Let G denote a complex, semisimple, simply-connected group. We identify the equivariant quantum differential equation for the cotangent bundle to the flag variety of G with the affine Knizhnik-Zamolodchikov connection of Cherednik and Matsuo. This recovers Kim's description of quantum cohomology of the flag variety itself as a limiting case. A parallel result is proven for resolutions of the Slodowy slices. Extension to arbitrary symplectic resolutions is discussed.
From our viewpoint, the operators of quantum multiplication form a family (parametrized by @math ) of maximal abelian subalgebras in a certain geometrically constructed Yangian. Long before geometric representation theory, Yangians appeared in mathematical physics as symmetries of integrable models of quantum mechanics and quantum field theory. The maximal abelian subalgebra there is the algebra of quantum integrals of motion, that is, of the operators commuting with the Hamiltonian. A profound correspondence between quantum integrable systems and supersymmetric gauge theories was discovered by Nekrasov and Shatashvili, see @cite_41 . It connects the two appearances of Yangians and, for example, correctly predicts that the eigenvalues of the operators of quantum multiplication are given by solutions of certain Bethe equations (well-known in quantum integrable systems, but probably quite mysterious to geometers). The integrable and geometric viewpoints are complementary in many ways and, no doubt, will lead to new insights on both sides of the correspondence.
@cite_41
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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.