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

Abstract

The space $Ω(G)$ of all based loops in a compact semisimple 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 $μ: Ω(G)\to (\t\times i\mathbb{R})^*$ be a moment map of the resulting $T\times S^1$ action. We show that all levels (that is, pre-images of points) of $μ$ are connected subspaces of $Ω(G)$ (or empty). The result holds if in the definition of $Ω(G)$ 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).