D=4 supergravity

[[!redirects 4-dimensional supergravity]]

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## Idea

[[supergravity]] in [[dimension]] 4.

The maximally supersymmetric $N = 8$-version arises from [[type II supergravity]] in 10 dimension by [[Kaluza-Klein mechanism|compactification]] on a 6-[[torus]].

The $N=1$-version arises from [[KK-reduction]] via [[M-theory on G2-manifolds]]. This hosts the [[super 2-brane in 4d]].

## Related entries

* [[string theory results applied elsewhere]]

  * [[KLT relations]]

* [[11-dimensional supergravity]]

* 10-dimensional [[type II supergravity]], [[heterotic supergravity]]

* [[7-dimensional supergravity]]

* [[5-dimensional supergravity]]

* **4-dimensional supergravity**

  * [[M-theory on G2-manifolds]], [[G2-MSSM]]

  * [[Freund-Rubin compactification]]

* [[3-dimensional supergravity]]

## References

Minmal 4d Supergravity was the first [[supergravity]] theory to be constructed, in

* {#FreedmanNieuwenhuizenFerrara76} [[Daniel Freedman]], [[Peter van Nieuwenhuizen]], [[Sergio Ferrara]], _Progress toward a theory of supergravity_, Phys. Rev. D13 (1976) 3214 ([doi.org/10.1103/PhysRevD.13.3214](https://doi.org/10.1103/PhysRevD.13.3214))

See also at _[supergravity -- History](supergravity#History)_.

### Geometric construction

Discussion in the [[D'Auria-Fré formulation of supergravity]] includes

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter III.3.5 and III.4 and V.4 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Riccardo D'Auria]], Sergio Ferrara, Mario Trigiante, _Supersymmetric completion of M-theory 4D-gauge algebra from twisted tori and fluxes_,   JHEP0601:081, 2006 ([arXiv:hep-th/0511158](https://arxiv.org/abs/hep-th/0511158))

The role of 2-form fields (tensor multiplets, via the [[4d supergravity Lie 2-algebra]] incarnated via its dual [[Chevalley-Eilenberg algebras]], "FDA"s) is discussed in

* [[José de Azcárraga]], J. M. Izquierdo, _Minimal $D=4$ supergravity from the superMaxwell algebra_, Nucl. Phys. B 885, 34-45 (2014) ([arXiv:1403.4128](https://arxiv.org/abs/1403.4128))

* {#AndrianopoliDAuriaSommovigo07} [[Laura Andrianopoli]], [[Riccardo D'Auria]], Luca Sommovigo, _$D=4$, $N=2$ Supergravity in the Presence of Vector-Tensor Multiplets and the Role of higher $p$-forms in the Framework of Free Differential Algebras_ ([arXiv:0710.3107](http://arxiv.org/abs/0710.3107))

* {#AAST11} [[Laura Andrianopoli]], [[Riccardo D'Auria]], Luca Sommovigo, Mario Trigiante, _$D=4$, $N=2$ Gauged Supergravity coupled to Vector-Tensor Multiplets_, Nucl.Phys.B851:1-29,2011 ([arXiv:1103.4813](http://arxiv.org/abs/1103.4813))

based on

* [[Murat Gunaydin]], S. McReynolds, M. Zagermann, _Unified $N=2$ Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions_, JHEP 0509:026,2005 ([arXiv:hep-th/0507227](https://arxiv.org/abs/hep-th/0507227))

Discussion of the splitting-decomposition analogous to that for the [[M-theory super Lie algebra]]

* D. M. Peñafiel, [[Lucrezia Ravera]], _On the Hidden Maxwell Superalgebra underlying D=4 Supergravity_, Fortschr. Phys. 65 (2017) no. 9, 1700005 ([arXiv:1701.04234](https://arxiv.org/abs/1701.04234))

* [[Lucrezia Ravera]], _Hidden Role of Maxwell Superalgebras in the Free Differential Algebras of D=4 and D=11 Supergravity_ ([arXiv:1801.08860](https://arxiv.org/abs/1801.08860))

See also

* Salih Kibaroğlu, Oktay Cebecioğlu, _$D=4$ supergravity from the Maxwell-Weyl superalgebra_ ([arXiv:1812.09861](https://arxiv.org/abs/1812.09861))

### On $N=1$ $d = 4$ supergravity

There are two different off-shell formulations, the "old minimal"

* [[Kellogg Stelle]] and [[Pete West]], Phys. Lett. 74B (1978) 330;

* S. Ferrara and [[Peter van Nieuwenhuizen]], Phys. Lett. 74B (1978) 333

and the "new minimal" supergravity

* V. Akulov, [[Dmitry Volkov]] and V. Soroka, _Generally covariant theories of gauge fields on superspace_, Theor. Math. Phys. 31 (1977) 285 ([doi:10.1007/BF01041233](https://doi.org/10.1007/BF01041233))

* M.F. Sohnius and P.C. West, idem. Phys. Lett. 105B (1981) 353; idem. Nucl. Phys. B198 (1982) 493.

* M.F. Sohnius and P.C. West, `The New Minimal Formulation of N = 1 Supergravity and its Tensor Calculus', Nueld Workshop, 1981:0187 (London, England, Aug. 1981).

* [[Jim Gates]], [[Martin Rocek]] and [[Warren Siegel]], Nucl. Phys. B198 (1982) 113

These two versions were later understood to be two different [[gauge fixings]] of N=1 d=4 coformal supergravity. Yet other gauge fixings are discussed in

* [[Jim Gates]], Jr., Hitoshi Nishino, _Will the Real 4D, $N=1$ SG Limit of Superstring/M-Theory Please Stand Up?_, Phys.Lett.B492:178-186,2000 ([arXiv:hep-th/0008206](http://arxiv.org/abs/hep-th/0008206))

See also

* Nicolas Boulanger, Mboyo Esole, _A Note on the uniqueness of $D = 4$, $N=1$ supergravity_, Class.Quant.Grav. 19 (2002) 2107-2124 ([arXiv:gr-qc/0110072](http://arxiv.org/abs/gr-qc/0110072))

Textbook accounts:

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 2, chapter III.2, III.3.5, III.3.10 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

Exposition:

* Robin Ducrocq, Michel Rausch de Traubenberg, Mauricio Valenzuela, *A pedagogical discussion of $N = 1$ four-dimensional supergravity in superspace* ([arXiv:2104.06671](https://arxiv.org/abs/2104.06671))

Survey of the formulation in [[superspace]]:

* S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, *Superspace approaches to $\mathcal{N} = 1$ supergravity*, *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.17088](https://arxiv.org/abs/2210.17088)]

See also:

* Wikipedia, _[Supergravity -- 4d N=1](http://en.wikipedia.org/wiki/Supergravity#4D_N_.3D_1_SUGRA)_

### On $N = 2$, $d = 4$ supergravity

More on the case [[number of supersymmetries|N=2]]:

* [[Laura Andrianopoli]], M. Bertolini, A. Ceresole, [[Riccardo D'Auria]], [[Sergio Ferrara]], [[Pietro Fré]], T. Magri, _$N = 2$ supergravity and $N = 2$ super Yang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings and the momentum map_. J. Geom. Phys. 23, 111–189, 1997 ([arXiv:hep-th/9605032](https://arxiv.org/abs/hep-th/9605032))

* Edoardo Lauria, [[Antoine Van Proeyen]], _$\mathcal{N}=2$ Supergravity in $D=4,5,6$ Dimensions_ ([arXiv:2004.11433](https://arxiv.org/abs/2004.11433))

Review of the [[superspace]] formulations:

* S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, *Covariant superspace approaches to $\mathcal{N}=2$ supergravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.11162](https://arxiv.org/abs/2211.11162)]

### On $N=8$ $d=4$ supergravity

#### Construction

The maximal $N=8$ supergravity in 4d was obtained by [[KK-reduction]] of [[11-dimensional supergravity]] on a [[torus|7-torus]] in

* {#CremmerJulia79} [[Eugene Cremmer]], [[Bernard Julia]], _The $SO(8)$ Supergravity_, Nucl. Phys. B 159 (1979) 141 ([spire](http://inspirehep.net/record/140465?ln=en))

* [[Eugene Cremmer]], [[Bernard Julia]], Phys. Lett. 80B (1978) 48; Nucl. Phys. B159 (1979) 141.

Its $SO(8)$-[[gauged supergravity|gauged]] version was obtained in

* [[Bernard de Wit]], [[Hermann Nicolai]], _$N=8$ supergravity with local $SO(8)\times SU(8)$ invariance_, Phys. Lett. 108 B (1982) 285 (doi:10.1016/0370-2693(82)91194-7)

* [[Bernard de Wit]]. [[Hermann Nicolai]], _$N = 8$ supergravity_,  Nucl. Phys. B208 (1982) 323 (doi:10.1016/0550-3213(82)90120-1)

and further gaugings by non-compact gauge groups in

* [[Chris Hull]], Phys. Rev. D30 (1984) 760;

* [[Chris Hull]], Phys. Lett. 142B (1984)

* [[Chris Hull]], Phys. Lett. 148B (1984) 297;

* [[Chris Hull]], Physica 15D (1985) 230; Nucl. Phys. B253 (1985) 650.

* [[Chris Hull]], Class. Quant. Grav. 2 (1985) 343.

* [[Chris Hull]], _New Gauged $N=8$, $D=4$ Supergravities_, _Class.Quant.Grav.20:5407-5424,2003_ ([arXiv:hep-th/0204156](https://arxiv.org/abs/hep-th/0204156))

#### $N = 8$ Perturbative quantum supergravity
 {#PerturbativeQuantumSupergravityN8}

For early results on [[loop order|2-loop]] finiteness of [[perturbative quantum gravity|perturbative quantum supergravity]] see [there](quantum+gravity#ReferencesNonRenormalizability).

Evidence for high [[loop order]] finiteness of $N=8$ 4d supergravity as as [[perturbative quantum field theory]] ([[perturbative quantum gravity]]) is discussed in

* [[Zvi Bern]], [[Lance Dixon]], [[Radu Roiban]], _Is $N = 8$ Supergravity Ultraviolet Finite?_, Phys.Lett.B644:265-271,2007 ([arXiv:hep-th/0611086](http://arxiv.org/abs/hep-th/0611086))

* [[Zvi Bern]], J. J. Carrasco, [[Lance Dixon]], H. Johansson, [[David Kosower]], [[Radu Roiban]], _Three-Loop Superfiniteness of N=8 Supergravity_, Phys.Rev.Lett.98:161303,2007 ([arXiv:hep-th/0702112](http://arxiv.org/abs/hep-th/0702112))

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Lance Dixon]], [[Henrik Johansson]], [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_, Fortschr. Phys. **59** (2011) 7-8 [[arXiv:1103.1848](http://arxiv.org/abs/1103.1848), [doi:10.1002/prop.201100037](https://doi.org/10.1002/prop.201100037)]

and via [[KLT relations]]:

* {#BCDJR11} [[Zvi Bern]], John Joseph Carrasco, [[Lance Dixon]], Henrik Johansson, [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_ ([arXiv:1103.1848](http://arxiv.org/abs/1103.1848))

surveyed in

* [[Radu Roiban]], _Is Perturbative $\mathcal{N}= 8$ Supergravity Finite?_ ([arXiv:hep-th/0702112](http://arxiv.org/abs/hep-th/0702112))

* [[Lance Dixon]], _Ultraviolet Behavior of $N=8$ Supergravity_ ([arXiv:1005.2703](http://arxiv.org/abs/1005.2703))

* {#FerraraMarrani12} [[Sergio Ferrara]], Alessio Marrani, _Quantum Gravity Needs Supersymmetry_ ([arXiv:1201.4328](http://arxiv.org/abs/1201.4328))

* {#Kallosh14} [[Renata Kallosh]], _An Update on Perturbative $N=8$ Supergravity_ ([arXiv:1412.7117](https://arxiv.org/abs/1412.7117))

Arguments for finiteness from [[E7]] [[U-duality]] is discussed in

* N. Beisert, H. Elvang, D. Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, $E_{7(7)}$ Constraints on Counterterms in N= 8 Supergravity_, Phys. Lett. B694, 265 (2010).

