# 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 ${\displaystyle \mathbf {A} }$, a gauge symmetry

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

where ${\displaystyle \alpha }$ is any arbitrary fixed 0-form and ${\displaystyle d}$ is the exterior derivative, and a gauge-invariant vector current ${\displaystyle \mathbf {J} }$ with density 1 satisfying the continuity equation

${\displaystyle d*\mathbf {J} =0}$

where * is the Hodge dual.

Alternatively, we may express ${\displaystyle \mathbf {J} }$ as a (${\displaystyle d-1}$)-closed form, but we do not consider that case here.

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

${\displaystyle \mathbf {F} }$ satisfies the equation of motion

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

(this equation obviously implies the continuity equation).

This can be derived from the action

${\displaystyle S=\int _{M}\left[{\frac {1}{2}}\mathbf {F} \wedge *\mathbf {F} -\mathbf {A} \wedge *\mathbf {J} \right]}$

where ${\displaystyle M}$ is the spacetime manifold.

## p-form Abelian electrodynamics

We have a p-form ${\displaystyle \mathbf {B} }$, a gauge symmetry

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

where ${\displaystyle \alpha }$ is any arbitrary fixed (p-1)-form and ${\displaystyle d}$ is the exterior derivative,

and a gauge-invariant p-vector ${\displaystyle \mathbf {J} }$ with density 1 satisfying the continuity equation

${\displaystyle d*\mathbf {J} =0}$

where * is the Hodge dual.

Alternatively, we may express ${\displaystyle \mathbf {J} }$ as a (d-p)-closed form.

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

${\displaystyle \mathbf {B} }$ satisfies the equation of motion

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

(this equation obviously implies the continuity equation).

This can be derived from the action

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

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

• Henneaux; Teitelboim (1986), "p-Form electrodynamics", Foundations of Physics 16 (7): 593-617, doi:10.1007/BF01889624
• Bunster, C.; Henneaux, M. (2011). "Action for twisted self-duality". Physical Review D. 83 (12). arXiv:1103.3621. Bibcode:2011PhRvD..83l5015B. doi:10.1103/PhysRevD.83.125015.
• Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", J. Math. Phys. 53, 102501 (2012) doi:10.1063/1.4754817