TITLE: Yang-Mills and Chern-Simons functionals as Morse functions
QUESTION [13 upvotes]: Can the Yang-Mills or Chern-Simons action functionals be considered as [possibly perfect] Morse functions? I assume we would be in an equivariant scenario due to considering the configuration spaces with gauge-groups/transformations. Or at least how far away are they from a Morse-Bott function (and from being perfect)?

REPLY [11 votes]: There's a generalization of Morse-Bott called Morse-Bott-Kirwan 
that you can read about in Kirwan's book. Basically this condition
guarantees that the unstable sets are manifolds, but not the stable sets, 
so the negative of a function that's Morse-Bott-Kirwan may not be.
If one defines a "Yang-Mills functional" very generally to be the
norm-square of a moment map, Kirwan proves that for $M$ finite-dimensional,
this norm-square is a perfect Morse-Bott-Kirwan function. (Of course the
original example of such is on $M$ a space of connections, where
Atiyah-Bott did the same, as I recall.)

REPLY [8 votes]: This got too long for a comment.
Atiyah and Bott showed that the Yang-Mills functional on a Riemann surface is equivariantly perfect, i.e. it's perfect for gauge-equivariant (integral) cohomology.  To be a little more precise, they showed that a certain stratification (the Harder-Narasimhan stratification) of the space of connections is perfect in this sense, and Daskalopoulos showed (using Uhlenbeck compactness among other things) that this stratification does in fact agree with stable manifolds of the Yang-Mills functional.  (Atiyah-Bott had conjectured this, but did not prove it in their paper.  Note that Uhlenbeck's compactness theorem came just after Atiyah-Bott.)
For non-orientable surfaces, the situation is different: in some cases the YM functional is "anti-perfect" in a certain sense, and in some cases it's neither perfect nor anti-perfect.  These ideas are discussed in recent work of Melissa Liu and Nan-Kuo Ho, and also in recent work of Tom Baird.