What is known about the Yang-Mills stratification over 3-manifolds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T06:40:55Z http://mathoverflow.net/feeds/question/100029 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100029/what-is-known-about-the-yang-mills-stratification-over-3-manifolds What is known about the Yang-Mills stratification over 3-manifolds? Dan Ramras 2012-06-19T18:41:38Z 2012-06-19T18:41:38Z <p>Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if <code>$E\rightarrow M$</code> is a <code>$U(n)$</code>-bundle over a 3-manifold, then the gradient flow of the Yang-Mills functional <code>$L$</code> on the space <code>$A(E)$</code> of (Sobolev) connections on <code>$E$</code> is well-defined and, for any starting connection <code>$A$</code>, the flow line converges to a critical point (i.e. a Yang-Mills connection) that I'll call <code>$A_\infty$</code>. 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 <code>$A(E)$</code> given by partitioning $A(E)$ according to the relation <code>$A\sim B$</code> if <code>$L(A_\infty) = L(B_\infty)$</code>. If $t\in \mathbb{R}$ is a critical value of <code>$L$</code>, 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.</p> <p>I'm curious if anything at all is known about this stratification. For instance:</p> <ol> <li><p>Are the subsets <code>$C_t$</code> actually submanifolds? I believe this would follow from general principals if <code>$L$</code> satisfied the Palais-Smale Condition C, but it does not. (However, <code>$L$</code> does satisfy a version of Condition C after modding out gauge transformations; I'm not sure how helpful that is.)</p></li> <li><p>If the <code>$C_t$</code> are submanifolds, is anything known about their codimensions?</p></li> </ol> <p>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. </p> <p>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).</p>