P-form electrodynamics

Ordinary (via. one-form) Abelian electrodynamics

We have a 1-form \mathbf{A}, a gauge symmetry :\mathbf{A} \rightarrow \mathbf{A} + d\alpha , 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 :d{\star}\mathbf{J} = 0 , 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} = d\mathbf{A}.

\mathbf{F} satisfies the equation of motion :d{\star}\mathbf{F} = {\star}\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 {\star}\mathbf{F} - \mathbf{A} \wedge {\star}\mathbf{J}\right] , where M is the spacetime manifold.

''p''-form Abelian electrodynamics

We have a p-form \mathbf{B}, a gauge symmetry :\mathbf{B} \rightarrow \mathbf{B} + d\mathbf{\alpha}, where \alpha is any arbitrary fixed (p − 1)-form and d is the exterior derivative, and a gauge-invariant p-vector \mathbf{J} with density 1 satisfying the continuity equation :d{\star}\mathbf{J} = 0 , 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{C} = d\mathbf{B}.

\mathbf{B} satisfies the equation of motion :d{\star}\mathbf{C} = {\star}\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 {\star}\mathbf{C} +(-1)^p \mathbf{B} \wedge {\star}\mathbf{J}\right] where M 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 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.

References

• Henneaux; Teitelboim (1986), "p-Form electrodynamics", Foundations of Physics 16 (7): 593-617,
• Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", J. Math. Phys. 53, 102501 (2012)