Symplectic and Poisson Geometry on Loop Spaces of Manifolds and Nonlinear Equations

TL;DR

This paper explores geometric structures on loop spaces of manifolds to provide Hamiltonian frameworks for various nonlinear equations in physics and field theory, demonstrating integrability and new geometric insights.

Contribution

It introduces new classes of Poisson and symplectic structures on loop spaces that yield Hamiltonian formulations for complex nonlinear equations and proves integrability of certain hydrodynamic systems.

Findings

Established Hamiltonian structures for nonlinear sigma models with torsion

Proved integrability of some hydrodynamic type systems

Connected nonlinear PDEs in topological field theories to integrable systems

Abstract

We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian and multihamiltonian representations for some important nonlinear equations of mathematical physics and field theory such as nonlinear sigma models with torsion, degenerate Lagrangian systems of field theory, systems of hydrodynamic type, N-component systems of Heisenberg magnet type, Monge-Amp\`ere equations, the Krichever-Novikov equation and others. In addition, we shall prove integrability of some class of nonhomogeneous systems of hydrodynamic type and give a description of nonlinear partial differential equations of associativity in $2D$ topological field theories (for some special type solutions of the Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) system) as integrable nondiagonalizable weakly nonlinear homogeneous systems of hydrodynamic type.