Symplectic geometry of the loop space of a Riemannian manifold
This is a rigorous development of the classical symplectic geometry of the \(\text{Diff} (S^1)\)-action on loop spaces, an important example of infinite-dimensional geometry. The \(\text{Diff} (S^1)\)-action is shown to always come with a smooth ``moment'' map \(\Phi\) which has most of the properties possessed by the classical moment maps in the finite- dimensional setting, in spite of the fact that the (weakly) symplectic form has degeneracy and the action is neither differentiable (in any strong sense) nor proper. It is demonstrated that e.g. the kernel of the differential \(D \Phi\) is the skew-complement to the orbit and the image of \(D \Phi\) in a point \(\gamma\) is the annihilator of the isotropy algebra of \(\gamma\). A simple geometric description is found of the generic orbit structure: the \(\Phi\)-fibers and the orbits are shown to intersect in a circle and their tangent spaces add up, together with a 1- dimensional slice, to the whole tangent space. This allows for a description of the reduced spaces, i.e. the ``symplectic quotient of the loop space by the action''.
infinite dimensional symplectic geometry
scientific article; zbMATH DE number 780907
scientific article