# Why does a principal G-bundle with a discrete structure group G have a unique flat connection?

I'm reading the Dijkgraaf–Witten paper Topological gauge theories and group cohomology (Comm. Math. Phys. 129 (1990) pp 393–429, doi:10.1007/BF02096988) and on page 395, 2nd paragraph they write

Suppose we choose a discrete group $$G$$. Every principal $$G$$-bundle has a unique, flat connection, and corresponds to a homomorphism $$\lambda : \pi_1(M) \rightarrow G$$

There are two parts to this question: why must such a bundle have a flat connection that's unique? And what exactly is the significance of the second part of the statement (the part about the homomorphism)?

• What have you thought about so far? Did you pick a discrete group and draw a picture? Nov 21, 2019 at 0:47
• @WillSawin I'm stuck at the difficulty arising from discreteness, let alone how to prove the statement itself! How does one even define a connection on a discrete G-bundle, if the underlying manifold is smooth? If a change in the position on the manifold results in a discrete change in g, then what about if I change the position by half as much? there's no notion of smoothness on the fiber. Perhaps the manifold being smooth is a faulty assumption? Nov 21, 2019 at 5:13
• Within any local trivialisation of the bundle, a curve in the base manifold will lift to a curve in the trivialisation with a constant $g$ value. However patching together local trivialisations along a closed curve in the base that is homotopically non-trivial can produce a non-trivial holonomy - think for example of the bundle $p:U(1)\to U(1)$ given by $z\mapsto z^2$, considered as a $\mathbb{Z}_2$-bundle over $U(1)$. Nov 21, 2019 at 5:26
• In physicists' notation, a connection expressed in local coordinates is represented by some field $A$, called a gauge field, valued in the Lie algebra. If the Lie algebra is zero, then $A=0$, unique. Nov 21, 2019 at 10:34
• So if I understand correctly, a general $G$-bundle can admit many nonequivalent flat connections, but by passing to a specific $G^\delta$-reduction (where $G^\delta$ is just $G$ with the discrete topology) a single flat connection is chosen? In other words the equivalence classes of $G^\delta$-reductions are in bijection with the equivalence classes of flat connections? Jul 28, 2021 at 8:56

Tsemo's answer addresses the first part of your question. To answer the second, for $$G$$ discrete, a $$G$$-bundle $$p:P\to M$$ is in particular a covering space for $$M$$, and so every path in $$M$$ has a unique lift (which happens to be the horizontal lift for the flat connection discussed in Tsemo's answer). So a closed path $$\gamma:[0,1]\to M$$ based at $$x\in M$$ lifts to the path $$\tilde\gamma:[0,1]\to P$$ (starting at some fixed $$q\in p^{-1}(x)$$). Since $$\tilde\gamma(0)$$ and $$\tilde\gamma(1)$$ lie in the same fibre, they differ by an element of $$G$$, so we can write $$\tilde\gamma(1) = \tilde\gamma(0)\cdot \Lambda(\gamma)$$. In fact, homotopies on $$M$$ also lift to $$P$$, and so $$\Lambda$$ factors through $$\pi_1(M)$$ to give a map $$\lambda:\pi_1(M)\to G$$. It is not too hard to show that $$\lambda$$ is a homomorphism. $$\lambda$$ essentially defines $$P$$ (for $$M$$ connected): if $$\tilde M$$ denotes the universal cover of $$M$$, then $$P \simeq \tilde{M}\times_\lambda G$$.
You can find a nice discussion of covering spaces in Hatcher's Algebraic Topology (particularly Section 1.3, Lifting Properties). The lifting arguments above apply more generally to principal bundles with flat connections (even if $$G$$ is not discrete): a discussion of flat bundles and holomorphy homorphisms is contained in Section 2.1.4 of Morita's "Geometry of Characteristic Classes".
Let $$p:P\rightarrow M$$ be a $$G$$-principal bundle, there exists a good covering $$(U_i)$$ which is a trivialization of $$p$$, the transition functions are defined by $$g_{ij}:U_i\cap U_j\rightarrow G$$, if $$G$$ is discrete, $$g_{ij}$$ is constant. On $$U_i\times G$$, we can define a distribution tangent to $$U_i\times\{g\}, g\in G$$ which gives rise to a connection on $$P$$, this connection is unique since by definition, a connection on $$U_i\times G$$ is a distribution transverse to the fibres of $$U_i\times G\rightarrow U_i$$, the dimension of this distribution is $$dimM$$ and for evry point of $$y$$ $$U_i\times G$$ there exists a unique vector subspace of $$U_i\times G$$ whose dimension is $$dimM$$ and it is $$T_y(U_i\times G)$$ since $$G$$ is discrete, this connection is flat since it is tagent to $$U_i\times\{g\}$$ which is integrable.