P-form electrodynamics

Short description: Generalization of electrodynamics

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

## Contents

* 1 Ordinary (via. one-form) Abelian electrodynamics
* 2 p-form Abelian electrodynamics
* 3 Non-abelian generalization
* 4 References

## Ordinary (via. one-form) Abelian electrodynamics

We have a one-form 𝐀, a gauge symmetry

𝐀->𝐀+dα,

where α 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 equation

d⋆𝐉=0,

where ⋆ is the Hodge star operator.

Alternatively, we may express 𝐉 as a closed (n − 1)-form, but we do not consider that case here.

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

𝐅 satisfies the equation of motion

d⋆𝐅=⋆𝐉

(this equation obviously implies the continuity equation).

This can be derived from the action

S=∫M[12𝐅∧⋆𝐅−𝐀∧⋆𝐉],

where M is the spacetime manifold.

## p-form Abelian electrodynamics

We have a p-form 𝐁, a gauge symmetry

𝐁->𝐁+dα,

where α is any arbitrary fixed (p − 1)-form and d is the exterior derivative, and a gauge-invariant p-vector 𝐉 with density 1 satisfying the continuity equation

d⋆𝐉=0,

where ⋆ is the Hodge star operator.

Alternatively, we may express 𝐉 as a closed (n − p)-form.

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

𝐁 satisfies the equation of motion

d⋆𝐂=⋆𝐉

(this equation obviously implies the continuity equation).

This can be derived from the action

S=∫M[12𝐂∧⋆𝐂+(−1)p𝐁∧⋆𝐉]

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.

## 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): 125015. doi:10.1103/PhysRevD.83.125015. Bibcode: 2011PhRvD..83l5015B.
* Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", J. Math. Phys. 53, 102501 (2012) doi:10.1063/1.4754817

* v
* t
* e

String theory

Background|

* Strings
* History of string theory
* First superstring revolution
* Second superstring revolution
* String theory landscape

Theory|

* Nambu–Goto action
* Polyakov action
* Bosonic string theory
* Superstring theory
* Type I string
* Type II string
* Type IIA string
* Type IIB string
* Heterotic string
* N=2 superstring
* F-theory
* String field theory
* Matrix string theory
* Non-critical string theory
* Non-linear sigma model
* Tachyon condensation
* RNS formalism
* GS formalism

String duality|

* T-duality
* S-duality
* U-duality
* Montonen–Olive duality

Particles and fields|

* Graviton
* Dilaton
* Tachyon
* Ramond–Ramond field
* Kalb–Ramond field
* Magnetic monopole
* Dual graviton
* Dual photon

Branes|

* D-brane
* NS5-brane
* M2-brane
* M5-brane
* S-brane
* Black brane
* Black holes
* Black string
* Brane cosmology
* Quiver diagram
* Hanany–Witten transition

Conformal field theory|

* Virasoro algebra
* Mirror symmetry
* Conformal anomaly
* Conformal algebra
* Superconformal algebra
* Vertex operator algebra
* Loop algebra
* Kac–Moody algebra
* Wess–Zumino–Witten model

Gauge theory|

* Anomalies
* Instantons
* Chern–Simons form
* Bogomol'nyi–Prasad–Sommerfield bound
* Exceptional Lie groups (G2, F4, E6, E7, E8)
* ADE classification
* Dirac string
* p-form electrodynamics

Geometry|

* Kaluza–Klein theory
* Compactification
* Why 10 dimensions?
* Kähler manifold
* Ricci-flat manifold
* Calabi–Yau manifold
* Hyperkähler manifold
* K3 surface
* G2 manifold
* Spin(7)-manifold
* Generalized complex manifold
* Orbifold
* Conifold
* Orientifold
* Moduli space
* Hořava–Witten domain wall
* K-theory
* Twisted K-theory

Supersymmetry|

* Supergravity
* Superspace
* Lie superalgebra
* Lie supergroup

Holography|

* Holographic principle
* AdS/CFT correspondence

M-theory|

* Matrix theory
* Introduction to M-theory

String theorists|

* Aganagić
* Arkani-Hamed
* Atiyah
* Banks
* Berenstein
* Bousso
* Cleaver
* Curtright
* Dijkgraaf
* Distler
* Douglas
* Duff
* Ferrara
* Fischler
* Friedan
* Gates
* Gliozzi
* Gopakumar
* Green
* Greene
* Gross
* Gubser
* Gukov
* Guth
* Hanson
* Harvey
* Hořava
* Gibbons
* Kachru
* Kaku
* Kallosh
* Kaluza
* Kapustin
* Klebanov
* Knizhnik
* Kontsevich
* Klein
* Linde
* Maldacena
* Mandelstam
* Marolf
* Martinec
* Minwalla
* Moore
* Motl
* Mukhi
* Myers
* Nanopoulos
* Naqvi
* Năstase
* Nekrasov
* Neveu
* Nielsen
* van Nieuwenhuizen
* Novikov
* Olive
* Ooguri
* Ovrut
* Polchinski
* Polyakov
* Rajaraman
* Ramond
* Randall
* Randjbar-Daemi
* Roček
* Rohm
* Scherk
* Schwarz
* Seiberg
* Sen
* Shenker
* Siegel
* Silverstein
* Sơn
* Staudacher
* Steinhardt
* Strominger
* Sundrum
* Susskind
* 't Hooft
* Townsend
* Trivedi
* Turok
* Vafa
* Veneziano
* Verlinde
* Verlinde
* Wess
* Witten
* Yau
* Yoneya
* Zamolodchikov
* Zamolodchikov
* Zaslow
* Zumino
* Zwiebach

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