P-form electrodynamics

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

Ordinary (viz. one-form) Abelian electrodynamics

We have a one-form A, a gauge symmetry

\mathbf{A} \rightarrow \mathbf{A} + d\alpha

where α is any arbitrary fixed 0-form and d is the exterior derivative, and a gauge-invariant vector current J with density 1 satisfying the continuity equation

d*\mathbf{J}=0

where * is the Hodge dual.

Alternatively, we may express J as a (d − 1)-closed form.

F is a gauge invariant 2-form defined as the exterior derivative \mathbf{F}=d\mathbf{A}.

A satisfies the equation of motion

d*\mathbf{F}=*\mathbf{J}

(this equation obviously implies the continuity equation).

This can be derived from the action

S=\int_M \left[\frac{1}{2}\mathbf{F}\wedge *\mathbf{F} - \mathbf{A} \wedge *\mathbf{J}\right]

where M is the spacetime manifold.

p-form Abelian electrodynamics

We have a p-form B, a gauge symmetry

\mathbf{B} \rightarrow \mathbf{B} + d\mathbf{\alpha}

where α is any arbitrary fixed (p-1)-form and d is the exterior derivative,

and a gauge-invariant p-vector J with density 1 satisfying the continuity equation

d*\mathbf{J}=0

where * is the Hodge dual.

Alternatively, we may express J as a (d-p)-closed form.

C is a gauge invariant (p+1)-form defined as the exterior derivative \mathbf{C}=d\mathbf{B}.

B satisfies the equation of motion

d*\mathbf{C}=*\mathbf{J}

(this equation obviously implies the continuity equation).

This can be derived from the action

S=\int_M \left[\frac{1}{2}\mathbf{C}\wedge *\mathbf{C} +(-1)^p \mathbf{B} \wedge *\mathbf{J}\right]

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 11d 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.

References