In theoretical physics, -form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
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 equation obviously implies the continuity equation).
This can be derived from the action
where is the spacetime manifold.
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 equation obviously implies the continuity equation).
This can be derived from the action
where is the spacetime manifold.
Other sign conventions do exist.
The KalbâÂÂRamond field is an example with in string theory; the RamondâÂÂRamond 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.
Just as we have non-abelian generalizations of electrodynamics, leading to YangâÂÂMills theories, we also have nonabelian generalizations of -form electrodynamics. They typically require the use of gerbes.