Arguments against finiteness to all orders include

* [[Michael Green]], [[Hirosi Ooguri]] and [[John Schwarz]], _Nondecoupling Supergravity from the Superstring_, Phys. Rev. Lett. 99 (2007) 041601.

 * [[Tom Banks]], _Why I don't Believe N= 8 SUGRA is Finite_, talk at _[Workshop "Supergravity versus Superstring Theory in the Ultraviolet"](http://www.gravity.psu.edu/events/supergravity/program.shtml)_, PennState Univ, PA USA, August 27-30 2009.

See also

* [[Renata Kallosh]], _The Ultraviolet Finiteness of $N=8$ Supergravity_, JHEP 1012:009,2010 ([arXiv:1009.1135](http://arxiv.org/abs/1009.1135))

* [[Jacques Distler]], _Decoupling $N = 8$ supergravity_ ([blog post](http://golem.ph.utexas.edu/~distler/blog/archives/001235.html))

Discussion along the lines of [[twistor string theory]] with [[scattering amplitudes]] encoded by [[rational maps]] from a [[complex curve]] two [[twistor space]]:

* {#CachazoSkinner12} Freddy Cachazo, [[David Skinner]], *Gravity from Rational Curves*, Phys. Rev. Lett. 110, 161301, 2012 ([arXiv:1207.0741](https://arxiv.org/abs/1207.0741), [arXiv:10.1103/PhysRevLett.110.161301](https://doi.org/10.1103/PhysRevLett.110.161301))

* {#Adamo15} Tim Adamo, *Gravity with a cosmological constant from rational curves*, JHEP 1511: 098, 2015 ([arXiv:1508.02554](https://arxiv.org/abs/1508.02554))

#### On gravitino phenomenology

A proposal for super-heavy [[gravitinos]] as [[dark matter]], by embedding [[D=4 N=8 supergravity]] into [[E10]]-[[U-duality]]-invariant [[M-theory]]:

* {#MeissnerNicolai18a} [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Standard Model Fermions and Infinite-Dimensional R-Symmetries_, Phys. Rev. Lett. 121, 091601 (2018) ([arXiv:1804.09606](https://arxiv.org/abs/1804.09606))

* {#MeissnerNicolai18b} [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Planck Mass Charged Gravitino Dark Matter_, Phys. Rev. D 100, 035001 (2019) ([arXiv:1809.01441](https://arxiv.org/abs/1809.01441))

following the proposal towards the end of

* {#GellMann83} [[Murray Gell-Mann]], introductory talk at _[Shelter Island II](https://en.wikipedia.org/wiki/Shelter_Island_Conference)_, 1983 ([[Gell-Mann_ShelterIslandII_1983.pdf:file]])

  in: _Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics_. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

### Gauged 4d supergravity
 {#ReferencesGauged}

Discussion of [[gauged supergravity]] in 4d originates around ([Cremmer-Julia 79](#CremmerJulia79) (where the [[E7]]-[[U-duality]] group was first seen)

Discussion of reduction from [[string theory]] includes

* L. Andrianopoli, [[Riccardo D'Auria]], S. Ferrara, M. A. Lledo, _4-D gauged supergravity analysis of Type IIB vacua on $K_3 \times T^2 / \mathbb{Z}_2$_, JHEP 0303:044,2003 ([arXiv:hep-th/0302174](http://arxiv.org/abs/hep-th/0302174))

Perturbative finiteness properties of gauged 4d supergravity from $N = 8$ ungauged 4d supergravity is discussed in [BCDJR 11, p. 24](#BCDJR11):

> Another question is whether $N = 8$ supergravity might point the way to other, more realistic finite (or well behaved) theories of quantum gravity, having less supersymmetry and (perhaps) chiral fermions. One step in this direction could be to examine the multiloop behavior of theories that can be thought of as spontaneously broken gauged $N = 8$ supergravity [73], which are known to have improved ultraviolet behavior at one loop [74].

and [Ferrara-Marrani 12, p. 12](#FerraraMarrani12):

> Another interesting aspect [21] which should be implied by UV finiteness of $N = 8, 6, 5$ supergravity in $D = 4$ dimensions is that their gauged versions should be possibly UV finite, as well. Roughly speaking, this is related to the fact that gauging may be regarded as a spontaneous soft breaking of an unbroken gauge symmetry, and UV properties should not be affected by such a spontaneous breaking, as it happens in the Standard Model of electro-weak interactions.

### Lift to string theory and M-theory

Descent of 4d $\mathcal{N} = 2$ Sugra from [[type IIA string theory]] is reviewed for instance in

* Thomas Wyder, section 1.3 of _Split attractor flow trees and black hole entropy in type II string theory_ ([spire](https://inspirehep.net/record/1397778/))

Discussion of lifts of [[gauged supergravity|gauged]] 4d supergravity to [[string theory]]/[[M-theory]] includes

* Walter H. Baron, _Uplifting Maximal Gauged Supergravities_ ([arXiv:1512.05567](http://arxiv.org/abs/1512.05567))

[[!redirects 4d supergravity]]

[[!redirects d=4 N=1 supergravity]]
[[!redirects D=4 N=1 supergravity]]
[[!redirects N=1 d=4 supergravity]]
[[!redirects N=1 D=4 supergravity]]

[[!redirects d=4 N=2 supergravity]]
[[!redirects D=4 N=2 supergravity]]
[[!redirects N=2 d=4 supergravity]]
[[!redirects N=2 D=4 supergravity]]

[[!redirects d=4 N=3 supergravity]]
[[!redirects D=4 N=3 supergravity]]

[[!redirects D=4 N=4 supergravity]]
[[!redirects d=4 N=4 supergravity]]

[[!redirects N=8 d=4 supergravity]]
[[!redirects N=8 D=4 supergravity]]
[[!redirects D=4 N=8 supergravity]]
[[!redirects d=4 N=8 supergravity]]


D=5 Chern-Simons theory

[[!redirects 5d Chern-Simons theory]]

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## Idea

[[higher dimensional Chern-Simons theory]] in [[dimension]] 5.

## Properties

### Moduli of fields (abelian case)

[[!include moduli of higher lines -- table]]

## Related concepts

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * **5d Chern-Simons theory**

    * [[5d supergravity]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[11d Chern-Simons theory]]

  * [[AKSZ sigma-models]]

  * [[string field theory]]

  * [[infinite-dimensional Chern-Simons theory]]

* [[self-dual Yang-Mills theory]]

* [[D=5 gravity]]

* [[D=5 supergravity]]

## References

A general account of the structure is in

* Olivera Miskovic, Ricardo Troncoso, [[Jorge Zanelli]], _Canonical sectors of five-dimensional Chern-Simons theories_ ([arXiv:hep-th/0504055](http://arxiv.org/abs/hep-th/0504055))

Remarks on applications in [[phenomenology|phenomemenological]] [[model (in theoretical physics)|models]] are in

* C. Hill, _Physics of $D = 5$ Chern-Simons term_ ([pdf](http://theory.fnal.gov/seminars/slides/2006/CHill.pdf))

The relation to [[5-dimensional supergravity]] is discussed in

* A. Ceresole, A. Lerda, and P. van Nieuwenhuizen , _Five-dimensional gravitational super Chern-Simons terms_, Phys. Rev. D 34, 1744–1748 (1986)

* Yuji Tachikawa, _Five-dimensional Chern-Simons terms and Nekrasov's instanton counting_ ([arXiv:hep-th/0401184](http://arxiv.org/abs/hep-th/0401184))

and via [[KK-compactification]] of the [[supergravity C-field]] of [[11-dimensional supergravity]] in

* Federico Bonetti, [[Thomas Grimm]], [[Stefan Hohenegger]], _One-loop Chern-Simons terms in five dimensions_ ([arXiv:1302.2918](http://arxiv.org/abs/1302.2918))

More on supersymmetric 5d CS theory is in

* Sergei M. Kuzenko, Joseph Novak, _On supersymmetric Chern-Simons-type theories in five dimensions_, JHEP 1402 (2014) 096 ([arXiv:1309.6803](http://arxiv.org/abs/1309.6803))

See also:

* Danhua Song, Mengyao Wu, Ke Wu, Jie Yang, *Higher Chern-Simons based on (2-)crossed modules*, JHEP **2023** 207 (2023) [[arXiv:2212.04667](https://arxiv.org/abs/2212.04667), doi:10.1007/JHEP07(2023)207]

[[!redirects 5-dimensional Chern-Simons theory]]


D=5 Einstein-Maxwell theory

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## Idea

[[Einstein-Maxwell theory]] in 5 [[dimensions]] (...)

## Related concepts

* [[black ring]]

* [[D=5 Maxwell theory]]

* [[D=5 super Yang-Mills theory]]

* [[Kaluza-Klein theory]]

## Related concepts

* [[Randall-Sundrum model]]

## References

* [[Paul Wesson]], _Space-Time-Matter: Modern Kaluza-Klein Theory_, World Scientific 1989 ([doi:10.1142/3889](https://doi.org/10.1142/3889))

* [[Paul Wesson]], James M. Overduin, _Principles of Space-Time-Matter: Cosmology, Particles and Waves in Five Dimensions_, World Scientific 2018 ([doi:10.1142/10871](https://doi.org/10.1142/10871))

[[!redirects 5d Einstein-Maxwell theory]]


D=5 gravity

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## Idea

[[gravity]] in [[spacetimes]] of [[dimension]] 5.

## Properties

### Black ring solutions

While impossible in 4 dimensions, 5-dimensional gravity has [[soliton|solitonic]] solutions with non-[[n-sphere|spherical]] [[horizons]], specifically with horizons [[homeomorphism|homeomorphic]] to a [[torus]]: "[[black rings]]" ([Emparan-Reall 02](#EmparanReall02))

## Related concepts

* [[Kaluza-Klein mechanism]], [[dilaton]]

* [[D=5 supergravity]]

* [[D=5 Chern-Simons theory]]

* [[D=2 gravity]]

## References

On [[black rings]] in 5d gravity:

* {#EmparanReall02} [[Roberto Emparan]], [[Harvey Reall]], _A rotating black ring in five dimensions_, Phys. Rev. Lett.88:101101, 2002 ([arXiv:hep-th/0110260](https://arxiv.org/abs/hep-th/0110260))

[[!redirects 5d gravity]]


D=5 Maxwell theory

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## Idea

The [[Maxwell equations]] were originally formulated on 4-dimensional [[Minkowski spacetime]], but in their formulation where the [[Faraday tensor]] is regarded as a [[differential 2-form]] $F$ with [[Hodge dual]] $\star F$ and [[de Rham differentials]]

$$R e
  \array{
    d F & = 0
    \\
    \star d F & = J
  }
$$

they immediately generalize to make sense on [[Minkowski spacetime]] of any dimension $D$, and in fact to any [[pseudo-Riemannian manifold]]. The case of $D =5$-Maxwell theory (hence [[D=5 Yang-Mills theory]] with [[U(1)]]-[[gauge group]])
plays a role mostly as a subsector of [[D=5 Einstein-Maxwell theory]] and/or of [[D=5 super Yang-Mills theory]], as well as a pre-image under [[Kaluza-Klein compactification]] of [[massive Yang-Mills theory]] in 4-dimensions.

