In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
We have a one-form A {\displaystyle \mathbf {A} } , a gauge symmetry
where α α --> {\displaystyle \alpha } is any arbitrary fixed 0-form and d {\displaystyle d} is the exterior derivative, and a gauge-invariant vector current J {\displaystyle \mathbf {J} } with density 1 satisfying the continuity equation
where ⋆ ⋆ --> {\displaystyle {\star }} is the Hodge star operator.
Alternatively, we may express J {\displaystyle \mathbf {J} } as a closed (n − 1)-form, but we do not consider that case here.
F {\displaystyle \mathbf {F} } is a gauge-invariant 2-form defined as the exterior derivative F = d A {\displaystyle \mathbf {F} =d\mathbf {A} } .
F {\displaystyle \mathbf {F} } satisfies the equation of motion
(this equation obviously implies the continuity equation).
This can be derived from the action
where M {\displaystyle M} is the spacetime manifold.
We have a p-form B {\displaystyle \mathbf {B} } , a gauge symmetry
where α α --> {\displaystyle \alpha } is any arbitrary fixed (p − 1)-form and d {\displaystyle d} is the exterior derivative, and a gauge-invariant p-vector J {\displaystyle \mathbf {J} } with density 1 satisfying the continuity equation
Alternatively, we may express J {\displaystyle \mathbf {J} } as a closed (n − p)-form.
C {\displaystyle \mathbf {C} } is a gauge-invariant (p + 1)-form defined as the exterior derivative C = d B {\displaystyle \mathbf {C} =d\mathbf {B} } .
B {\displaystyle \mathbf {B} } satisfies the equation of motion
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 eleven-dimensional supergravity or M-theory, we have a 3-form electrodynamics.
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.
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