Appears in collection : Vasily Pestun - Quantum gauge theories and integrable system

Seiberg-Witten theory maps supersymmetric four-dimensional gauge theories with extended supersymmetry to algebraic completely integrable systems. For large class of such integrable systems the phase space is the moduli space of solutions of self-dual hyperKahler equations and their low-dimensional descendants. In particular, the list of such integrable systems includes Hitchin systems defined on Riemann surfaces with singularities at marked points (two-dimensional PDE), monopoles on circle bundles over surfaces (three-dimensional PDE or circle-valued Hitchin system) and instantons on torically fibered hyperKahler manifolds (four-dimensional PDE or elliptically valued Hitchin system). Deformations of four-dimensional gauge theory by curved backgrounds correspond to the quantization of the associated algebraic integrable systems. Quantization of Hitchin systems has relation to geometric Langlands correspondence and to the Toda two-dimensional conformal theory with Wg-algebra symmetry. Quantization of g-monopole and g-instanton moduli spaces relates to the representation theory of Drinfeld-Jimbo quantum affine algebras (and their rational and elliptic versions, Yangians and elliptic groups), associated respectively to g in the monopole case (circle-valued Hitchin) and to the central extension of the loop algebra of g in the instanton case (elliptically valued Hitchin). It is expected that there exists an analogue of geometric Langlands correspondence for quantization of the monopole and instanton algebraic integrable system (circle-valued and elliptically-valued Hitchin).