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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)?

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  • $\begingroup$ What have you thought about so far? Did you pick a discrete group and draw a picture? $\endgroup$
    – Will Sawin
    Nov 21, 2019 at 0:47
  • $\begingroup$ @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? $\endgroup$ Nov 21, 2019 at 5:13
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    $\begingroup$ 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 $U(1)$. $\endgroup$
    – user17945
    Nov 21, 2019 at 5:26
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    $\begingroup$ 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. $\endgroup$
    – Ben McKay
    Nov 21, 2019 at 10:34
  • $\begingroup$ 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? $\endgroup$
    – NDewolf
    Jul 28, 2021 at 8:56

2 Answers 2

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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".

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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.

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    $\begingroup$ Sorry for a question that's probably dumb, but how does being tangent to an integrable manifold imply flatness? $\endgroup$ Nov 21, 2019 at 23:20
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    $\begingroup$ Because flatness is equavalent to integrability. $\endgroup$ Nov 22, 2019 at 3:14

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