# Linking the de Rham bundle/complex over spacetime to the gauge bundle

In some textbooks, the Maxwell equations are stated in a very simple mathematical form (up to multiplicative constants coming from the system of units being used):

$$\begin{array} \mbox{d}F =0, \\ \delta F = j, \end{array}$$

where $$d$$ and $$\delta$$ are the exterior differential and co-differential on Minkowski spacetime, respectively. This form carries over to curved spacetime.

However, we know (see for example here) that F is not simply a two-form field in the cotangent bundle of the Minkowski spacetime, but is a section of a U(1) associated bundle also over Minkowski spacetime.

So why does $$F$$ actually carry Lorentz indices, which are typical for de Rham bundle/complex, and why is it therefore acted on by the $$d$$ and $$\delta$$?

The curvature form on a $$G$$-principal bundle $$P\to M$$ is a $$\mathfrak{g}$$-valued 2-form that is equivariant and horizontal, and hence descends to a well-defined global $$\mathfrak{g}$$-valued 2-form on $$M$$. Since it is a 2-form, it has 2 ordinary Lorentz indices.
To see this, pick any trivializations $$\phi_i : U_i\to M$$ and identity sections $$s_i : U_i \to U_i\times G\subset P, x\mapsto (x,1)$$ over them and try to glue the pullbacks $$s_i^\ast F$$ together to a form on $$M$$. You will see that equivariance and horizontality make it so that the gluing is consistent. For this reason equivariant horizontal forms are also called basic or tensorial forms.
• I am just noting here that the curvature form descends to local $\mathfrak g$ valued forms on each trivialization that are related by the images of the transition functions in the adjoint representation or it descends globally into an $\mathrm{Ad}(P)$ valued 2-form, but the Lie valued form is only global if the bundle is globally trivial. May 25, 2020 at 21:47
If it were a group $$G$$ other that $${\rm U}(1)$$, $$F$$ would carry other indices and have the form $$F= \frac 12 \hat \lambda_a F^a_{\mu \nu} \,dx^\mu \wedge dx^\nu$$ where $$\hat \lambda_a$$ are generators of the Lie algebra of the group $$G$$. As a result $$F$$ is Lie-algebra-valued two form, and is the curvature form of the of the principal $$G$$-bundle $$P$$. If one chooses a represesentation space $$V$$ for $$G$$, the abstract generators $$\hat \lambda_a$$ become matrices acting on $$V$$ and $$F$$ now becomes the curvature on the associated bundle $$V\times_G P$$.
Because $${\rm U}(1)$$ is a Abelian, and hence all its irrreps are one dimensional, the single matrix $$\lambda$$ is just a numbers and we usually don't bother to include an explicit $$\hat \lambda$$ in the formula for $$F$$. Instead people write $$qF$$ where $$q$$ is the charge of the particle. The integer valued charge $$q$$ labels the $${\rm U}(1)$$ representation $$e^{i\theta}\mapsto e^{iq\theta}$$.