Mokhov O.L. Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds, and Integrable Equations

Harwood Academic Publ, 2008. — 134 p. — (Reviews in Mathematics and Mathematical Physics, Volume 11, Part 2). — ISBN10: 190486872X.This survey paper is devoted to Hamiltonian geometry of integrable partial differential equations and the theory of infinite-dimensional symplectic and Poisson structures. Infinite-dimensional symplectic and Poisson geometry is closely connected with modern mathematical physics, field theory, and the theory of integrable systems of partial differential equations. In particular, structures studied in the present paper, namely, symplectic and Poisson structures of special differential-geometric type on loop spaces of manifolds, generate Hamiltonian representations for a number of important non-linear systems of mathematical physics and field theory, for example, such as systems of hydrodynamic type (these systems arise not only in Euler hydrodynamics and gas dynamics but also, in particular, if we apply the Whitham averaging procedure to the equations of the soliton theory, equations of associativity in topological field theory (these equations also play one of the key roles in the theory of Gromov-Witten invariants, which is being developed at present, the theory of quantum cohomology, and certain classic enumerative problems of algebraic geometry, non-linear sigmamodels, Heisenberg magnets, Monge-Ampère equations, and many others.Differential geometry of symplectic structures on loop spaces of smooth manifolds
Complexes of homogeneous forms on loop spaces of smooth manifolds and their cohomology groups
Local and non-local Poisson structures of differential-geometric type
Equations of associativity in two-dimensional topological field theory and nondiagonalizable integrable systems of hydrodynamic type