title: P-form electrodynamics
text: # P-form electrodynamics
  
  In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
  
  ## Ordinary (via. one-form) Abelian electrodynamics
  
  We have a one-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 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 p-form electrodynamics. They typically require the use of gerbes.
  
  