Connectivity properties of moment maps on based loop groups

Harada, Megumi¹ ; Holm, Tara S² ; Jeffrey, Lisa C³ ; Mare, Augustin-Liviu⁴

¹ Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada
² Department of Mathematics, 589 Malott Hall, Cornell University, Ithaca, NY 14850-4201, USA
³ Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
⁴ Department of Mathematics and Statistics, University of Regina, College West 307.14, Regina, Saskatchewan S4S 0A2, Canada

Geometry & topology, Tome 10 (2006) no. 3, pp. 1607-1634.

Résumé

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.

Détail

DOI : 10.2140/gt.2006.10.1607

Keywords: loop group, moment map, connectivity property

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