Connectedness of levels for moment maps on various classes of loop groups

Abstract

The space $\Omega(G)$ of all based loops in a compact simply connected Lie group $G$ has an action of the maximal torus $T \subset G$ (by pointwise conjugation) and of the circle $S^{1}$ (by rotation of loops). Let $\mu\colon \Omega(G) \to (\mathfrak{t} \times i\mathbb{R})^{*}$ be a moment map of the resulting $T \times S^{1}$ action. We show that all levels (that is, pre-images of points) of $\mu$ are connected subspaces of $\Omega(G)$ (or empty). The result holds if in the definition of $\Omega(G)$ loops are of class $C^{\infty}$ or of any Sobolev class $H^{s}$, with $s \ge 1$ (for loops of class $H^{1}$ connectedness of regular levels has been proved by Harada, Holm, Jeffrey, and the author in [3]).