Symplectic geometry of the loop space of a Riemannian manifold
type: Item
instance of: scholarly article
title: Symplectic geometry of the loop space of a Riemannian manifold
zbMATH Open document ID: 0826.58006
DOI: 10.1016/0393-0440(94)00033-Z
author: Tilmann Wurzbacher
published in: Journal of Geometry and Physics
publication date: 1995-08-03T00:00:00Z
review text: 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''.
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 58D19
zbMATH DE Number: 780907
zbMATH Keywords: infinite dimensional symplectic geometry
zbMATH Keywords: diffeomorphism action on loops
reviewed by: Dušan Repovš
MaRDI profile type: Publication
Wikidata QID: Q115363107
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description: scientific article; zbMATH DE number 780907