P form electrodynamics

Ordinary (viz. one-form) Abelian electrodynamics

We have a one-form A , a gauge symmetry

where α is any arbitrary fixed 0-form and d is the exterior derivative, and a gauge-invariant vector current J with density 1 satisfying the continuity equation

where * is the Hodge dual.

Alternatively, we may express J as a ( d − 1 )-closed form, but we do not consider that case here.

F is a gauge invariant 2-form defined as the exterior derivative F = d A .

F satisfies the equation of motion

(this equation obviously implies the continuity equation).

This can be derived from the action

where M is the spacetime manifold.

p-form Abelian electrodynamics

We have a p-form B , a gauge symmetry

where α is any arbitrary fixed (p-1)-form and d is the exterior derivative,

and a gauge-invariant p-vector J with density 1 satisfying the continuity equation

where * is the Hodge dual.

Alternatively, we may express J as a (d-p)-closed form.

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

B satisfies the equation of motion

(this equation obviously implies the continuity equation).

This can be derived from the action

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 11d 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.