Painleve Field Theory

TL;DR

This paper introduces multidimensional Painlevé field theories linked to isomonodromy problems, exploring their reductions, limits, and connections to integrable systems in various dimensions.

Contribution

It develops novel multidimensional Painlevé field theories related to isomonodromy problems and explores their reductions, limits, and integrable system connections.

Findings

Derived nonlocal Painlevé equations in 2+1 dimensions.

Connected Painlevé field theories to integrable Euler-Arnold tops.

Explored limits leading to classical and integrable systems.

Abstract

We propose multidimensional versions of the Painlev\'e 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 Painlev\'e 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 Painlev\'e VI field theory in D=1+1 four constants of the Painlev\'e 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 systems. In this way we derive two and three dimensional integrable nonlocal versions of the integrable Euler-Arnold tops.