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Painleve Field Theory

We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the zero modes leads to ${\rm SL}(N, {\mathbb C})$ monodromy preserving equations. The latter coincide with the Painlevé VI equation for $N\!=\!2$. We consider two types of the bundles. In the first one the group of automorphisms is the centrally and cocentrally extended loop group $L({\rm SL}(N, {\mathbb C}))$ or some multiloop group. In the case of the Painlevé VI field theory in D=1+1 four constants of the Painlevé VI equation become dynamical fields. The second type of bundles are defined by the group of automorphisms of the noncommutative torus. They lead to the equations in dimension 2+1. In both cases we consider trigonometric, rational and scaling limits of the theories. Generically (except some degenerate cases) the derived equations are nonlocal. We consider Whitham quasiclassical limit to integrable syst