Connectivity properties of moment maps on based loop groups

Abstract

For a compact, connected, simply-connected Lie group $G$, the loop group $LG$ is the infinite-dimensional Hilbert Lie group consisting of $H^1$–Sobolev maps $S^1\to G.$ The geometry of $LG$ and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops $\Omega \left(G\right)$ is an example of a homogeneous space of $LG$ and has a natural Hamiltonian $T\times S^1$ action, where $T$ is the maximal torus of $G$. We study the moment map $\mu$ for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian $T$–space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image $\mu \left(\Omega \left(G\right)\right)$ is convex. We also show that for the energy functional $E$, which is the moment map for the $S^1$ rotation action, each non-empty preimage is connected.