# What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow of the Yang-Mills functional $L$ on the space $A(E)$ of (Sobolev) connections on $E$ is well-defined and, for any starting connection $A$, the flow line converges to a critical point (i.e. a Yang-Mills connection) that I'll call $A_\infty$. In particular, there are no finite time singularities (i.e. no bubbling) as there would be in 4 dimensions. This means that there is a well-defined "stratification" of $A(E)$ given by partitioning $A(E)$ according to the relation $A\sim B$ if $L(A_\infty) = L(B_\infty)$. If $t\in \mathbb{R}$ is a critical value of $L$, I'll write $C_t$ for the associated stratum. Rade showed that the gradient flow defines a deformation retraction from $C_t$ to its subset $L^{-1} (t)$ of Yang-Mills connections.

1. Are the subsets $C_t$ actually submanifolds? I believe this would follow from general principals if $L$ satisfied the Palais-Smale Condition C, but it does not. (However, $L$ does satisfy a version of Condition C after modding out gauge transformations; I'm not sure how helpful that is.)

2. If the $C_t$ are submanifolds, is anything known about their codimensions?

In 2-d Yang-Mills theory, Daskalopoulos provided detailed answers to both questions (building on ideas of Atiyah-Bott). But his arguments make heavy use of complex analytic methods, using the equivalence between Hermitian connections and complex structures in the 2-d case.

Surely one needs to be a little careful about the exact Sobolev regularity of the connections used, but I'm not terribly concerned about that (i.e. I'm happy to assume extra regularity if it helps anything).

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This probably doesn't help, but the critical levels themselves need not be manifolds, eg the flat moduli space is not typically a manifold. Presumably $C_0$ is open, though. –  Paul Jun 20 '12 at 3:00
Right, there's no reason to think the critical levels are smooth. But in the Riemann surface case, this somehow doesn't cause a problem. –  Dan Ramras Jun 20 '12 at 5:09
This reminds me of work by Graeme Segal & John Jones. The aim of this project was to construct a spectrum for Floer homology. –  Bruce Westbury Jun 20 '12 at 8:31