## Properties

[[!include relation between 5d Maxwell theory and self-dual 3-forms in 6d -- section]]

\linebreak

### Relation between 5d Maxwell theory and massive vector mesons in 4d
 {#RelationBetween5dMaxwellTheoryAndMassiveVectorMesons}

In an abelian version of [[hadron Kaluza-Klein theory]], hence the way the [[Skyrmion]]-model, coupled to the tower of [[vector mesons]], appears in [[holographic QCD]], the [[Kaluza-Klein reduction]] of [[D=5 Maxwell theory]] to 4d yields a theory of a tower of massive vector particles which may be interpreted as [[vector mesons]] ([Sutcliffe 10, Sec. 3](#Sutcliffe10)):

#### Flat space model

Following [Sutcliffe 10, Sec. 3](#Sutcliffe10), we first consider a toy example where the 5d metric is that of flat [[Minkowski spacetime]].

Consider 4d- and 5d-dimensional [[Minkowski spacetime]] equipped with global [[orthonormal basis|orthonormal]] [[coordinate charts]]  $\{x^\mu\}$, $\{x^\kappa\}$, respectively, adapated to an [[isometry|isometric]] [[embedding of smooth manifolds|embedding]]

$$
  \array{
   &
    \mathbb{R}^{3,1}
    &\overset{\;\;\;\iota_5\;\;\;}{\hookrightarrow}&
    \mathbb{R}^{4,1}
    \\
    \mu = &   0, 1, 2, 3\;
    \\
    \kappa = & 0, 1, 2, 3, && 4
  }
$$

With this notation, the [[pullback of differential forms]] along this embedding is notationally implicit.

Assuming that the 5d [[vector potential]] $\widehat A \in \Omega^1(\mathbb{R}^{4,1})$ is [[square integrable function|square-integrable]] along the $x^4$-drection, we may expand it in a [[linear basis]] $\{h_n\}_{n \in \mathbb{N}}$ of [[Hermite functions]] as a sequence of  [[vector potentials]] $V^{(n)} \in \Omega^1(\mathbb{R}^{4,1})$ that do not depend on $x^4$ anymore.

\[
  \label{HermiteExpansionOf5dVectorPotential}
  \widehat{A}
  \big(
    \{x^\mu\}, x^4
  \big)
  \;=\;
  \underset{n}{\sum}
    h_n(x^4)
    \,
    V^{(n)}(\{x^\mu\})
\]

We assume now (*not* following [Sutcliffe 10](#Sutcliffe10) in this) that $V^{(n)}_4 = 0$, hence that the $V^{(n)}$ are [[pullback of differential forms|pulled back]] from 4d spacetime, and we take these 4d [[vector potentials]] to be in 4d [[Lorenz gauge]]

\[
  \label{4dLorenzGauge}
  d \star_4 V^{(n)}
  \;=\;
  0
  \,.
\]

By the differential recursion relation of [[Hermite functions]] (see [there](Hermite+polynomial#eq:DifferentialRecursionRelationForHermiteFunctions))

\[
  \label{DifferentialRecursionRelationOfHermiteFunctions}
  \frac{d}{ d (x^4)^n}
  h_n(x^4)
  \;=\;
  \sqrt{
    \tfrac
      {n}
      {2}
  }
  h_{n-1}(x^4)
  -
  \sqrt{
    \tfrac
      {n-1}
      {2}
  }
  h_{n+1}(x^4)
\]

it follows that the corresponding [[field strength]] is expanded as

\[
  \label{HermiteExpansionOfFieldStrength}
  d \widehat{A}
  \;=\;
  \underset{n}{\sum}
  h_n
  \Big(
    d V^{(n)}
    +
    \big(
      \sqrt{\tfrac{n+1}{2}}
      V^{(n+1)}
      -
      \sqrt{\tfrac{n}{2}}
      V^{(n-1)}
    \big)
    \wedge
    d x^4
  \Big)
  \,.
\]

The [[Hodge dual]] of that is:

$$
  \star_5 d \widehat{A}
  \;=\;
  \underset{n}{\sum}
  h_n
  \Big(
    \star_4 d V^{(n)} \wedge d x^4
    +
    \sqrt{\tfrac{n}{2}}
    \star_4 V^{(n-1)}
    -
    \sqrt{\tfrac{n+1}{2}}
    \star_{4} V^{(n+1)}
  \Big)
  \,.
$$

(Notice the sign reversal of the last two terms. A quick way to see this is to notice the [[Hodge pairing]]-relation (see [there](Hodge+star+operator#HodgePairing)) and the fact that commuting the [[differential 3-form|3-form]] $\star_4 V^{(n)}$ past the [[differential 1-form|1-form]] $d x^4$ produces a sign.)

Now using once more the differential recursion relation (eq:DifferentialRecursionRelationOfHermiteFunctions) of the [[Hermite functions]], the [[de Rham differential]] of this Hode dual field strength is

\[
  \label{SystemOfCoupledEquationsIn4d}
  \begin{aligned}
    & d \star_5 d \widehat{A}
    \\
    &
  \begin{aligned}
  =
  \underset{n}{\sum}
  h_n
  \Big(
    &
    d \star_4 d V^{(n)} \wedge d x^4
    +
    \sqrt{\tfrac{n}{2}}
    \underset{
      = 0
    }{
      \underbrace{
        d \star_4 V^{(n-1)}
      }
    }
    -
    \sqrt{\tfrac{n+1}{2}}
    \underset{
      = 0
    }{
      \underbrace{
        d \star_{4} V^{(n+1)}
      }
    }
    \\
    &
    +
    \;\;
    \sqrt{\tfrac{n+1}{2}}
    \big(
      \star_4 d V^{(n+1)} \wedge d x^4
      +
      \sqrt{\tfrac{n+1}{2}}
      \star_4 V^{(n)}
      -
      \sqrt{\tfrac{n+2}{2}}
      \star_{4} V^{(n+2)}
    \big)
    \wedge d x^4
    \\
    &
    -
    \;\;
    \sqrt{\tfrac{\;n\;}{2}}
    \big(
      \star_4 d V^{(n-1)} \wedge d x^4
      +
      \sqrt{\tfrac{n-1}{2}}
      \star_4 V^{(n-2)}
      -
      \sqrt{\tfrac{\;n\;}{2}}
      \star_{4} V^{(n)}
    \big)
    \wedge d x^4
  \Big)
  \end{aligned}
  \\
  & =
  \underset{n}{\sum}
  h_n
  \Big(
    d \star_4 d V^{(n)} \wedge d x^4
    +
    \left(
      n + \tfrac{1}{2}
    \right)
    \star_{4} V^{(n)}
    -
    \sqrt{\tfrac{n+2}{2}\tfrac{n+1}{2}}
    \star_{4} V^{(n+2)}
    -
    \sqrt{\tfrac{n}{2}\tfrac{n-1}{2}}
    \star_4 V^{(n-2)}
  \Big)
  \wedge d x^4
  \end{aligned}
\]

Here the terms over the braces vanish because we have chosen [[Lorenz gauge]] in (eq:4dLorenzGauge).

Hence under the expansion (eq:HermiteExpansionOf5dVectorPotential) the 5d [[Maxwell equations]] $d \star_5 d \widehat A = 0$ become equivalently the following system of coupled equations of [[massive Yang-Mills theory]] in 4d (using, with [this Prop.](Hodge+star+operator#HodgeStarFollowedByHodgeStar) that the [[Hodge star operator]] on 3-forms in 4d squares to unity):

$$
  \star_4
  d \star_4 d V^{(n)}
  \;=\;
  -
    \left(
      n + \tfrac{1}{2}
    \right)
    V^{(n)}
  +
    \sqrt{\tfrac{n+2}{2}\tfrac{n+1}{2}}
    V^{(n+2)}
  +
    \sqrt{\tfrac{n}{2}\tfrac{n-1}{2}}
    V^{(n-2)}
  \Big)
$$

Truncating this to the first term yields the [[Klein-Gordon equation]] for a [[field (physics)|field]] of [[mass]] $1/\sqrt{2}$ (by [this Prop.](Hodge+star+operator#WaveOperatorOnMinkowskiSpacetime)):

$$
  \partial^\mu \partial_\mu V^{(1)}
  \;=\;
  \tfrac{1}{2}
  V^{(1)}
  \,.
$$

More generally, truncating at some higher number $N$ of modes, one may find a [[linear transformation]] among the fields $V^{(n)}$ for $1 \leq n \leq N$ which diagonalizes (eq:SystemOfCoupledEquationsIn4d) to obtain a collection of $N$ [[Klein-Gordon equations]] for $N$ massive fields.

\linebreak

#### Sakai-Sugimoto model
 {#SakaiSugimotoModel}

We discuss 5d Maxwell theory on the 5d spacetime that appears in the [[Sakai-Sugimoto model]] (from [[D4/D8-brane intersections]]), following [Bolognesi-Sutcliffe 13](#BolognesiSutcliffe13), [Sutcliffe 15](#Sutcliffe15).

\linebreak

Consider 4d- and 5d-dimensional [[smooth manifolds]] [[diffeomorphism|diffeomorphic]] to [[Cartesian space]] and equipped with global [[orthonormal basis|orthonormal]] [[coordinate charts]]  $\{\{x^\mu\}, z\}$, , respectively, adapated to an [[isometry|isometric]] [[embedding of smooth manifolds|embedding]]

\[
  \label{SakaiSugimotoSpacetime}
  \array{
   &
    \mathbb{R}^{3,1}
    &\overset{\;\;\;\iota_5\;\;\;}{\hookrightarrow}&
    \mathbb{R}^{4,1}
    \\
    \mu = &   0, 1, 2, 3\;
    \\
    \kappa = & 0, 1, 2, 3, && z
  }
\]

+-- {: .num_prop #SakaiSugimotoMetric}
###### Definition
**(Sakai-Sugimoto metric)**

On the 5d manifold (eq:SakaiSugimotoSpacetime) the _Sakai-Sugimito metric_ is the [[metric tensor]]

\[
  \label{SakaiSugimotoMetric}
  g
  \;\coloneqq\;
  H^{+1} \, \eta_{\mu \nu} d x^\mu \otimes d x^\nu
    +
  H^{-1} d z \otimes d z
\]

where

* $(\eta_{\mu \nu}) = diag(-1,+1,+1,+1)$ is the metric of 4d [[Minkowski spacetime]],

* $H$ is the [[smooth function]] given by

  $$
    H(\{x^\mu\}, z)
    \;\coloneqq\;
    H(z)
    \;\coloneqq\;
    \left(
      1 + \frac{z^2}{L^2}
    \right)^{2/3}
  $$

  for some [[real number]] $L$.

=--

This metric is highlighted in this form in ([Bolognesi-Sutcliffe 13, (2.1), (2.2)](#BolognesiSutcliffe13) [Sutcliffe 15, (39), (40)](#Sutcliffe15)).

