P-form electrodynamics

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

Ordinary (viz. one-form) Abelian electrodynamics[edit]

We have a one-form , a gauge symmetry

where is any arbitrary fixed 0-form and is the exterior derivative, and a gauge-invariant vector current with density 1 satisfying the continuity equation

where * is the Hodge dual.

Alternatively, we may express as a ()-closed form, but we do not consider that case here.

is a gauge invariant 2-form defined as the exterior derivative .

satisfies the equation of motion

(this equation obviously implies the continuity equation).

This can be derived from the action

where is the spacetime manifold.

p-form Abelian electrodynamics[edit]

We have a p-form , a gauge symmetry

where is any arbitrary fixed (p-1)-form and is the exterior derivative,

and a gauge-invariant p-vector with density 1 satisfying the continuity equation

where * is the Hodge dual.

Alternatively, we may express as a (d-p)-closed form.

is a gauge invariant (p+1)-form defined as the exterior derivative .

satisfies the equation of motion

(this equation obviously implies the continuity equation).

This can be derived from the action

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[edit]

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[edit]

  • 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