Is there any work on Morse-Bott homology for infinite-dimensional manifolds (e.g. Hilbert manifolds). I am particularly interested in the case where we have a locally trivial fiber bundle and the Morse-Bott function is the function on the total space which is a pulled back Morse function on the base.

See Atiyah and Bott's papers on Yang-Mills for Riemann surfaces.

The essence of it is that the Yang-Mills functional acts like a
perfect Morse functional after quotienting by gauge transformations.
So, look at it before the quotient to get your Morse-Bott function.

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