# Which connections annihilate the curvature of a given connection?

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.

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This appears to be a pointwise linear algebra question to me, because $d_{\bar{A}}F = 0$ if and only if $(d_{\bar{A}} - d_A)F = 0$. Since the first order term is the same for both operators, you're left with a zero-th order equation, roughly $[\bar{A} - A, F] = 0$. – Deane Yang Apr 28 '11 at 14:20
That is correct, with a wedge product built in. I am looking for (roughly) the size of the space of allowed $\bar A$ (or $\bar A - A$). The answer ought to be straightforward, but I cannot figure out how to go about calculating it. I was hoping that $d_{\bar A} F = 0$ would have more easily calculable solutions. – Amitabha Lahiri Apr 29 '11 at 7:28