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

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

2007-02-26

arxiv.org/abs/math/0702792v4

Mathematics

Differential Geometry

15 pages; minor changes to the proof of Proposition 4.2; references added

Scientific paper

The space $\Omega(G)$ 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 $\mu : \Omega(G)\to (\t\times i\mathbb{R})^*$ be a moment map of the resulting $T\times S^1$ action. We show that all levels (that is, pre-images of points) of $\mu$ 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).

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