P-form electrodynamics


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

Ordinary (via. one-form) Abelian electrodynamics

We have a 1-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 star operator.
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 -form, a gauge symmetry
where is any arbitrary fixed -form and is the exterior derivative, and a gauge-invariant -vector with density 1 satisfying the continuity equation
where is the Hodge star operator.
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 is the spacetime manifold.
Other sign conventions do exist.
The Kalb–Ramond field is an example with in string theory; the Ramond–Ramond fields whose charged sources are D-branes are examples for all values of. 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 -form electrodynamics. They typically require the use of gerbes.