersymmetric gauge theories and quantum integrable systems was formulated in full generality. The novelty of this correspondence is the identification of the Planck constant of the quantum integrable system with a twisted mass parameter on the gauge theory side.…”

“…In [4][5][6] explicit formulas for W eff for a very large class of theories were given, including the properly two dimensional theories, three dimensional theories compactified on a circle, and four dimensional theories compactified on the two-torus or subject to the two dimensional Ω-deformation [14]; the interested reader can consult these papers for details. Our notations follow mostly the conventions from these papers.…”

“…In [4][5][6] one can find the expressions for the effective twisted superpotentials of the four dimensional N = 1 theory, compactified on a torus T 2 or the four dimensional N = 2 theory subject to a special Ω-background. These formulas are much more complicated and we shall not write them here.…”

“…The correspondence between supersymmetric gauge theory and quantum integrable system was discovered, in an example, in [1], further explored in [2,3], where it was speculated that such a correspondence should be a general property of a larger class of gauge theories in various space-time dimensions, and finally formulated and explained in [4][5][6] . Prior to [1] a connection of two dimensional pure Yang-Mills theory with massive matter to spin and many-body systems systems with long-range interaction appeared in [7][8][9][10] (the embedding of pure two dimensional Yang-Mills theory into the supersymmetric theory was discussed in [11]).…”