# Why discrete gauge fields must be flat?

I found in some papers, for example "Generalized Global Symmetries" and "Generalized Symmetries in Condensed Matter", that the gauge field of a discrete symmetry must be flat, i.e. $$dA=0$$. However, I do not understand this statement in a precise way.

I can give an intuitive idea of why this should be the case. Given that, discrete symmetries identify different points on the principal bundle, which are not related by a continuous deformation, the connection (i.e. the gauge field) does not carry any local information, such as the curvature, and therefore must be flat.

But, I am not satisfied with this argument, since it is a little imprecise. Can anyone give another explanation for why discrete gauge fields must be flat?

That's a nice question. The cleanest answer (that I know of) relies on a result of Milnor [1].

But first, let's rephrase things a bit. For a group $$G$$, let's define a flat $$G$$-bundle to be a principal $$G$$-bundle with a flat connection.

Your physics question translates to the following question in mathematics

If $$G$$ is a discrete group, why is every principal $$G$$-bundle flat?

Milnor's characterisation of flat bundles [1] states that

[Milnor's criterion] A principal $$G$$-bundle on $$X$$ is flat if and only if it is induced from the universal covering bundle of $$X$$ by a homomorphism $$\pi_1(X)\to G$$.

So now, let's show that $$G$$-bundles with discrete structure group satisfy Milnor's criterion.

Suppose you want to build a $$G$$-gauge theory over $$X$$, i.e. you want to construct a principal $$G$$-bundle over $$X$$. Such a bundle is classified by a map $$\newcommand{\B}{\mathrm{B}}$$ $$X\longrightarrow\B G,\tag{1}$$ where $$\B G$$ is the classifying space of $$G$$. In other words, there is a principal $$G$$-bundle $$\xi:\mathrm{E}G\to \B G$$$$(*)$$, such that all principal $$G$$-bundles on $$X$$ are pullbacks of $$\xi$$. A nice explanation of this is given in this math.SE answer.

Oh, but now we're in business! For a discrete group, $$\B G \cong K(G,1)$$ is an Eilenberg-MacLane space, so it holds that $$\pi_1(\B G)\cong G$$, where $$\pi_1$$ is the first homotopy group. Cool! So I can take $$\pi_1$$ of (1) and say that for a discrete group all principal $$G$$-bundles are classified by a map $$\pi_1(X)\longrightarrow G.\tag{2}$$ Therefore all principal $$G$$-bundles for a discrete group $$G$$ satisfy Milnor's criterion and are, therefore, flat.

In contrast, this wouldn't work for a continuous, say, group, because $$\B G$$ would not be an Eilenberg-MacLane space and it would not, generally, hold that $$\pi_1(G)\cong G$$.

To make some connection (pun not really intended) with physics, note that, for a discrete group the set of maps (2): $$\operatorname{Hom}(\pi_1(X),G)$$ is isomorphic to the first cohomology group $$\mathrm{H}^1(X;G)$$ (see e.g. this phys.SE answer of mine), therefore elements of $$\operatorname{Hom}(\pi_1(X),G)$$ define connection one-forms with zero curvature, i.e. flat connection one-forms, $$A\in\mathrm{H}^1(X;G)$$, which is how we usually think of gauge fields of discrete groups in physics.

Finally, there is another way to show the same result, by using the constancy of transition functions and tangency of the connection to an integrable manifold as explained in this MathOverflow answer.

References: [1] John Milnor, On the existence of a connection with curvature zero , dx.doi.org/10.5169/seals-25344

$$(*)$$$$\mathrm{E} G$$ is a weakly contractible space, defined in the Wikipedia article linked above

• Thank you for the answer Commented Nov 13, 2022 at 17:54
• Is that for any continuous group G, the principal G-bundle necessarily be non-flat? For example, the gauge field of the U(1) group is non-flat as well? Commented Apr 18 at 11:33

There is a simple physical reason. Recall that Wilson loops take values in (a given representation of) the gauge group, and in the continuum, the field strength of a gauge field can be defined from an infinitesimal Wilson loop. As a Wilson loop is shrunk to nothing (assuming it is defined on a contractible loop), it must continuously deform to the identity. But for a discrete gauge group, this is only possible if the Wilson loop is the identity for any contractible loop. Thus, the field strength must be locally flat.

With this reasoning, it is also easy to understand why non-flat gauge fields can exist in lattice gauge theories even when the gauge group is discrete. If spacetime is discretized, then a Wilson loop cannot be shrunk to nothing, so a field strength (not necessarily flat) can be defined on each plaquette of the lattice.