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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)?

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5 Answers 5

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The problem with the CS functional is that the Morse indices of its critical points are infinite. In particular, this functional cannot be perfect. The Floer complex does not compute the homology of any particular space (though it might compute the homology of a certain spectrum).

On 4-manifolds the YM functional has some analytic deficiencies: it lacks Palais-Smale condition. This lack of Palais-Smale manifests itself in the form of "bubbling" which is a nagging issue to be taken seriously when defining Donaldson invariants.

On 2-manifolds it was investigated thoroughly by Atiyah-Bott. In that paper they describe how equivariant cohomology can be used to establish some forms of perfectness.

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In particular, whether the YM functional os Palais-Smale or not is determined by whether boundedness of the functional, which is essentially a Sobolev norm, implies continuity of the connection. And the "bubbling" phenomenon occurs exactly at the so-called critical case, when you just lose continuity and Palais-Smale. This is also when the functional is invariant under scaling. It is this setting, where you just barely lose Palais-Smale, where there can be bubbling, i.e., loss of topology but at only a finite number of points. –  Deane Yang Apr 15 '12 at 21:46
It is a bit more subltle. For example on a $4$-manifold, both Nolinear equations $$\Delta u +u^3 =0 $$ and $$\Delta u =u^3 $$ involve critical Sobolev exponents. The bubbling occurs only in the second equation. What prevents bubbling from forming in the1st equation is a monotonicity feature. A similar monotonicity feature is responsible for the compactness of the moduli space in Seiberg-Witten theory. –  Liviu Nicolaescu Apr 15 '12 at 23:33

Let's consider CS functional for concreteness. The problem is that CS is neither Morse nor Morse-Bott (because its critical points are flat connections and character varieties of 3-manifold groups could be rather bad). The trick is to perturb CS to a Morse function. This was done first by Taubes ("Casson's invariant and gauge theory") and then developed into Floer theory.

How far is CS from being Morse-Bott? If you consider $SU(2)$ connections then for Seifert manifolds the character variety can have quadratic singularities, so it's not a manifold. There are examples of hyperbolic manifolds so that the $SU(2)$ character variety has cubic singularities. In fact, for $SO(3)$ flat connections over 3-manifolds the situation is much worse and you can have any singularity over ${\mathbb Z}$. I suspect the same happens even in $SU(2)$ case but it's harder to prove. So, perturbation to a Morse function is the only way to go.

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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.)

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Here is a non-technical answer to your question, which I bring up only to illustrate that your question is deep and well-studied. A good cartoon picture of the homology groups Floer assigned to 3-manifolds is that they are the "Morse homology" of Chern–Simons as a "function" (really, closed 1-form with integer periods) on the stack of principal bundles with connection over the manifold. Choosing a metric on the manifold picks out a metric on this stack, and for that metric the gradient flows are the anti-self-dual pure Yang–Mills instantons on the infinite cylinder over the manifold. This is, I think, well explained in Atiyah's paper 1988 paper "New invariants of 3- and 4-dimensional manifolds", given at the Hermann Weyl centennial conference.

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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.

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