stabilizer of gauge group

Question

Suppose $A$ is a flat connection on a fiber bundle $V$ over a manifold $M$, with fiber $G$. What is the stabilizer of the action of the gauge group on the space of all flat connections (i.e. $g(x)\cdot A=A$)? Can we conclude that $g(x)$ is a constant map?

Answer 1

The answer is given in lemma 6.1.4 of Rudolph & Schmidt's, "Differential Geometry and Mathematical Physics: Part II", and the theorem that follows. See there for details. It states the following:

Let $v_0\in V$, and let $A$ be a connection on $V$ and $g\in\mathcal{G}$ a gauge transformation. Then $g\cdot A=A$ if and only if the restriction of $g$ to the holonomy bundle $P_{p_0}(A)$ is constant.

They go on to prove the following:

The isotropy group $\mathcal{G}_A$ is isomorphic to the centraliser of the holonomy group in $G$.

Note that, for a flat connection, the holonomy group is necessarily discrete, by the Ambrose-Singer theorem. However, its centraliser is not necessarily all of $G$.