Hello, I am looking for an answer to the following question:

Given a gauge connection $A$ with curvature $F$, what is the space of gauge connections $\bar A$ such that $d_{\bar A} F = 0$? (In this notation $d_A F = 0$ is the Bianchi identity, so $F$ is not simply covariantly constant w.r.t. $\bar A$.)

More specifically, is it a discrete set? Or is it the same as the space of all connections modulo some discrete set? Modulo some continuous symmetry? None of these?

Many thanks in advance for any help or pointers.