My question is about what we know, in dimension $4$, about the loss of compactness of Yang–Mills connections with $L^2$-bounded curvature. My background is more analytical than geometrical and it is hard for me to go through the modern literature on the subject, I apologize in advance if the miss references.
More precisely, I consider $P$ a $SU(2)$-bundle over $M$ a compact simply connected $4$-manifold and a sequence of Yang–Mills connection $\nabla_n$ such that $\Vert F_{\nabla_n}\Vert_2\leq M<+\infty$.
From Uhlenbeck's work we know that, if the sequence is not compact, it must concentrate at a finite number of points.
From Taubes's work, we know that such examples exist. In fact there is some condition to the existence work of Taubes. First he proves the existence of self-dual connections (which is obviously stronger than just Yang–Mills).
In his 1982 paper Self-dual Yang-Mills connections on non-self-dual 4-manifolds, in JDG, he proves that if the intersection form is definite he can glue an instanton everywhere on $M$.
In his 1984 paper Self-dual connections on 4-manifolds with indefinite intersection matrix, also in JDG, he proves that if $-c_2 \geq \max (4/3 b_-,1)$ if $b_-\not=2$ or $-c_2 \geq 4$ if $b_-=2$ then there is a self-dual connection. Here $c_2$ is the second Chern class of $P$ and $b_-=1/2(rank(Q)-sign(Q))$ where $Q$ is the intersection form.
The connections are obtained by perturbing some connections which concentrate $k$ instantons, but in the '84 case he needs to solves an obstruction. Which he did by some topological arguments. In particular we do not know where are the points of concentration.
My first question is for self-dual connection: do we expect that a sequence of connections should concentrate to a particular point (or configuration of points) of the based manifold (with respect to the metric)? Is there some refinement of Uhlenbeck's work in this direction when $Q$ is not definite?
My second question is for Yang–Mills connections: Taubes's hypothesis are optimal for certain configurations of $b_-$ and $c_2$. Can we expect to lower them if we relax the self-dual condition to Yang–Mills?
Any recent reference on the (analytic) development of the blow-up analysis of Yang–Mills in dimension $4$ will be welcome.
Thx in advance.
