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

1995/03/11 by O. I. Mokhov, Mokhov, Oleg · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.hep-th/9503076

openalex publication_date 1995/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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-Ampère 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.

Cited by

• On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension

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