Description 
SeibergWitten theory maps supersymmetric fourdimensional 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 selfdual hyperKahler equations and their lowdimensional descendants. In particular, the list of such integrable systems includes Hitchin systems defined on Riemann surfaces with singularities at marked points (twodimensional PDE), monopoles on circle bundles over surfaces (threedimensional PDE or circlevalued Hitchin system) and instantons on torically fibered hyperKahler manifolds (fourdimensional PDE or elliptically valued Hitchin system). Deformations of fourdimensional 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 twodimensional conformal theory with Wgalgebra symmetry. Quantization of gmonopole and ginstanton moduli spaces relates to the representation theory of DrinfeldJimbo quantum affine algebras (and their rational and elliptic versions, Yangians and elliptic groups), associated respectively to g in the monopole case (circlevalued 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 (circlevalued and ellipticallyvalued Hitchin).
