Connectivity properties of moment maps on based loop groups

Harada, Megumi; Holm, Tara S; Jeffrey, Lisa C; Mare, Augustin-Liviu

doi:10.2140/gt.2006.10.1607

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Connectivity properties of moment maps on based loop groups

Megumi Harada, Tara S Holm, Lisa C Jeffrey and Augustin-Liviu Mare

Geometry & Topology 10 (2006) 1607–1634

DOI: 10.2140/gt.2006.10.1607

arXiv: math.SG/0503684

Abstract

For a compact, connected, simply-connected Lie group $G$ , the loop group $L G$ is the infinite-dimensional Hilbert Lie group consisting of $H^{1}$ –Sobolev maps $S^{1} \to G .$ The geometry of $L G$ and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops $Ω (G)$ is an example of a homogeneous space of $L G$ and has a natural Hamiltonian $T \times S^{1}$ action, where $T$ is the maximal torus of $G$ . We study the moment map $μ$ 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 $μ (Ω (G))$ 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.

Keywords

loop group, moment map, connectivity property

Mathematical Subject Classification 2000

Primary: 53D20

Secondary: 22E65

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Publication

Received: 4 April 2005

Revised: 15 July 2005

Accepted: 6 September 2006

Published: 28 October 2006

Proposed: Ralph Cohen

Seconded: Haynes Miller, Frances Kirwan

Authors

Megumi Harada
Department of Mathematics and Statistics McMaster University 1280 Main Street West Hamilton Ontario L8S 4K1 Canada

Tara S Holm
Department of Mathematics 589 Malott Hall Cornell University Ithaca NY 14850-4201 USA

Lisa C Jeffrey
Department of Mathematics University of Toronto Toronto Ontario M5S 2E4 Canada

Augustin-Liviu Mare
Department of Mathematics and Statistics University of Regina College West 307.14 Regina Saskatchewan S4S 0A2 Canada

Volume 10, issue 3 (2006)