In theoretical physics, -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 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.
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 equation obviously implies the continuity equation).
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 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 -form electrodynamics. They typically require the use of gerbes.
References
Henneaux; Teitelboim (1986), "-Form electrodynamics", Foundations of Physics 16 (7): 593-617,
Navarro; Sancho (2012), "Energy and electromagnetism of a differential -form ", J. Math. Phys. 53, 102501 (2012)