P-form electrodynamics

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

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 $p = 2$ 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.

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P-form electrodynamics

p-form Abelian electrodynamics

Non-abelian generalization

References

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