+-- {: .num_prop #SakaiSugimotoEquation}
###### Proposition
**(Sakai-Suimoto equation)**

Let $A \in \Omega^1(\mathbb{R}^{4,1})$ be a [[differential 1-form]] on the spacetime (eq:SakaiSugimotoSpacetime) with vanishing $z$-component and in 4d [[Lorenz gauge]]:

\[
  \label{VanishingzComponentOfVectorPotentialOnSSSpacetime}
  A_z = 0
\]

\[
 \label{LorenzGaugeForVectorPotentialOnSSSpacetime}
  \partial_\mu \eta^{\mu \nu} A_\nu = 0
 \,.
\]

Then the 5d vacuum [[Maxwell equations]] for the [[vector potential]] $A$ with respect to the [[metric tensor]] (eq:SakaiSugimotoMetric) are equivalent to

\[
 \label{SakaiSugomotoEquation}
  \begin{aligned}
    &
    \star d \star d A
    \;=\;
    0
    \\
    \Leftrightarrow
    \;\;\;
    &
    \eta^{\mu \nu}\partial_\mu \partial_\nu A
    \;+\;
    H^{1/2}
    \partial_z
    \left(
      H^{3/2} \partial_z A
    \right)
    \;=\;
    0
  \end{aligned}
  \,.
\]

=--

This equation is highlighted in [Bolognesi-Sutcliffe 13, (4.14)](#BolognesiSutcliffe13), [Sutcliffe 15, (48)](#Sutcliffe15).

+-- {: .proof}
###### Proof

Here is a somewhat lengthy computation:

First notice the general component formula for the operator $\star d \star d$ applied to 1-forms, akin to, but different from, the component formula  for the [[Laplace-Beltrami operator]] applied to 0-forms/functions (see [there](Laplace+operator#LaplaceOperatorInComponents))

$$
  \begin{aligned}
    \star d \star d A
    & =
    \star d \star (\partial_{j_1} f) A_{j_2} d x^{j_1} \wedge d x^{j_2}
    \\
    & =
    \star d
    \left(
      \tfrac{1}{ 2 \color{green}  (D-2)! }
      \sqrt{
        \left\vert
           det\big( (g_{i j}) \big)
        \right\vert
      }
      \,
      g^{ i_1  j_1 }
      g^{ i_2  j_2 }
      (\partial_{j_1} A_{j_2})
      \,
      \epsilon_{
        i_1 i_2
        {\color{green} k_3 \cdots k_{D} }
      }
      d x^{
        \color{green} k_3
      }
      \wedge \cdots \wedge
      d x^{
        \color{green} k_{D}
      }
    \right)
    \\
    & =
    \star
    \partial_{ \color{magenta} k_2}
    \left(
      \tfrac{1}{ 2 \color{green} (D-2)! }
      \sqrt{
        \left\vert
           det\big( (g_{i j}) \big)
        \right\vert
      }
      \,
      g^{ i_1 j_1 }
      g^{ i_2 j_2 }
      (\partial_{j_1} A_{j_2} )
      \,
      \epsilon_{
        i_1 i_2
        {\color{green} k_3 \cdots k_{D} }
      }
      d x^{ \color{magenta} k_2 }
      \wedge
      d x^{
        \color{green} k_3
      } \wedge \cdots \wedge
      d x^{
        \color{green} k_{D}
      }
    \right)
    \\
    & =
    \sqrt{
      \left\vert
        det\big( (g_{i j}) \big)
      \right\vert
    }
    \underset{
      =
      \det\big( (g_{i j})^{-1}  \big)
      \tfrac{1}{2}
      \delta^{ \color{magenta} k_1 k_2 }_{i_1 i_2}
    }{
    \underbrace{
    \tfrac{
      (-1)^{D-1}
    }{
       { \color{orange} (D-1)! }
       2 { \color{green} (D-2)! }
    }
    \epsilon_{
      l_1
      { \color{orange} l_2 \cdots l_{D} }
    }
    g^{
      { \color{orange} l_1 }
      { \color{magenta} k_1  }
    }
    g^{
      { \color{orange} l_2 }
      { \color{magenta} k_2  }
    }
    g^{
      { \color{orange} l_3 }
      { \color{green} k_3  }
    }
    \cdots
    g^{
      { \color{orange} l_D}
      { \color{green} k_D }
    }
    \epsilon_{
      i_1 i_2
      {\color{green} k_3 \cdots k_{D} }
    }
    }
    }
    \,
    \partial_{ \color{magenta} k_2 }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{ i_1  j_1 }
      g^{ i_2  j_2 }
      (\partial_{j_1} A_{j_2})
    \right)
    \,
    g_{ {\color{magenta} k_1} r }
    d x^{ \color{magenta} r  }
    \\
    & =
    \frac{
      (-1)^{D-1}
    }{
       \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
    }
    \tfrac{1}{2}
    \delta^{ \color{magenta} k_1 k_2 }_{i_1 i_2}
    \partial_{ \color{magenta} k_2 }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{ i_1  j_1 }
      g^{ i_2  j_2 }
      (\partial_{j_1} A_{j_2})
    \right)
    \,
    g_{ {\color{magenta} k_1 } r }
    d x^{  r }
  \end{aligned}
$$

Now specialize this to the the case at hand, where

1. $A = A_\mu d x^{\mu} A_z d^z $

1. $A_z = 0$

1. $\partial_\mu \eta^{\mu \nu} A_\nu = 0$

1. $g_{\mu \nu} = H \eta_{\mu \nu}$

1. $\partial_\mu H = 0$.

We get:

$$
  \begin{aligned}
    &
    \frac{
      (-1)^{D-1}
    }{
       \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
    }
    \tfrac{1}{2}
    \delta^{ \color{magenta} k_1 k_2 }_{i_1 i_2}
    \partial_{ \color{magenta} k_2 }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{ i_1  j_1 }
      g^{ i_2  j_2 }
      (\partial_{j_1} A_{j_2})
    \right)
    \,
    g_{ {\color{magenta} k_1 } r }
    d x^{  r }
    \\
    & =
    \frac{
      (-1)^{D-1}
    }{
       \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
    }
    \tfrac{1}{2}
    \delta^{ \color{magenta} \mu_1 \mu_2 }_{i_1 i_2}
    \partial_{ \color{magenta} \mu_2 }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{ i_1  j_1 }
      g^{ i_2  j_2 }
      (\partial_{j_1} A_{j_2})
    \right)
    \,
    g_{ {\color{magenta} \mu_1 } r }
    d x^{  r }
    \\
    & \phantom{=}\;
    +
    \frac{
      (-1)^{D-1}
    }{
       \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
    }
    \tfrac{1}{2}
    \delta^{ \color{magenta} \mu_1 z }_{i_1 i_2}
    \partial_{ \color{magenta} z }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{ i_1  j_1 }
      g^{ i_2  j_2 }
      (\partial_{j_1} A_{j_2})
    \right)
    \,
    g_{ {\color{magenta} \mu_1 } r }
    d x^{  r }
    \\
    & \phantom{=}\;
    +
    \frac{
      (-1)^{D-1}
    }{
       \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
    }
    \tfrac{1}{2}
    \delta^{ \color{magenta} z \mu_2 }_{i_1 i_2}
    \partial_{ \color{magenta} \mu_2 }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{ i_1  j_1 }
      g^{ i_2  j_2 }
      (\partial_{j_1} A_{j_2})
    \right)
    \,
    g_{ {\color{magenta} z } r }
    d x^{  r }
    \\
    & =
    \tfrac{
      (-1)^{D-1}
    }{2}
    H^{-2}
    \big(
      \underset{
        = 0
      }{
      \underbrace{
      \partial_{\mu_2}
      \partial_{\mu_1}
      A^{\mu_2}
      }
      }
    \big)
    H^1
    d x^{\mu_1}
    -
    (-1)^{D-1}
    \tfrac{1}{2}
    H^{-2}
    \big(
      \partial_{\mu_2}
      \partial^{\mu_2}
    A^{\mu_1}
    \big)
    H^1
    d x^{\mu_1}
    \\
    & \phantom{=}\;
    (-1)^{D-1}
    \tfrac{1}{2}
    H^{-3/2}
    \partial_z
    \big(
      H^{3/2}
      \partial_{\mu_1}
      \underset{ = 0}{
      \underbrace{
      A_{z}
      }
      }
    \big)
    H^1
    d x^{\mu_1}
    -
    (-1)^{D-1}
    \tfrac{1}{2}
    H^{-3/2}
    \partial_z
    \left(
      H^{3/2}
      \partial_{z}
      A_{\mu_1}
    \right)
    H^1
    d x^{\mu_1}
    \\
    & \phantom{=}\;
    +
    (-1)^{D-1}
    \tfrac{1}{2}
    H^{-3/2}
    \underset{
      = 0
    }{
    \underbrace{
    \partial_{\mu_2}
    \left(
      H^{3/2}
      \partial_z
      A^{\mu_2}
    \right)
    }
    }
    H^{-1} d z
    -
    (-1)^{D-1}
    \tfrac{1}{2}
    H^{-3/2}
    \partial_{\mu_2}
    \big(
      H^{3/2}
      \partial_{\mu_2}
      \underset{
        = 0
      }{
      \underbrace{
        A_{z}
      }
      }
    \big)
    H^{-1} d z
    \\
    & =
    -
    (-1)^{D-1}
    \tfrac{1}{2}
    H^{-1}
    \,
    \Big(
      \eta^{\mu \nu}\partial_\mu \partial_\nu A
      +
      H^{1/2}
      \partial_z
      \left(
        H^{3/2} \partial_z A
      \right)
    \Big)
  \end{aligned}
$$

Here the terms over the braces vanish, alternatingly, by
the assumptions (eq:VanishingzComponentOfVectorPotentialOnSSSpacetime) and (eq:LorenzGaugeForVectorPotentialOnSSSpacetime).

=--

\linebreak

(...)

\linebreak

## Related concepts

* [[D=5 Einstein-Maxwell theory]]

* [[D=5 super Yang-Mills theory]]

* [[Randall-Sundrum model]]

## References

General:

* [[Paul Wesson]], Chapter 5 of: _Space-Time-Matter: Modern Kaluza-Klein Theory_, World Scientific 1989 ([doi:10.1142/3889](https://doi.org/10.1142/3889))

* [[Paul Wesson]], James M. Overduin, Chapter 6 of: _Principles of Space-Time-Matter: Cosmology, Particles and Waves in Five Dimensions_, World Scientific 2018 ([doi:10.1142/10871](https://doi.org/10.1142/10871))

Relation to [[vector mesons]] and the [[Skyrmion model]] via the [[Sakai-Sugimoto model]] and variants:

* {#Sutcliffe10} [[Paul Sutcliffe]], Section 3 of: _Skyrmions, instantons and holography_, JHEP 1008:019, 2010 ([arXiv:1003.0023](https://arxiv.org/abs/1003.0023))

* {#BolognesiSutcliffe13} [[Stefano Bolognesi]], [[Paul Sutcliffe]], _The Sakai-Sugimoto soliton_, JHEP 1401:078, 2014 ([arXiv:1309.1396](https://arxiv.org/abs/1309.1396))

* {#Sutcliffe15} [[Paul Sutcliffe]], _Holographic Skyrmions_, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 ([spire:1383608](http://inspirehep.net/record/1383608))

[[!redirects 5d Maxwell theory]]

[[!redirects D=5 electromagnetism]]
[[!redirects 5d electromagnetism]]


D=5 super Yang-Mills theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[super Yang-Mills theory]] on [[spacetimes]] of [[dimension]] $D=5$, hence the [[supersymmetry|supersymmetric]] version of [[D=5 Yang-Mills theory]].

## Properties

### As the worldvolume theory of the D4-brane

$D = 5$ SYM may be regarded as the [[worldvolume]] field theory on the [[D4-brane]] in [[type II string theory]].

### Via reduction from the M5-brane / $D=6$ SCFT

For relation to the [[D=6 N=(2,0) SCFT]] via [[KK-compactification]] on a [[circle]] [[fiber]], hence as [[worldvolume]] theory of the [[double dimensional reduction]] of the [[M5-brane]] (see also at [[Perry-Schwarz Lagrangian]]), see the references [below](#ReferencesRelationToD=6SCFT).

