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Abstract:
We first review the classical applications of Morse theory in the
context of Hamiltonian group actions. The examples which motivated
much of the theory were infinite-dimensional examples: The space of
loops on a compact Lie group, with the Morse function given by the
moment map of the natural circle action given by rotation, motivated
the main theorems in the case of abelian group actions, while the work
of Atiyah and Bott on the action of the gauge group on the space of
connections on a two-manifold motivated the later work of Kirwan on
Morse theory in the context of Hamiltonian actions of nonabelian
groups.
The formal structures arising in these contexts are paralleled in many
ways in the case of group actions on hyperkahler manifolds. Here
again some important examples occur in infinite dimensions: These are
the spaces of Higgs bundles in two dimensions, and the spaces of
connections on hyperkahler four-manifolds. However, in the
hyperkahler context, no analog of Morse theory is known to hold in
general, even in the finite dimensional case, and the results which
parallel the work of Morse, Bott, Atiyah-Bott, and Kirwan for
symplectic manifolds are only known in special cases. We review the
analogies between the symplectic and hyperkahler cases and the
differences between them, and present some recent work, done in
collaboration with Daskalopoulos and Wilkin, which we hope will shed
some light on this problem.
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