How many distinct flat connections are there on a flat bundle?

Question

Given a flat smooth vector bundle (i.e. with constant transition functions), how many distinct flat connections could we put on it? If the flat connection is not unique, is it unique up to gauge equivalence (i.e. automorphism of bundles with connection)?

Answer 1

On any given vector bundle $E$ there is a space of flat connections on it modulo gauge, as you say. If the structure group of the bundle is $G$, it is an appropriate subset of the space $\text{Hom}(\pi_1 X, G)/\sim$, where here I'm modding out by conjugacy; each such representation gives rise to a vector bundle, so you restrict to the subset such that the vector bundle has the right isomorphism type.

There is no general way of saying what this is for an arbitrary vector bundle over an arbitrary manifold. Instead I'll stick with comments about specific cases.

1) Complex line bundles. $E$ supports a flat connection of and only if $c_1(E)$ is torsion. Suppose for convenience $X$ has a CW decomposition with a single 0-cell. Given the corresponding representation $f: \pi_1(X) \to U(1)$, you can extract $c_1(E)$ on the level of cellular cochains by, for each loop in the 1-skeleton, picking a lift $\tilde f(e)$ of $f(e)$ to $\Bbb R$; and if $a$ is a 2-cell, the boundary of $a$ can be written as some word $w$ in the $e$; define $c_1(a) = \tilde f(w)$. It is straightforward to verify that this defines an integer-valued cocycle, and slightly less straightforward to see it determines the same cohomology class as $c_1(E)$. So you can determine the space of flat connections on a complex line bundle completely at the level of representations.

2) If $D$ is a flat connection on a bundle $E$, the linearization of the flat connection equation at $D$ is $Da = 0$, $a \in \Omega^1(\mathfrak g_E)$. The linearization of modding out by gauge is modding out by the image of $D$, so we see that the dimension "should be" $\dim H^1(\mathfrak g_E,D)$; more precisely this is the Zariski tangent space of the moduli space of flat connections at $D$. (Note that if $E$ is a line bundle the induced connection on $\mathfrak g_E$ is trivial, and in that case this is just the same as the usual $b_1(X)$ in the complex case, an 0 in the real case.) In general, the euler characteristic $\chi(E)$ of the bundle is independent of the choice of flat connection on it. (I never worked this out by hand, but one side of the Atiyah-Singer index theorem should be $\chi(\mathfrak g_E)$, and the other should be $\chi(D)$, the Euler characteristic of the de Rham complex of $\mathfrak g_E$ with the operator induced by $D$.) This can be a helpful tool for understanding what $b_1(D)$ is. As an example, take the trivial $SU(2)$-bundle over a surface. In this case, $\chi(E) = 3\chi(\Sigma_g)$, and Poincare duality says we only need to know $b_0(D)$ and $b_1(D)$. $b_0(D)$ is the space of parallel sections of $E$ with respect to $D$; if the connection is reducible to a connection on a $U(1)$-bundle, $b_0(D) = 1$; if it's reducible to a connection on a real line bundle, $b_0(D) = 3$; and if it's irreducible, $b_0(D) = 0$. So $b_1(D)$ is completely determined in this case by whether or not the connection $D$ is reducible, and 'how reducible' it is.