## Related entries

* [[D=4 super Yang-Mills theory]]

* $D=5$ super Yang-Mills theory

  * [[D=5 Yang-Mills theory]]

  * [[D=5 Maxwell theory]]

  * [[D=5 Einstein-Maxwell theory]]

* [[D=6 super Yang-Mills theory]]

* [[D=10 super Yang-Mills theory]]

[[!include superconformal symmetry -- table]]

## References

### General

* [[Nathan Seiberg]], _Five Dimensional SUSY Field Theories, Non-trivial Fixed Points and String Dynamics_, Phys. Lett. B388:753-760, 1996 ([arXiv:hep-th/9608111](https://arxiv.org/abs/hep-th/9608111))

* Arthur Hebecker, _5D Super Yang-Mills Theory in 4D Superspace, Superfield Brane Operators, and Applications to Orbifold GUTs_ ([arXiv:hep-ph/0112230](https://arxiv.org/abs/hep-ph/0112230))

* [[Clay Cordova]], [[Daniel Jafferis]], _Five-Dimensional Maximally Supersymmetric Yang-Mills in Supergravity Backgrounds_ ([arXiv:1305.2886](https://arxiv.org/abs/1305.2886))

* Shuichi Yokoyama, _Supersymmetry Algebra in Super Yang-Mills Theories_, JHEP09(2015)211 ([arXiv:1506.03522](https://arxiv.org/abs/1506.03522))

* I.L. Buchbinder, E.A. Ivanov, I.B. Samsonov, _Low-energy effective action in $5D$, $\mathcal{N}=2$ supersymmetric gauge theory_, Nuclear Physics B
Volume 940, March 2019, Pages 54-62 ([arXiv:1812.07206](https://arxiv.org/abs/1812.07206))

* Lakshya Bhardwaj, _On the classification of 5d SCFTs_ ([arXiv:1909.09635](https://arxiv.org/abs/1909.09635))

[[instantons]]:

* Dongsu Bak, [[Andreas Gustavsson]], _One dyonic instanton in 5d maximal SYM theory_ ([arXiv:1305.3637](https://arxiv.org/abs/1305.3637))

[[topological twists]]:

* Louise Anderson, _Five-dimensional topologically twisted maximally supersymmetric Yang-Mills theory_ ([arXiv:1212.5019](http://arxiv.org/abs/1212.5019))

The [[perturbation theory]] is considered in

* [[Zvi Bern]], John Joseph Carrasco, [[Lance Dixon]], [[Michael Douglas]], Matt von Hippel, Henrik Johansson, _$D = 5$ maximally supersymmetric Yang-Mills theory diverges at six loops_ ([arXiv:1210.7709](http://arxiv.org/abs/1210.7709))

### From $D = 6$ SCFT
 {#ReferencesRelationToD=6SCFT}

Relation to the [[D=6 N=(2,0) SCFT]] via [[KK-compactification]] on a [[circle]] [[fiber]], hence as [[worldvolume]] theory of the [[D4-brane]] [[double dimensional reduction]] of the [[M5-brane]] (see also at [[Perry-Schwarz Lagrangian]]):

* [[Nathan Seiberg]], Sec. 7 of _Notes on Theories with 16 Supercharges_, Nucl. Phys. Proc. Suppl. 67:158-171, 1998 ([arXiv:hep-th/9705117](https://arxiv.org/abs/hep-th/9705117))

* {#Douglas11} [[Michael Douglas]], _On D=5 super Yang-Mills theory and (2,0) theory_, JHEP 1102:011, 2011 ([arXiv:1012.2880](https://arxiv.org/abs/1012.2880))

* {#LPS11} [[Neil Lambert]], Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, _M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills_, JHEP 1101:083, 2011 ([arXiv:1012.2882](https://arxiv.org/abs/1012.2882))

* {#Witten11} [[Edward Witten]], Sections 4 and 5 of _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))

* Moritz McGarrie, _5D Maximally Supersymmetric Yang-Mills in 4D Superspace: Applications_ ([arXiv:1303.4534](https://arxiv.org/abs/1303.4534))

* [[Yuji Tachikawa]], Section 6.4 of: *$\mathcal{N}=2$ supersymmetric dynamics for pedestrians*, Lecture Notes in Physics, vol. 890, 2014 ([arXiv:1312.2684](https://arxiv.org/abs/1312.2684), [doi:10.1007/978-3-319-08822-8](https://link.springer.com/book/10.1007%2F978-3-319-08822-8), [web version](http://www-hep.phys.s.u-tokyo.ac.jp/~yujitach/tmp/dummies/dummies.html#dummiesse7.html))

* [[Chris Hull]], [[Neil Lambert]], _Emergent Time and the M5-Brane_, JHEP06(2014)016 ([arXiv:1403.4532](https://arxiv.org/abs/1403.4532))

* Chiung Hwang, Joonho Kim, Seok Kim, Jaemo Park, Section 3.4.2 of: _General instanton counting and 5d SCFT_ ([arxiv:1406.6793](https://arxiv.org/abs/1406.6793))

* [[Andreas Gustavsson]], _Five-dimensional Super-Yang-Mills and its Kaluza-Klein tower_. JHEP01(2019)222 ([arXiv:1812.01897](https://arxiv.org/abs/1812.01897))

* [[Neil Lambert]], Sec. 3.1 and 3.4.3. of _Lessons from M2's and Hopes for M5's_, _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html), August 2018_ Fortschritte der Physik, 2019 ([arXiv:1903.02825](https://arxiv.org/abs/1903.02825), [slides pdf](http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4))

* Lakshya Bhardwaj, Patrick Jefferson, Hee-Cheol Kim, Houri-Christina Tarazi, [[Cumrun Vafa]], _Twisted Circle Compactification of 6d SCFTs_ ([arXiv:1909.11666](https://arxiv.org/abs/1909.11666))

* Lakshya Bhardwaj, _More 5d KK theories_ ([arXiv:2005.01722](https://arxiv.org/abs/2005.01722))

* Max Hubner, _5d SCFTs from $(E_n, E_m)$ Conformal Matter_ ([arXiv:2006.01694](https://arxiv.org/abs/2006.01694))

* Lakshya Bhardwaj, _Flavor Symmetry of 5d SCFTs, Part 1: General Setup_ ([arXiv:2010.13230](https://arxiv.org/abs/2010.13230))

* Lakshya Bhardwaj, _Flavor Symmetry of 5d SCFTs, Part 2: Applications_ ([arXiv:2010.13235](https://arxiv.org/abs/2010.13235))

* Hirotaka Hayashi, Hee-Cheol Kim, [[Kantaro Ohmori]], _6d/5d exceptional gauge theories from web diagrams_ ([arXiv:2103.02799](https://arxiv.org/abs/2103.02799))

### From M-theory on Calabi-Yau 3-folds

From [[M-theory on Calabi-Yau 3-folds]]:

* {#CDS18} Cyril Closset, Michele Del Zotto, Vivek Saxena, _Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective_ ([arXiv:1812.10451](https://arxiv.org/abs/1812.10451))

* Vivek Saxena, _Rank-two 5d SCFTs from M-theory at isolated toric singularities: a systematic study_, High Energ. Phys. 2020, 198 (2020)  ([arXiv:1911.09574](https://arxiv.org/abs/1911.09574))

### Realization on $(p,q)$5-brane webs

Realization ([[geometric engineering of quantum field theory|geometric engineering]]) on [[(p,q)5-brane webs]]:

* {#AharonyHanany97} [[Ofer Aharony]], [[Amihay Hanany]], _Branes, Superpotentials and Superconformal Fixed Points_, Nucl. Phys. B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170))

* {#AharonyHananyKrol97} [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116))

* [[Oren Bergman]], Gabi Zafrir, _Lifting 4d dualities to 5d_, JHEP04 (2015) 141 ([arXiv:1410.2806](https://arxiv.org/abs/1410.2806))

Further developments:

* [[Taro Kimura]], Rui-Dong Zhu, Section 2 of _Web Construction of ABCDEFG and Affine Quiver Gauge Theories_ ([arXiv:1907.02382](https://arxiv.org/abs/1907.02382))

[[!redirects  D=5 sYM]]

[[!redirects  D=5 SYM]]

[[!redirects D=5 N=1 super Yang-Mills theory]]
[[!redirects D=5 N=1 SYM]]

[[!redirects 5d super Yang-Mills theory]]


D=5 supergravity

[[!redirects 5-dimensional supergravity]]
[[!redirects 5-dimensional supergravity]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[supergravity]] in [[dimension]] 5. For $N = 1$ this arises from [[11-dimensional supergravity]] by [[KK-compactification]] on a [[Calabi-Yau manifold]] of complex dimension 3 (see at _[[M-theory on Calabi-Yau manifolds]]_), hence serves as the low-energy [[effective field theory]] of the strong-coupling version of Calabi-Yau compactifications of [[type IIA string theory]] (see [[supersymmetry and Calabi-Yau manifolds]])

$$
  \array{
    11d \; SuGra\; on\; S^1 \times Y_6 \times X_4
      &\longrightarrow&
    5d \; SuGra\; on\; S^1 \times X_4 && strong \; coupling
    \\
    \downarrow && \downarrow
    \\
    10d\; tpye \;IIA\; Sugra\; on \; Y_6 \times X_4
      &\longrightarrow&
    10d\; Sugra\; on \; X_4 && weak \; coupling
  }
$$

## Properties

### 5d Chern-Simons term
 {#5dChernSimonsTerm}

This theory has a 2-form field strength $F_2$, locally $F_2 = d A$, with a [[5d Chern-Simons theory]] [[action functional]] locally of the form $\propto \int_X F_2 \wedge F_2 \wedge A$ (e.g. [Castellani-D'Auria-Fre (III.5.70)](#CastellaniDAuriaFre), [Gauntlet-Myers-Townsend  98, p. 3](#GauntlettMyersTownsend98), [GGHPR 02 (2.1)](#GGHPR02), [Bonetti-Grimm-Hohenegger 13](#BonettiGrimmHohenegger13)). Hence its [[equation of motion]] is of the non-linear form

$$
  d F_3 = F_2 \wedge F_2
$$

with $F_3 \coloneqq \star F_2$ the [[Hodge dual]] of $F_2$ ([GGHPR 02 (2.2)](#GGHPR02)).

This is reflected in the corresponding cochains on [[super Minkowski spacetime]]

$$
  \mu_{D0,5d} = \overline{\psi}_A \psi_A
  \phantom{AAA}
  \mu_{string,5d} = \overline{\psi}_A\Gamma_a \psi_A  \wedge e^a
$$

satisfying

$$
  d \mu_{string,5d} = \mu_{D0,5d} \wedge \mu_{D0,5d}
  \,.
$$

due to the [[Fierz identity]] in [Castellani-D'Auria-Fré 91 (III.5.50a)](#CastellaniDAuriaFre), [this example](Fierz+identity#QuadraticFierzIdentitiesIn5d):

\begin{imagefromfile}
  "file_name": "FierzIdentitiesForBraneCocyclesIn5d.png",
  "width": 600
\end{imagefromfile}

(the other Fierz identity (III.5.50a) gives the  cocycle for the [[membrane]] ([[super 2-brane in 5d]]) $\mu_{membrane,5d} \coloneqq \frac{i}{2}\overline{\psi}_A \Gamma_{a b}  \psi \wedge e^a \wedge e^b$, $d \mu_{membrane,5d} = 0$, that appears already in the old [[brane scan]]. )

This is a lower dimensional analogue to the situation for the [[C-field]] $G_4$ (locally $G_4 = d C$) in [[11-dimensional supergravity]], which has a Chern-Simons term locally of the form $\propto \int G_4 \wedge G_4 \wedge C$ and hence the equation of motion

$$
  d G_7
  \;=\;
  -\tfrac{1}{2}G_4 \wedge G_4
$$

with $G_7 = \star G_4$.

