What does the de Rham complex have to do with physics?

You start with real-valued functions over \(\mathbb{R}^n\). What do you get when you apply a differential operator? Why, a vector field of course, the gradient field. But not all vector fields are gradients of a scalar field. Those that are are called exact.

What happens when you apply a differential operator to a vector field? In three dimensions, there are two common ways to do this: you can compute the divergence, or you can compute the curl. The generalization of the divergence is the interior derivative. The curl gets a little bit more complicated, because the method commonly used to compute it in three dimensions is unique to three dimensions. However, we can arrange the result in the form of a matrix:

\[\left[\begin{matrix}0&\frac{\partial v^x}{\partial y}-\frac{\partial v^y}{\partial x}&\frac{\partial v^x}{\partial z}-\frac{\partial v^z}{\partial x}\\\frac{\partial v^y}{\partial x}-\frac{\partial v^x}{\partial y}&0&\frac{\partial v^y}{\partial z}-\frac{\partial v^z}{\partial y}\\\frac{\partial v^z}{\partial x}-\frac{\partial v^x}{\partial z}&\frac{\partial v^z}{\partial y}-\frac{\partial v^y}{\partial z}&0\end{matrix}\right].\]

This, unlike the cross product, can be generalized to any dimension n. The matrix we get is a totally antisymmetric n×n matrix. However, not all fields of 2-dimensional totally antisymmetric matrices are the result of applying the exterior derivative (which is what this is called) to a vector field. Those that are are called, you guessed it: exact.

This process can be continued. Applying the exterior derivative to a matrix we get a field of n×n×n tensors. This is called a 3-form. The previous field of matrices was a 2-form, a vector field, a 1-form, whereas the scalar field that we started with is a 0-form.

This can go on beyond 3-forms... for a while. Beyond a tensor of rank n, the result will be identically zero.

The result is also identically zero when the exterior derivative is applied twice in a row. This is the generalization of a classic result from vector calculus: namely, that the curl of a gradient is always zero.

We call a form for which the exterior derivative is zero closed. The previous result can then be rephrased: all exact forms are closed.

The question then arises: are all closed forms exact? The answer, surprisingly, is in the negative: there are closed forms that are not exact.

Another theorem from vector calculus is Stokes' theorem, or its various cousins. The basic idea is that the integral of a field over a closed surface is identical to the integral of the divergence of the field over the volume that the surface encloses. Intuitively, if you think electric charges, the idea is that when you count the number of electric field lines that cross a closed surface with the arrow pointing inward, and subtract the number of lines that cross the surface with the arrow pointing outward, whatever remains is the number of electric charges that are inside the enclosed volume.

Stokes' theorem fails for manifolds on which closed forms exist that are not exact.

To me, the most profound implication is that in certain manifolds, you can have something that, for all practical intents and purposes, appears indistinguishable from an electric charge to a distant observer, yet there are no charges within, only topological anomalies of the manifold itself. If this idea could only be reconciled with what I know about quantum and/or relativistic physics, it may give rise to the possibility that what we perceive as matter is really pure geometry.

And what has this to do with the de Rham complex? Why, the series of forms starting with 0-forms (scalar fields), 1-forms (vectors), etc, collectively form what is called the de Rham complex. And, as it turns out, there are other complexes with similar properties, all of which are collectively called elliptic complexes.

Incidentally, the way the matrix above was formed is formally the same as applying the wedge product between the derivative operator and the vector v: ∇ /\ v. The wedge product is the generalization of the cross-product that is well known from three-dimensional vector algebra. You guessed it: it is a product that's formed by multiplying individual terms in all possible combinations, and applying alternating signs. The result of the wedge product of a p-form and a q-form is a p+q-form.