P-form electrodynamics explained

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

A , a gauge symmetryA → A + d\alpha , where \alpha is any arbitrary fixed 0-form and d is the exterior derivative, and a gauge-invariant vector current J with density 1 satisfying the continuity equationd{\star} J = 0 , where {\star} is the Hodge star operator.

Alternatively, we may express

J as a closed -form, but we do not consider that case here.F is a gauge-invariant 2-form defined as the exterior derivative F = d A .F satisfies the equation of motiond{\star} F = {\star} J (this equation obviously implies the continuity equation).

This can be derived from the action

S = \int _{M} \left[ (1)/(2) F \wedge {\star} F - A \wedge {\star} J \right] , where M is the spacetime manifold.

p-form Abelian electrodynamics

We have a -form

B , a gauge symmetryB → B + d \alpha , where \alpha is any arbitrary fixed -form and d is the exterior derivative, and a gauge-invariant -vector J with density 1 satisfying the continuity equationd{\star} J = 0 , where {\star} is the Hodge star operator.

Alternatively, we may express

J as a closed -form.C is a gauge-invariant -form defined as the exterior derivative C = d B .B satisfies the equation of motiond{\star} C = {\star} J (this equation obviously implies the continuity equation).

This can be derived from the action

S = \int _{M} \left[ (1)/(2) C \wedge {\star} C + ( -1 ) ^{p} B \wedge {\star} J \right] 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 eleven-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,
• Bunster . C. . Henneaux . M. . 10.1103/PhysRevD.83.125015 . Action for twisted self-duality . Physical Review D . 83 . 12 . 2011 . 125015 . 1103.3621 . 2011PhRvD..83l5015B . 119268081 .
• Navarro; Sancho (2012), "Energy and electromagnetism of a differential -form ", J. Math. Phys. 53, 102501 (2012)