### Black holes and black rings

The first [[black ring]] solution in 5d sugra was found in ([Elvang-Emparan-Mateos-Reall 04](#ElvangEmparanMateosReall04), [Elvang-Emparan-Mateos-Reall 05](#ElvangEmparanMateosReall05)).

Supersymmetric black holes exist precisely only in dimensions 4 and 5 ([Gauntlett-Myers-Townsend 98](#GauntlettMyersTownsend98)). These play a key role in the discussion of [[black holes in string theory]].

(There are supersymmetric particle-like solutions of $d \gt 5$ supergravity theories that are sometimes called black holes, but these are always singular. There are also supersymmetric black holes in $d = 3$, but
the spacetime in that case is asymptotically [[anti-de Sitter spacetime]] rather than asymptotically flat. Of course, there are non-singular supersymmetric [[black brane]] solutions in various $d \geq 4$ supergravity theories but these are neither 'particle-like' nor, strictly speaking,
asymptotically flat.)

### Via Calabi-Yau compactification of 11d supergravity
 {#ViaCYCompactificationOf11dSupergravity}

Discussion of 5d supergravity as a [[KK-compactification]] of [[11-dimensional supergravity]] on a [[Calabi-Yau manifold]] of complex dimension 3 ([[M-theory on Calabi-Yau manifolds]]) is discussed in

([Hull-Townsend 95, p.30-31](#HullTownsend95), [Cadavid-Ceresole-D'Auria-Ferrara 95](#CadavidCeresoleDAuriaFerrara95) [Ferrara-Khuria-Minasian 96](#FerraraKhuriaMinasian96), [Ferrara-Minasian-Sagnotti 96](#FerraraMinasianSagnotti96)). See also ([Mizoguchi-Ohta 98](#MizoguchiOhta98)).

### U-duality

[[!include U-duality -- table]]

### $N = 2$ supersymmetry

Consider super Lie algebra cocoycles on $N =2$ 5d [[super-Minkowski spacetime]] (as in the [[brane scan]]).

With the notation as used at _[super Minkowski spacetime -- Canonical coordinates](super%20Minkowski%20spacetime#CanonicalCoordinates)_, there are now two copies of spinor-valued 1-forms, denoted $\psi_1$ and $\psi_2$. We use indices of the form $A,B, \cdots$ for these. Then the non-trivial bit of the [[Chevalley-Eilenberg algebra]] differential for $N = 2$, $d = 5$ [[super Minkowski spacetime]] is

$$
  d_{CE} e^a = - \tfrac{i}{2} \overline{\psi}_A \wedge \Gamma^a \psi_A
$$

where summation over repeated indices is understood.

There is a [[Fierz identity]]

$$
  \overline{\psi}_A \wedge \psi_A \wedge \overline{\psi}_B \wedge \psi_B
  \;=\;
  \overline{\psi}_A \wedge \Gamma_a \psi_A
  \wedge \overline{\psi}_B \wedge \Gamma^a \psi_B
  \,.
$$

([Castellani-D'Auria-Fré (III.5.50a)](#CastellaniDAuriaFre))

This implies that

$$
  d_{CE} (\overline{\psi}_A \Gamma^a \psi_A \wedge e_a)
  \;\propto\;
  (\overline{\psi}_A \wedge \Gamma^a \psi_A)
    \wedge
  (\overline{\psi}_B \wedge \Gamma^a \psi_B)
  \,.
$$

There is a 4-cocycle of the form

$$
  \mu_2
    =
  \epsilon^{A B} \overline{\psi}_A \wedge \Gamma^{a b} \psi_B
  \wedge e_a \wedge e_b
  \,.
$$

([Castellani-D'Auria-Fré (III.5.50b), (III.5.53c)](#CastellaniDAuriaFre))

## Related concepts

* [[D=5 gravity]]

* [[D=5 Chern-Simons theory]]

* [[11-dimensional supergravity]]

* 10-dimensional [[type II supergravity]], [[heterotic supergravity]]

* [[7-dimensional supergravity]]

* **5-dimensional supergravity**

  * [[super 2-brane in 5d]]

* [[4-dimensional supergravity]]

* [[3-dimensional supergravity]]

## References

Construction of 5d [[gauged supergravity]] via [[D'Auria-Fré formulation of supergravity]] is in

* {#ACFG01} [[Laura Andrianopoli]], Francesco Cordaro, [[Pietro Fré]], Leonardo Gualtieri, _Non-Semisimple Gaugings of D=5 N=8 Supergravity and FDA.s_, Class. Quant. Grav. 18 (2001) 395-414 [[arXiv:hep-th/0009048](http://arxiv.org/abs/hep-th/0009048)]

surveyed in

* [[Laura Andrianopoli]], Francesco Cordaro, [[Pietro Fré]], _Non-Semisimple Gaugings of D=5 N=8 Supergravity_, Fortsch.Phys.49:511-518,2001 ([arXiv:hep-th/0012203](http://arxiv.org/abs/hep-th/0012203))

### General

General discussion:

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter III.5 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

  (in [[D'Auria-Fre formulation of supergravity]])

* [[William Linch III]], Markus A. Luty, J. Phillips, _Five dimensional supergravity in $N = 1$ superspace_, Phys.Rev.D68:025008,2003 ([arXiv:hep-th/0209060](https://arxiv.org/abs/hep-th/0209060))

* {#GGHPR02} [[Jerome Gauntlett]], [[Jan Gutowski]], [[Christopher Hull]], Stathis Pakis, [[Harvey Reall]], _All supersymmetric solutions of minimal supergravity in five dimensions_, Class. Quant. Grav. 20 (2003) 4587-4634 [[arXiv:hep-th/0209114](http://arxiv.org/abs/hep-th/0209114), [doi:10.1088/0264-9381/20/21/005](https://doi.org/10.1088/0264-9381/20/21/005)]

* Sorin Cucu, _From M-theory to $D=5$ supergravity and duality-symmetric theories_ ([arXiv:hep-th/0310105](http://arxiv.org/abs/hep-th/0310105))

* [[Eric Bergshoeff]], Sorin Cucu, Tim de Wit, Jos Gheerardyn, Stefan Vandoren, [[Antoine Van Proeyen]], _$N=2$ supergravity in five dimensions revisited_ ([arXiv:hep-th/0403045](http://arxiv.org/abs/hep-th/0403045))

* [[Katrin Becker]], [[Melanie Becker]], Daniel Butter, [[William Linch III]], Stephen Randall, _Five-dimensional Supergravity in N = 1/2 Superspace_ ([arXiv:1909.09208](https://arxiv.org/abs/1909.09208))

* Edoardo Lauria, [[Antoine Van Proeyen]], _$\mathcal{N}=2$ Supergravity in $D=4,5,6$ Dimensions_ ([arXiv:2004.11433](https://arxiv.org/abs/2004.11433))

See also:

* Andrew Beckett, [[José Figueroa-O'Farrill]], *Killing superalgebras for lorentzian five-manifolds* ([arxiv:2105.05775](https://arxiv.org/abs/2105.05775))

* Soumya Adhikari, Bindusar Sahoo, $\mathcal{N}=2$ conformal supergravity in five dimensions* [[arXiv:2312.01879](https://arxiv.org/abs/2312.01879)]

### Via M-theory on Calabi-Yau 3-folds

Discussion via [[KK-compactification]] as [[M-theory on Calabi-Yau manifolds]] includes

* {#HullTownsend95} [[Chris Hull]], [[Paul Townsend]], pages 30 and 31 of _Unity of Superstring Dualities_, Nucl.Phys.B438:109-137,1995 ([arXiv:hep-th/9410167](https://arxiv.org/abs/hep-th/9410167))

* {CadavidCeresoleDAuriaFerrara95} A.C. Cadavid, A. Ceresole, [[Riccardo D'Auria]], [[Sergio Ferrara]], _11-Dimensional Supergravity Compactified on Calabi-Yau Threefolds_ ([arXiv:hep-th/9506144](https://arxiv.org/abs/hep-th/9506144))

* {#FerraraKhuriaMinasian96} [[Sergio Ferrara]], Ramzi R. Khuria,  [[Ruben Minasian]], _M-theory on a Calabi-Yau manifold_, Phys.Lett.B375:81-88,1996 ([arXiv:hep-th/9602102](https://arxiv.org/abs/hep-th/9602102))

* {#FerraraMinasianSagnotti96} [[Sergio Ferrara]], [[Ruben Minasian]], [[Augusto Sagnotti]], _Low-Energy Analysis of M and F Theories on Calabi-Yau Threefolds_, Nucl.Phys. B474 (1996) 323-342 ([arXiv:hep-th/9604097](https://arxiv.org/abs/hep-th/9604097))

* {#MizoguchiOhta98} S. Mizoguchi, N. Ohta, _More on the Similarity between $D=5$ Simple Supergravity and M Theory_, Phys.Lett. B441 (1998) 123-132 ([arXiv:hep-th/9807111](https://arxiv.org/abs/hep-th/9807111))

Further discussion of the [[5d Chern-Simons theory|5d Chern-Simons term]] includes

* {#BonettiGrimmHohenegger13} Federico Bonetti, [[Thomas Grimm]], [[Stefan Hohenegger]], _One-loop Chern-Simons terms in five dimensions_ ([arXiv:1302.2918](http://arxiv.org/abs/1302.2918))

(one-loop corrections).

### Via type IIB theory

* Arnaud Baguet, [[Olaf Hohm]], [[Henning Samtleben]], _Consistent Type IIB Reductions to Maximal 5D Supergravity_ ([arXiv:1506.01385](https://arxiv.org/abs/1506.01385))

### Gauged sugra
 {#ReferencesGauged}

The maximal 5d [[gauged supergravity]] was first constructed in

*  M. Pernici, K. Pilch, [[Peter van Nieuwenhuizen]], _Gauged $N=8$ $D=5$ Supergravity_, Nucl.Phys. B259 (1985) 460 ([spire](https://inspirehep.net/record/16067?ln=en))

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Gauged $N = 8$ Supergravity in Five Dimensions_, Phys. Lett. 154B, (1985) 268 ([spire:207663](http://inspirehep.net/record/207663), doi:10.1016/0370-2693(85)90361-2)

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Compact and Non–Compact Gauged Supergravity Theories in Five Dimensions_, Nucl. Phys. B272 (1986) 598 ([spire:219727](http://inspirehep.net/record/219727), doi:10.1016/0550-3213(86)90237-3)

See ([ACFG 01](#ACFG01)).

