Connectivity and Convexity Properties of the Momentum Map for Group Actions on Hilbert Manifolds

Abstract

In the early $1980$s a landmark result was obtained by Atiyah and independently Guillemin and Sternberg: the image of the momentum map for a torus action on a compact symplectic manifold is a convex polyhedron. Atiyah's proof makes use of the fact that level sets of the momentum map are connected. These proofs work in the setting of finite-dimensional compact symplectic manifolds. One can ask how these results generalize. A well-known example of an infinite-dimensional symplectic manifold with a finite-dimensional torus action is the based loop group. Atiyah and Pressley proved convexity for this example, but not connectedness of level sets. A proof of connectedness of level sets for the based loop group was provided by Harada, Holm, Jeffrey and Mare in $2006$. In this thesis we study Hilbert manifolds equipped with a strong symplectic structure and a finite-dimensional group action preserving the strong symplectic structure. We prove connectedness of regular generic level sets of the momentum map. We use this to prove convexity of the image of the momentum map.