(University of California at Santa Cruz)
The geometry of Hamiltonian Loop Group Spaces
Abstract: We study Hamiltonian actions of the loop group LG, where G is a compact Lie group, on symplectic Banach manifolds. In the case where the moment map for the action is proper, we prove an infinite-dimensional analog of Kirwan's surjectivity theorem, which shows that the equivariant cohomology ring of the manifold gives a set of generators for the cohomology ring of the reduced space. Examples where this theorem apply arise from Yang-Mills theory in two dimensions as well as from coadjoint orbits of the loop group.
The method of proof involves the study of an appropriate generalization of Morse-Kirwan theory to infinite dimensions, and the construction of equivariantly perfect Morse-Kirwan functions on Hamiltonian LG-spaces. In the case of the smallest coadjoint orbit of the loop group, our function coincides with the energy function whose perfection plays a key role in the classical proof of Bott periodicity. In the case of Hamiltonian LG-spaces arising from Yang-Mills theory, the function we study is closely related to the Yang-Mills functional whose Morse-theoretic properties were first studied by Atiyah and Bott.
(Joint work with R. Bott and S. Tolman)