* [[Murat Gunaydin]], Marco Zagermann, _The Gauging of Five-dimensional, $N=2$ Maxwell-Einstein Supergravity Theories coupled to Tensor Multiplets_,  	Nucl.Phys.B572:131-150,2000 ([arXiv:hep-th/9912027](https://arxiv.org/abs/hep-th/9912027))

* [[Murat Gunaydin]], Marco Zagermann, _The Vacua of 5d, $N=2$ Gauged Yang-Mills/Einstein/Tensor Supergravity: Abelian Case_, Phys.Rev.D62:044028,2000 ([arXiv:hep-th/0002228](http://arxiv.org/abs/hep-th/0002228))

* A. Ceresole, [[Gianguido Dall'Agata]], _General matter coupled $N=2$, $D=5$ gauged supergravity_, Nucl.Phys. B585 (2000) 143-170 ([arXiv:hep-th/0004111](http://arxiv.org/abs/hep-th/0004111))

* [[John Ellis]], [[Murat Gunaydin]], Marco Zagermann, _Options for Gauge Groups in Five-Dimensional Supergravity_, JHEP 0111:024,2001 ([arXiv:hep-th/0108094](http://arxiv.org/abs/hep-th/0108094))

### Horava-Witten compactification
 {#ReferencesHWCompactification}

Discussion of [[KK-compactification]] on $S^1/(\mathbb{Z}/2)$-orbifolds (the version of [[Horava-Witten theory]] after dimensional reduction) is discussed in

* Filipe Paccetti Correia, Michael G. Schmidt, Zurab Tavartkiladze, _4D Superfield Reduction of 5D Orbifold SUGRA and Heterotic M-theory_ ([arXiv:hep-th/0602173](https://arxiv.org/abs/hep-th/0602173))

### Black hole solutions

Discussion of lifts of 4d black holes to 5d black holes and [[black rings]] and embedding as [[black holes in string theory]] includes

* {#GauntlettMyersTownsend98} [[Jerome Gauntlett]], [[Robert C. Myers]], [[Paul Townsend]], *Black Holes of $D=5$ Supergravity*, Class.Quant.Grav. **16** (1999) 1-21 [[arXiv:hep-th/9810204](http://arxiv.org/abs/hep-th/9810204), [doi:10.1088/0264-9381/16/1/001](https://doi.org/10.1088/0264-9381/16/1/001)]

* {#ElvangEmparanMateosReall04} [[Henriette Elvang]], [[Roberto Emparan]], [[David Mateos]], [[Harvey Reall]], _A supersymmetric black ring_, Phys. Rev. Lett. 93:211302,2004 ([arXiv:hep-th/0407065](http://arxiv.org/abs/hep-th/0407065))

* {#ElvangEmparanMateosReall05} [[Henriette Elvang]], [[Roberto Emparan]], [[David Mateos]], [[Harvey Reall]], _Supersymmetric 4D Rotating Black Holes from 5D Black Rings_, JHEP0508:042,2005 ([arXiv:hep-th/0504125](http://arxiv.org/abs/hep-th/0504125))

* [[Iosif Bena]], [[Per Kraus]], _Microscopic description of black rings in AdS/CFT_ JHEP 12 (2004) 070 ([hep-th/0408186](http://arxiv.org/abs/hep-th/0408186))

* [[Iosif Bena]], [[Per Kraus]], _Microstates of the D1-D5-KK system_ Phys. Rev. D72 (2005) 025007 ([hep-th/0503053](http://arxiv.org/abs/hep-th/0503053))

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _5D black rings and 4D black holes_ JHEP 02 (2006) 023 ([hep-th/0504126](http://arxiv.org/abs/hep-th/0504126))

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _New connections between 4D and 5D black holes_, JHEP 02 (2006) 024 ([hep-th/0503217](https://arxiv.org/abs/hep-th/0503217))

* Alejandra Castro, Joshua L. Davis, [[Per Kraus]], Finn Larsen, _String Theory Effects on Five-Dimensional Black Hole Physics_ ([arXiv:0801.1863](http://arxiv.org/abs/0801.1863))

Review:

* [[Per Kraus]], _Lectures on black holes and the $AdS_3$/$CFT_2$ correspondence_ ([arXiv:hep-th/0609074](http://arxiv.org/abs/hep-th/0609074))

  _Stringy black holes in five dimensions_, 2007 ([pdf slides](http://hep.ps.uci.edu/~arajaram/Irvine.07.pdf))

Further [[defect branes]]:

* Eun Kyung Park, Pyung Seong Kwon, *NS-branes in 5d brane world models*, Phys. Rev. D82:046001, 2010 ([arXiv:1007.1290](https://arxiv.org/abs/1007.1290), [doi:10.1103/PhysRevD.82.046001](https://doi.org/10.1103/PhysRevD.82.046001))

* Minkyu Park, [[Masaki Shigemori]], _Codimension-2 Solutions in Five-Dimensional Supergravity_, JHEP 1510 (2015) 011 ([arXiv:1505.05169](http://arxiv.org/abs/1505.05169))

* [[Masaki Shigemori]], *Interpolating between multi-center microstate geometries*, JHEP 09 (2021) 010 ([arXiv:2105.11639](https://arxiv.org/abs/2105.11639))

[[!redirects 5d supergravity]]


D=5 Yang-Mills theory

#Contents#
* table of contents
{:toc}

## Idea

[[Yang-Mills theory]] on [[spacetimes]] of [[dimension]] $D=5$.

For [[U(1)]]-[[gauge group]] this reduces to [[D=5 Maxwell theory]]. Under [[KK-compactification]] this becomes [[massive Yang-Mills theory]] in 4 dimensions. Under [[supersymmetry|supersymmetrization]] it becomes [[D=5 super Yang-Mills theory]].

## Properties

### Hadrons as KK-modes of 5d Yang-Mills theory
 {#HadronsAsKKModesOf5dYangMillsTheoryReferences}

On [[Skyrmions]] in 4d as [[AdS/QCD|holographic]]/[[KK reduction|KK-theoretic]] reduction of [[instantons]] in [[D=5 Yang-Mills theory]]:

* {#AtiyahManton89} [[Michael Atiyah]], [[Nicholas Manton]], _Skyrmions from instantons_, Phys.  Lett.  B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

[[Skyrmions]] in 4 dimensional spacetime, with [[vector meson]]-contributiuons included, are the [[AdS-CFT|holographic]]/[[KK-theory]] reduction of [[instantons]] in [[D=5 Yang-Mills theory]] ([Sakai-Sugimoto 04, Section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, Section 3.3](#SakaiSugimoto05), reviewed in [Sugimoto 16, Section 15.3.4](#Sugimoto16), [Bartolini 17, Section 2](#Bartolini17).

This phenomenon is essentially the theorem of [Atiyah-Manton 89](#AtiyahManton89), this is highlighted and developed in [Sutcliffe 10](#Sutcliffe10), [Sutcliffe 15](#Sutcliffe15).

In this way Skyrmions (and hence [[baryons]] and [[atomic nuclei]], see [below](#AsAModelForBaryonsAndNuclei)) appear in the [[Witten-Sakai-Sugimoto model]], which realizes (something close to) [[non-perturbative quantum field theory|non-perturbative]] [[QCD]] as a [[D4-D8-brane intersection|D4/D8]]-[[intersecting D-brane model]] described by the [[AdS-QCD correspondence]] ("[[holographic QCD]]").

This already produces [[baryon]] [[mass]] spectra with moderate quantitative agreement with [[experiment]] ([HSSY 07](#HSSY07)):

> graphics from [Sugimoto 16](#Sugimoto16)

This way, via the equivalence between [[D4-D8-brane intersections]] with [[instantons]] in the [[D8-brane]]-[[worldvolume]], the Skyrme model becomes equivalent to a model of [[baryons]] by [[wrapped brane|wrapped]] [[D4-branes]] ([Sugimoto 16, 15.4.1](#Sugimoto16)):

\begin{center}

\end{center}

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

## Related concepts

* [[D=5 Maxwell theory]]

* [[D=5 super Yang-Mills theory]]

* [[Skyrme model]]

## References

### General

* [[Yuji Tachikawa]], Section 6.4 of: *$\mathcal{N}=2$ supersymmetric dynamics for pedestrians*, Lecture Notes in Physics, vol. 890, 2014 ([arXiv:1312.2684](https://arxiv.org/abs/1312.2684), [doi:10.1007/978-3-319-08822-8](https://link.springer.com/book/10.1007%2F978-3-319-08822-8), [web version](http://www-hep.phys.s.u-tokyo.ac.jp/~yujitach/tmp/dummies/dummies.html#dummiesse7.html))

[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]

[[!redirects 5d Yang-Mills theory]]


D=6 N=(1,0) SCFT

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}

## Idea

[[superconformal field theory]] in [[dimension]] 6 with [[number of supersymmetries]] equal to 1.

Arises from

* [[M5-branes]] at D-, and E-type [[ADE-singularities]]

* [[M-theory on S1/G_HW times H/G_ADE]]

(...)

## Related concepts

[[!include superconformal symmetry -- table]]

* [[D=6 N=(2,0) SCFT]]

* [[D=6 supergravity]]

* [[D=5 super Yang-Mills theory]]

* [[M5-brane elliptic genus]]

## References

### General

* [[Kantaro Ohmori]], _Six-Dimensional Superconformal Field Theories and Their Torus Compactifications_, Springer Theses 2018 ([springer:book/9789811330919](https://www.springer.com/gp/book/9789811330919))

Comprehensive review of expected [[top-down model building|top-down constructions]] of [[D=6 N=(2,0) SCFT|$D=6$ $\mathcal{N}=(2,0)$ SCFTs]] and [[D=6 N=(1,0) SCFT|$D=6$ $\mathcal{N}=(1,0)$ SCFTs]]

* [[Jonathan J. Heckman]], [[Tom Rudelius]], *Top Down Approach to 6D SCFTs*, J. Phys. A: Math. Theor. **52** 9 (2019) 093001 [[arXiv:1805.06467](https://arxiv.org/abs/1805.06467), [doi:10.1088/1751-8121/aafc81](https://doi.org/10.1088/1751-8121/aafc81)]

### Anomaly cancellation

Discussion of [[anomaly cancellation]] and [[Green-Schwarz mechanism]]:

* [[Michael Green]], [[John Schwarz]], [[Pete West]], _Anomaly-free chiral theories in six dimensions_, Nuclear Physics B Volume 254, 1985, Pages 327-348 (doi:10.1016/0550-3213(85)90222-6)

* [[Augusto Sagnotti]], _A Note on the Green - Schwarz Mechanism in Open - String Theories_, Phys. Lett. B294:196-203, 1992 ([arXiv:hep-th/9210127](https://arxiv.org/abs/hep-th/9210127))

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], _Anomaly polynomial of E-string theories_, J. High Energ. Phys. 2014, 2 (2014) ([arXiv:1404.3887](https://arxiv.org/abs/1404.3887))

  > (Green-Schwarz anomaly in 1.2)

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], [[Kazuya Yonekura]], _Anomaly polynomial of general 6d SCFTs_, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 ([arXiv:1408.5572](https://arxiv.org/abs/1408.5572))

  > (Green-Schwarz anomaly on tensor branch of M5-branes on MO9 in 2.18)

* [[Kenneth Intriligator]], _$6d$, $\mathcal{N}=(1,0)$ Coulomb Branch Anomaly Matching_, J. High Energ. Phys. 2014, 162 (2014) ([arXiv:1408.6745](https://arxiv.org/abs/1408.6745), doi:10.1007/JHEP10(2014)162)

  > (Green-Schwarz anomaly on Coulomb branch in 4.1)

* [[Hiroyuki Shimizu]], _Aspects of anomalies in 6d superconformal field theories_, Tokyo 2018 ([spire:1802462](https://inspirehep.net/literature/1802462), [doi:10.15083/00077898](https://doi.org/10.15083/00077898))

  > (Green-Schwarz anomaly on ADE-singularity in 7.2.8)

* [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], from p. 18 in: _2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories_ ([arXiv:2009.00138](https://arxiv.org/abs/2009.00138))

### Lagrangian description

A class of [[Lagrangian densities]] for [[D=6 N=(1,0) SCFTs]] with [[non-abelian group|non-abelian]] [[gauge group]] has been proposed in

* {#SSW11} [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, _$(1,0)$ superconformal models in six dimensions_,  J. High Energ. Phys. 2011, 62 (2011) ([arXiv:1108.4060](https://arxiv.org/abs/1108.4060), doi:10.1007/JHEP12(2011)062)

reviewed in

* [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, Linus Wulff, _New superconformal models in six dimensions: Gauge group and representation structure_, PoS CORFU2011 (2011) 071 ([arXiv:1204.0542](https://arxiv.org/abs/1204.0542), [spire:1102894](http://inspirehep.net/record/1102894))

