Representation of fundamental group and flat connections

Question

I read Differential Geometry Of Complex Vector Bundles by Kobayashi, and he says there that a vector bundle $E$ has flat connection is equivalent to $E$ being defined by a representation of $\pi_1$. But he doesn't prove this. There are any suggestion how to start to prove this, or anyone has reference with a proof of this statement (I couldn't find any such reference).

Answer 1

Complete and elementary proofs of this fact may be found in detail in the following books:

1. Morita's "Geometry of Characteristic Classes" Section 2.1.4 pg. 55 or
2. Taubes' "Differential Geometry" Section 13.9.1 pg. 162.

Answer 2

The category of representations of the fundamental group $π_1$ can be replaced by the equivalent category of representations of the fundamental groupoid, i.e., functors $\def\Vect{{\sf Vect}} π_{≤1}(M)→\Vect$, where $\Vect$ is the category of complex vector spaces and $π_{≤1}(M)$ is the fundamental groupoid of $M$.

Now one can define a functor from the category of vector bundles to the category of functors $π_{≤1}(M)→\Vect$: send a vector bundle $E→M$ to the functor $π_{≤1}(M)→\Vect$ that maps $m∈M$ to the fiber $E_m∈\Vect$ and a homotopy class of paths $m→m'$ to the parallel transport map $E_m→E_{m'}$, which is well defined because the connection on $E$ is flat.

Observe that the functor defined above is natural in $M$ and defines a morphism of stacks of categories over the site of smooth manifolds.

To show that a morphism between two such stacks is a weak equivalence of stacks, it suffices to show that for $\def\R{{\bf R}} M=\R^n$ (respectively ${\bf C}^n$ if we are working over complex manifolds) we get an equivalence of categories.

Indeed, the category of complex vector bundles with a flat connection on $\R^n$ is equivalent to the category $\Vect$ of vector spaces via the functor that takes the fiber over $0∈\R^n$.

Likewise, the category of functors $π_{≤1}(\R^n)→\Vect$ is equivalent to the category $\Vect$ of vector spaces via the functor that evaluates at $0∈π_{≤1}(\R^n)$.

Answer 3

If you have a vector bundle $E$ with a flat connection $A$ (so $F_A = 0$), then fix a point $x\in M$, and consider the fiber $E_x$. For any homotopy of loops $\gamma_1,\gamma_2$ based at $x$, described by a map $H:I^2\to M$, we'll have something like $$P^A_{\gamma_2} - P^A_{\gamma_1} = \int_{H(I^2)}F_A = 0. $$ (See http://www.deaneyang.com/papers/holonomy2.pdf.) So the parallel transport only depends on the homotopy classes of loops, and defines a representation $$P^A:\pi_1 \to Aut(E_x) = G$$

Conversely if one has a representation $\rho$ of $\pi_1$, one can construct a vector bundle $E$ by taking the trivial vector bundle on the universal cover, and then quotienting by $\pi_1$ and its representation. $$E = \hat M \times_\rho \mathbb C^k = \hat M \times \mathbb C^k/\sim$$ where $$(x,v) \sim (y,\rho(\gamma).v)$$ where $\gamma$ is the loop in $\pi_1$ which lifts to a path connecting $x$ to $y$ in $\hat M$.

I think one gets that vector bundles with flat connections mod gauge is equivalent to $G$-representations of $\pi_1$ mod $G$-conjugation