p-form electrodynamics

We have a 1-form $\backslash mathbf\{A\}$, a gauge symmetry

:$\backslash mathbf\{A\}\; \backslash rightarrow\; \backslash mathbf\{A\}\; +\; d\backslash alpha\; ,$

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

:$d\{\backslash star\}\backslash mathbf\{J\}\; =\; 0\; ,$

where $\{\backslash star\}$ is the Hodge star operator.

Alternatively, we may express $\backslash mathbf\{J\}$ as a closed {{math|(n − 1)}}-form, but we do not consider that case here.

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

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

:$d\{\backslash star\}\backslash mathbf\{F\}\; =\; \{\backslash star\}\backslash mathbf\{J\}$

(this equation obviously implies the continuity equation).

This can be derived from the action

:$S=\backslash int\_M\; \backslash left[\backslash frac\{1\}\{2\}\backslash mathbf\{F\}\; \backslash wedge\; \{\backslash star\}\backslash mathbf\{F\}\; -\; \backslash mathbf\{A\}\; \backslash wedge\; \{\backslash star\}\backslash mathbf\{J\}\backslash right]\; ,$

where $M$ is the spacetime manifold.

We have a {{math|p}}-form $\backslash mathbf\{B\}$, a gauge symmetry

:$\backslash mathbf\{B\}\; \backslash rightarrow\; \backslash mathbf\{B\}\; +\; d\backslash mathbf\{\backslash alpha\},$

where $\backslash alpha$ is any arbitrary fixed {{math|(p − 1)}}-form and $d$ is the exterior derivative, and a gauge-invariant p-vector $\backslash mathbf\{J\}$ with density 1 satisfying the continuity equation

:$d\{\backslash star\}\backslash mathbf\{J\}\; =\; 0\; ,$

where $\{\backslash star\}$ is the Hodge star operator.

Alternatively, we may express $\backslash mathbf\{J\}$ as a closed {{math|(n − p)}}-form.

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

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

:$d\{\backslash star\}\backslash mathbf\{C\}\; =\; \{\backslash star\}\backslash mathbf\{J\}$

(this equation obviously implies the continuity equation).

This can be derived from the action

:$S=\backslash int\_M\; \backslash left[\backslash frac\{1\}\{2\}\backslash mathbf\{C\}\; \backslash wedge\; \{\backslash star\}\backslash mathbf\{C\}\; +(-1)^p\; \backslash mathbf\{B\}\; \backslash wedge\; \{\backslash star\}\backslash mathbf\{J\}\backslash right]$

where {{math|M}} is the spacetime manifold.

Other sign conventions do exist.

The Kalb–Ramond field is an example with {{math|1=p = 2}} in string theory; the Ramond–Ramond fields whose charged sources are D-branes are examples for all values of {{math|p}}. In eleven-dimensional supergravity or M-theory, we have a 3-form electrodynamics.

Just as we have non-abelian generalizations of electrodynamics, leading to Yang–Mills theories, we also have nonabelian generalizations of {{math|p}}-form electrodynamics. They typically require the use of gerbes.

• Henneaux; Teitelboim (1986), "{{math|p}}-Form electrodynamics", Foundations of Physics 16 (7): 593-617, {{doi|10.1007/BF01889624}}
• {{Cite journal | last1 = Bunster | first1 = C. | last2 = Henneaux | first2 = M. | doi = 10.1103/PhysRevD.83.125015 | title = Action for twisted self-duality | journal = Physical Review D | volume = 83 | issue = 12 | year = 2011 | page = 125015 |arxiv = 1103.3621 |bibcode = 2011PhRvD..83l5015B | s2cid = 119268081 }}
• Navarro; Sancho (2012), "Energy and electromagnetism of a differential {{math|k}}-form ", J. Math. Phys. 53, 102501 (2012) {{doi|10.1063/1.4754817}}

{{String theory topics |state=collapsed}}

{{DEFAULTSORT:P-Form Electrodynamics}}

Category:Electrodynamics

Category:String theory