Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant

Question

This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the trivial SU(2) bundle over the 3-manifold M, modulo gauge transformation. In the stationary phase approximation as the level k-->infinity, we write this integral as the sum of functions f_i(k), where the sum is over the set {c_i} of critical points of the chern-simons functional (which is basically the log of the integrand). The critical points are the flat connections. In his analysis of the function f_tr(k) corresponding to the trivial flat connection (product connection), Rozansky (as well as previous authors, I believe) found the Casson invariant (as the second coefficient if you write f_i as a power series in 1/k, or something like that).

Now, at first glance it strikes me as odd that the casson invariant shows up here, because the casson invariant can, by definition, be thought of as a (signed) sum over ALL flat connections, EXCEPT the trivial connection. So my question is: what gives? Why does the contribution from the trivial connection give the casson invariant? Why isn't it instead, say, the sum of the 2nd coefficients of the contributions from all other flat connections? is there some explanation for this? Have contributions from other connections, say reducible connections for a homology 3-sphere, been analyzed? Does the casson invariant show up there as well?

If you've gotten this far, here's another question: right now, i can write |H_1(M)|*(Casson-Walker invariant of M), say for lens spaces M=L(p,q) as the sum over certain INTEGERS, one per conjugacy class of reducible flat connection on SU(2)XM, PLUS the signature of the 2-bridge knot b(p,q). I'm trying to identify this decomposition of the casson invariant with others in the literature, of which there are many, but I'm having the problem that none of these are (always) integral decompositions, even though they ADD UP to an integer. Well this is a long shot, but any ideas would be appreciated!

Answer 1

The Casson invariant is not the same sum or integral over connections that you would derive from the perturbative expansion Cherns-Simons quantum field theory at all flat connections. There is more than one way to rigorously interpret that expansion; one method uses Kontsevich's configuration space integrals. Dylan Thurston and I proved (in Perturbative 3-manifold invariants by cut-and-paste topology) that the configuration space invariant using only the flat connection is proportional to the Casson invariant, for any simple Lie group as the gauge group. (The configuration space integral is known as the theta invariant, because the Feynman-Jacobi diagram is a theta.) By contrast, Casson's invariant has been interpreted by Witten as a gauge theory with a certain Lie supergroup, whose underlying Lie group is SU(2).

So what gives? One answer is that the Casson invariant is the unique finite-type invariant of homology 3-spheres of degree 1, up to a scalar factor. (The Wikipedia link discusses finite-type invariants of knots, but as the page mentions briefly, there is an extension to 3-manifolds due to Garoufalidis, Habiro, Levine, and Ohtsuki.) As such, you should expect it to show up many times. The corresponding phenomenon at the level of link invariants is that all of the standard quantum link invariants that are polynomials in $q$ have the same second derivatives at $q=1$, again up to a scalar factor. In fact, Dylan and I didn't do anything directly with the Casson invariant; instead we showed that the theta invariant has finite-type degree 1.