P-form electrodynamics
# 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 11-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.

