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 \mathbf{A}, a gauge symmetry where \alpha is any arbitrary fixed 0-form and d is the exterior derivative, and a gauge-invariant vector current \mathbf{J} with density 1 satisfying the continuity equation where {\star} is the Hodge star operator. Alternatively, we may express \mathbf{J} as a closed (n − 1) -form, but we do not consider that case here. \mathbf{F} is a gauge-invariant 2-form defined as the exterior derivative. \mathbf{F} satisfies the equation of motion (this equation obviously implies the continuity equation). This can be derived from the action where M is the spacetime manifold.

p-form Abelian electrodynamics

We have a p -form \mathbf{B}, a gauge symmetry where \alpha is any arbitrary fixed (p − 1) -form and d is the exterior derivative, and a gauge-invariant [[p-vector| p -vector]] \mathbf{J} with density 1 satisfying the continuity equation where {\star} is the Hodge star operator. Alternatively, we may express \mathbf{J} as a closed (n − p) -form. \mathbf{C} is a gauge-invariant (p + 1) -form defined as the exterior derivative. \mathbf{B} satisfies the equation of motion (this equation obviously implies the continuity equation). 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 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 p -form electrodynamics. They typically require the use of gerbes.