Abstract
For a compact, connected, simply-connected Lie group , the loop group is the infinite-dimensional Hilbert Lie group consisting of –Sobolev maps The geometry of and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops is an example of a homogeneous space of and has a natural Hamiltonian action, where is the maximal torus of . We study the moment map 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 –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 is convex. We also show that for the energy functional , which is the moment map for the rotation action, each non-empty preimage is connected.
Citation
Megumi Harada. Tara S Holm. Lisa C Jeffrey. Augustin-Liviu Mare. "Connectivity properties of moment maps on based loop groups." Geom. Topol. 10 (3) 1607 - 1634, 2006. https://doi.org/10.2140/gt.2006.10.1607
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