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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

arXiv: math.SG/0503684


For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H1–Sobolev maps S1 G. The geometry of LG 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 LG and has a natural Hamiltonian T × S1 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 S1 rotation action, each non-empty preimage is connected.

loop group, moment map, connectivity property
Mathematical Subject Classification 2000
Primary: 53D20
Secondary: 22E65
Forward citations
Received: 4 April 2005
Revised: 15 July 2005
Accepted: 6 September 2006
Published: 28 October 2006
Proposed: Ralph Cohen
Seconded: Haynes Miller, Frances Kirwan
Megumi Harada
Department of Mathematics and Statistics
McMaster University
1280 Main Street West
Ontario L8S 4K1
Tara S Holm
Department of Mathematics
589 Malott Hall
Cornell University
NY 14850-4201
Lisa C Jeffrey
Department of Mathematics
University of Toronto
Ontario M5S 2E4
Augustin-Liviu Mare
Department of Mathematics and Statistics
University of Regina
College West 307.14
Saskatchewan S4S 0A2