P-form Electrodynamics

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 $\mathbf$, a gauge symmetry
:$\mathbf \rightarrow \mathbf + d\alpha ,$
where $\alpha$ is any arbitrary fixed 0-form and $d$ is the exterior derivative, and a gauge-invariant vector current $\mathbf$ with density 1 satisfying the continuity equation
:$d\mathbf = 0 ,$
where $$ is the Hodge star operator.

Alternatively, we may express $\mathbf$ as a closed -form, but we do not consider that case here.

$\mathbf$ is a gauge-invariant 2-form defined as the exterior derivative $\mathbf = d\mathbf$.

$\mathbf$ satisfies the equation of motion
:$d\mathbf = \mathbf$
(this equation obviously implies the continuity equation).

This can be derived from the action
:S=\int_M \left frac\mathbf \wedge \mathbf - \mathbf \wedge \mathbf\right,
where $M$ is the spacetime manifold.

''p''-form Abelian electrodynamics

We have a -form $\mathbf$, a gauge symmetry
:$\mathbf \rightarrow \mathbf + d\mathbf,$
where $\alpha$ is any arbitrary fixed -form and $d$ is the exterior derivative, and a gauge-invariant -vector $\mathbf$ with density 1 satisfying the continuity equation
:$d\mathbf = 0 ,$
where $$ is the Hodge star operator.

Alternatively, we may express $\mathbf$ as a closed -form.

$\mathbf$ is a gauge-invariant -form defined as the exterior derivative $\mathbf = d\mathbf$.

$\mathbf$ satisfies the equation of motion
:$d\mathbf = \mathbf$
(this equation obviously implies the continuity equation).

This can be derived from the action
:S=\int_M \left frac\mathbf \wedge \mathbf +(-1)^p \mathbf \wedge \mathbf\right/math>
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)

{{DEFAULTSORT:P-Form Electrodynamics
Electrodynamics
String theory