P-form electrodynamics

In theoretical physics, $p$-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.

Ordinary (via. one-form) Abelian electrodynamics
We have a one-form $$\mathbf{A}$$, a gauge symmetry
 * $$\mathbf{A} \rightarrow \mathbf{A} + d\alpha ,$$

where $$\alpha$$ is any arbitrary fixed 0-form and $$d$$ is the exterior derivative, and a gauge-invariant vector current $$\mathbf{J}$$ with density 1 satisfying the continuity equation
 * $$d{\star}\mathbf{J} = 0 ,$$

where $${\star}$$ is the Hodge star operator.

Alternatively, we may express $$\mathbf{J}$$ as a closed $(n − 1)$-form, but we do not consider that case here.

$$\mathbf{F}$$ is a gauge-invariant 2-form defined as the exterior derivative $$\mathbf{F} = d\mathbf{A}$$.

$$\mathbf{F}$$ satisfies the equation of motion
 * $$d{\star}\mathbf{F} = {\star}\mathbf{J}$$

(this equation obviously implies the continuity equation).

This can be derived from the action
 * $$S=\int_M \left[\frac{1}{2}\mathbf{F} \wedge {\star}\mathbf{F} - \mathbf{A} \wedge {\star}\mathbf{J}\right] ,$$

where $$M$$ is the spacetime manifold.

p-form Abelian electrodynamics
We have a $p$-form $$\mathbf{B}$$, a gauge symmetry
 * $$\mathbf{B} \rightarrow \mathbf{B} + d\mathbf{\alpha},$$

where $$\alpha$$ is any arbitrary fixed $(p − 1)$-form and $$d$$ is the exterior derivative, and a gauge-invariant $p$-vector $$\mathbf{J}$$ with density 1 satisfying the continuity equation
 * $$d{\star}\mathbf{J} = 0 ,$$

where $${\star}$$ is the Hodge star operator.

Alternatively, we may express $$\mathbf{J}$$ as a closed $(n − p)$-form.

$$\mathbf{C}$$ is a gauge-invariant $(p + 1)$-form defined as the exterior derivative $$\mathbf{C} = d\mathbf{B}$$.

$$\mathbf{B}$$ satisfies the equation of motion
 * $$d{\star}\mathbf{C} = {\star}\mathbf{J}$$

(this equation obviously implies the continuity equation).

This can be derived from the action
 * $$S=\int_M \left[\frac{1}{2}\mathbf{C} \wedge {\star}\mathbf{C} +(-1)^p \mathbf{B} \wedge {\star}\mathbf{J}\right]$$

where $M$ is the spacetime manifold.

Other sign conventions do exist.

The Kalb–Ramond field is an example with $p = 2$ in string theory; the Ramond–Ramond fields whose charged sources are D-branes are examples for all values of $p$. In eleven-dimensional supergravity or M-theory, we have a 3-form electrodynamics.

Non-abelian generalization
Just as we have non-abelian generalizations of electrodynamics, leading to Yang–Mills theories, we also have nonabelian generalizations of $p$-form electrodynamics. They typically require the use of gerbes.