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, a gauge symmetry
where is any arbitrary fixed 0-form and is the exterior derivative, and a gauge-invariant vector current with density 1 satisfying the continuity equation
where * is the Hodge dual.
Alternatively, we may express as a -closed form, but we do not consider that case here.
is a gauge-invariant 2-form defined as the exterior derivative.
satisfies the equation of motion
.
This can be derived from the action
where is the spacetime manifold.

p-form Abelian electrodynamics

We have a p-form, a gauge symmetry
where is any arbitrary fixed -form and is the exterior derivative,
and a gauge-invariant p-vector with density 1 satisfying the continuity equation
where * is the Hodge dual.
Alternatively, we may express as a -closed form.
is a gauge-invariant -form defined as the exterior derivative.
satisfies the equation of motion
.
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.