In terms of differential forms, in Minkowski spacetime, the Maxwell equations are

Here, \(F\), \(H\), and \(P\) are 2-forms, \(J\) is a 1-form, and \(\star\) is the Hodge star operation. In \(D\) spacetime dimensions, \(\star H\) is a \((D-2)\)-form, and \(\star J\) is a \((D-1)\)-form.

The first set of the Maxwell equations can be solved by introducing the potential \(A\), a 1-form:

If the Lorenz equation is satisfied,

then you can introduce a Hertz potential form \(B\) given by

Since \(A\) is a 1-form, \(\star A\) is a \((D-1)\)-form, and thus \(\star B\) is a \((D-2)\)-form. Then it follows that \(B\) is a 2-form in any number of spacetime dimensions.

Since \(\star J\) is conserved,

you also have a Lorenz equation for \(J\). You can introduce an electric Hertz current form \(Q\) given by

Since \(J\) is a 1-form, you have \(Q\) being a \((D-2)\)-form, and thus \(\star Q\) being a 2-form in any number of spacetime dimensions.

Let \(\circ\) be the inverse Hodge star operation (i.e. proportional to \(\star\)). Then it follows that

Thus

and

Finally, you get

Equivalently, you have

This equation is a second-order partial differential equation for \(B\) with \(Q\) and \(P\) as source terms.

## Magnetic Monopoles

In the presence of magnetic monopoles, the first set of the Maxwell equations take the form

Here \(\star K\) is a 3-form magnetic current. This means that \(K\) is a \((D-3)\)-form. Since you also have

you can introduce a magnetic Hertz current form \(R\) given by

Here \(\star R\) is a 2-form in any number of spacetime dimensions, and thus \(R\) is a \((D-2)\)-form.

Note that when \(\star K \neq 0\) you have \(F \neq \partial \wedge A\). Indeed, now you have

This result suggests that if you introduce a Hertz potential, then \(R\) will appear as a source for it in the second-order partial differential equation for the Hertz potential.

## Branes

A point-particle couples to a 1-form current. The point-particle is a 0-brane. A \(p\)-brane will couple to a \((p+1)\)-form current. The Maxwell equations are

If \(J\) is a \((p+1)\)-form, then \(\star J\) will be a \((D-p-1)\)-form, and \(\star F\) is a \((D-p-2)\)-form. Hence, \(F\) is a \((p+2)\)-form and \(\star K\) is a \((p+3)\)-form.

The Hertz current forms are introduced via

You have \(Q\) being a \((D-p-2)\)-form and thus \(\star Q\) being a \((p+2)\)-form. In a similar way, \(\star R\) is a \((p+2)\)-form and thus \(R\) is a \((D-p-2)\)-form.

The potential form \(A\) is introduced via

Since \(F\) is \((p+2)\)-form, \(A\) will be a \((p+1)\)-form. Thus, \(\star A\) will be a \((D-p-1)\)-form, and the Hertz potential form \(\star B\) will be a \((D-p-2)\)-form.

The point of this exercise was to show that \(\star B\), \(Q\) and \(R\) are algebraically compatible.