An attempt to understand the [SSW 11](#SSW11)-models as [[higher gauge theories]] for gauge-[[L-infinity algebras]] which are variants of the [[string Lie 2-algebra]] is due to:

* [[Christian Saemann]], [[Lennart Schmidt]], _The Non-Abelian Self-Dual String and the $(2,0)$-Theory_ ([arXiv:1705.02353](https://arxiv.org/abs/1705.02353))

* [[Christian Saemann]], [[Lennart Schmidt]], _Towards an M5-Brane Model I: A 6d Superconformal Field Theory_, J. Math. Phys. 59 (2018) 043502 ([arxiv:1712.06623](https://arxiv.org/abs/1712.06623))

A description induced form a non-[[Lorentz group|Lorentz]] invariant 5d Lagrangian:

* [[Neil Lambert]], Tristan Orchard, *Non-Lorentzian Avatars of $(1,0)$ Theories*, J. High Energ. Phys. **2021** 205 (2021) [[arXiv:2011.06968](https://arxiv.org/abs/2011.06968), doi:10.1007/JHEP02(2021)205]

* [[Neil Lambert]], Joseph Smith, *RG Flows and Symmetry Enhancement in Five-Dimensional Lifshitz Gauge Theories* [[arXiv:2212.07717](https://arxiv.org/abs/2212.07717)]

reviewed in:

* [[Neil Lambert]], *A non-Lorentzian View of the M5-brane*,  talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Lambert2023)]

### Geometric engineering on 5-branes

On [[D=6 N=(1,0) SCFTs]] via [[geometric engineering of QFT|geometric engineering]] on [[M5-branes]]/[[NS5-branes]] at D-, E-type [[ADE-singularities]], notably from [[M-theory on S1/G_HW times H/G_ADE]], hence from [[orbifolds]] of [[type I' string theory]] (see at [half NS5-brane](NS5-brane#NSHalfBranes)):

* [[Ori Ganor]], [[David Morrison]], [[Nathan Seiberg]], _Branes, Calabi-Yau Spaces, and Toroidal Compactification of the $N=1$ Six-Dimensional $E_8$ Theory_, Nucl. Phys. B487:93-127, 1997 ([arXiv:hep-th/9610251](https://arxiv.org/abs/hep-th/9610251))

  (emphasis on the [[KK-compactifications]])

* [[Michele Del Zotto]], [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP12(2014)003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], _$S^1/T^2$ Compactifications of 6d $\mathcal{N} = (1,0)$ Theories and Brane Webs_, J. High Energ. Phys. (2016) 2016: 24 ([arXiv:1509.03195](https://arxiv.org/abs/1509.03195))

* {#HKLY15} Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, _6d SCFTs, 5d Dualities and Tao Web Diagrams_, JHEP05 (2019)203 ([arXiv:1509.03300](https://arxiv.org/abs/1509.03300))

* Ibrahima Bah, Achilleas Passias, [[Alessandro Tomasiello]], _$AdS_5$ compactifications with punctures in massive IIA supergravity_, JHEP11 (2017)050 ([arXiv:1704.07389](https://arxiv.org/abs/1704.07389))

### KK-Compactification to $D=4$ $\mathcal{N} = 1$ SYM

The [[KK-compactification]] specifically of the [[D=6 N=(1,0) SCFT]] to [[D=4 N=1 super Yang-Mills]]:

* Ibrahima Bah, Christopher Beem, Nikolay Bobev, Brian Wecht, _Four-Dimensional SCFTs from M5-Branes_ ([arXiv:1203.0303](http://arxiv.org/abs/1203.0303))

* Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _$4d$ $\mathcal{N} = 1$ from $6d (1,0)$_, J. High Energ. Phys. (2017) 2017: 64 ([arXiv:1610.09178](https://arxiv.org/abs/1610.09178))

* Ibrahima Bah, [[Amihay Hanany]], Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, _$4d$ $\mathcal{N}=1$ from $6d$ $\mathcal{N}=(1,0)$ on a torus with fluxes_ ([arXiv:1702.04740](https://arxiv.org/abs/1702.04740))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _E-String Theory on Riemann Surfaces_, Fortsch. Phys. ([arXiv:1709.02496](https://arxiv.org/abs/1709.02496))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _D-type Conformal Matter and SU/USp Quivers_, JHEP06(2018)058 ([arXiv:1802.00620](https://arxiv.org/abs/1802.00620))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _Compactifications of ADE conformal matter on a torus_, JHEP09(2018)110 ([arXiv:1806.07620](https://arxiv.org/abs/1806.07620))

* Shlomo S. Razamat, Gabi Zafrir, _Compactification of 6d minimal SCFTs on Riemann surfaces_, Phys. Rev. D 98, 066006 (2018) ([arXiv:1806.09196](https://arxiv.org/abs/1806.09196))

* [[Jin Chen]], [[Babak Haghighat]], [[Shuwei Liu]], [[Marcus Sperling]], _4d $\mathcal{N}=1$ from 6d D-type $\mathcal{N}=(1,0)$_,  J. High Energ. Phys. **2020** 152 (2020) [[arXiv:1907.00536](https://arxiv.org/abs/1907.00536), doi:10.1007/JHEP01(2020)152]

[[!include flavor on heterotic M5-branes -- references]]

[[!redirects D=6 N=(1,0) SCFTs]]


D=6 N=(2,0) SCFT

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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Super-Geometry
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[[!include supergeometry - contents]]
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#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

According to the [classification of superconformal symmetry](supersymmetry#ClassificationSuperconformal), there should exist [[superconformal field theories]] in 6 dimensions...

[[!include superconformal symmetry -- table]]

...with $(2,0)$-[[supersymmetry]], that contain a [[self-dual higher gauge theory]] whose field configurations are [[connections on a 2-bundle]] (a [[circle n-bundle with connection|circle 2-bundle with connection]] in the abelian case).

This was derived by [Claus, Kallosh & van Proeyen 1997](#ClausKalloshProeyen97), in the abelian case and to low order, as the small fluctuations of the [[Green-Schwarz sigma-model]] of the [[M5-brane]] around the embedding in the [[asymptotic boundary]] of the [[AdS-spacetime]] that is the [[near-horizon geometry]] of the [[black brane|black]] M5-brane.

In accord with this the [AdS7-CFT6](AdS-CFT#AdS7CFT6) correspondence predicts that the nonabelian 6d theory is the corresponding theory for $N$ coincident M5-branes.

In the non-abelian case it is expected ([Witten 07](#Witten07)) that the compactification of such theories are at the heart of the phenomenon that leads to [[S-duality]] in [[super Yang-Mills theory]] and further to [[geometric Langlands duality]] ([Witten 09](#Witten09)).

Due to the [[self-dual higher gauge theory|self-duality]] a characterization of these theories by an [[action functional]] is subtle. Therefore more direct descriptions are still under investigation (for instance [SSW11](#SSW11)). Review includes ([Moore11](#Moore11), [Moore 12](#Moore12)).

## Properties

### Geometric engineering
  {#GeometricEngineering}

For [[geometric engineering]] of the [[6d (2,0)-superconformal QFT]], see at _[duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations -- Geometric engineering of 6d (2,0)-SCFT](duality+between+M-theory+on+Z2-orbifolds+and+type+IIB+string+theory+on+K3-fibrations#GeometricEngineeringOfThe6d2SuperconformalQFT)_.

### Holographic dual

Under [[AdS-CFT|AdS7/CFT6]] the 6d $(2,0)$-superconformal QFT is supposed to be dual to [[11-dimensional supergravity|M-theory]] on [[anti de Sitter spacetime]] $AdS_7 \times S^4$.

See [[AdS/CFT correspondence]] for more on this.

### Realization of quantum chromodynamics

See at _[[AdS-QCD correspondence]]_.

### Solitonic 1-branes

The 5d $(2,0)$-[[SCFT]] has tensionless 1-[[brane]] configurations. From the point of view of the ambient [[11-dimensional supergravity]] these are the boundaries of [[M2-branes]] ending on the [[M5-branes]]. ([GGT](#GGT))

### Compactification on a Riemann surface and AGT correspondence
 {#CompactificationOnARiemannSurface}

\begin{imagefromfile}
  "file_name": "6d_qft_graph.gif",
  "width": 600,
  "float": "right",
  "margin": {
    "top": 0,
    "right": 20,
    "bottom": 10,
    "left": 20,
    "unit": "px"
  },
  "alt": "Compactification diagram"
\end{imagefromfile}

> (graphics taken from ([Workshop 14](#Workshop14)))

The [[Kaluza-Klein mechanism|compactification]] of the 5-brane on a [[Riemann surface]] yields as [[worldvolume]] [[theory (physics)|theory]] [[N=2 D=4 super Yang-Mills theory]]. See at _[N=2 D=4 SYM -- Construction by compactification of 5-branes](N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes)_.

The _[[AGT correspondence]]_ relates  the [[partition function]] of $SU(2)^{n+3g-3}$-[[N=2 D=4 super Yang-Mills theory]] obtained by [[Kaluza-Klein mechanism|compactifying]] the $6d$ M5-brane theory on a [[Riemann surface]] $C_{g,n}$ of [[genus]] $g$ with $n$ punctures to 2d [[Liouville theory]] on $C_{g,n}$.

More generally, this kind of construction allows to describe the 6d (2,0)-theory as a "[[2d SCFT]] with values in [[super Yang-Mills theory|4d SYM]]". See at _[[AGT correspondence]]_ for more on this.

### Twistor space description

Famously the solutions to [[self-dual Yang-Mills theory]] in [[dimension]] 4 can be obtained as images of degree-2 cohomology classes under the [[Penrose-Ward twistor transform]]. Since the 6d QFT on the [[M5-brane]] contains a 2-form [[self-dual higher gauge field]] it seems natural to expect that it can be described by a higher analogy of the twistor transform. For references exploring this idea see at _[twistor space -- References -- Application to the self-dual 2-form field in 6d](twistor+space#ReferencesApplicationToSelfDual2FormField)_.

## Related entries

* [[little string theory]]

* [[D=6 N=(1,0) SCFT]]

* [[D=6 supergravity]]

* [[D=2 N=(2,0) SCFT]]

* [[D=5 super Yang-Mills theory]]

* [[M5-brane elliptic genus]]

[[!include gauge theory from AdS-CFT -- table]]

[[!include superconformal symmetry -- table]]

## References

### General

The first indication of a 6d theory with a self-dual 2-form field appears in

* {#Witten95} [[Edward Witten]], section 1 of _Some comments on string dynamics_ ([hepth/9507121](http://arxiv.org/abs/hep-th/9507121))

Derivation of the abelian 6d theory to low order as the small fluctuations of the [[Green-Schwarz sigma-model]] of the [[M5-brane]] around a solution embedding as the asymptotic boundary of the [[AdS-spacetime]] [[near-horizon geometry]] of a [[black brane|black]] 5-brane is due to

* {#ClausKalloshProeyen97} P. Claus, [[Renata Kallosh]], [[Antoine Van Proeyen]], _M 5-brane and superconformal $(0,2)$ tensor multiplet in 6 dimensions_, Nucl. Phys. B **518** (1998) 117-150 [doi:10.1016/S0550-3213(98)00137-0, [arXiv:hep-th/9711161](http://arxiv.org/abs/hep-th/9711161)]

General survey:

* {#Moore11} [[Greg Moore]], _On the role of six‐dimensional $(2,0)$-theories in recent developments in Physical Mathematics_, talk at [Strings 2011](http://www2.physics.uu.se/external/strings2011/) ([pdf slides](http://www.physics.rutgers.edu/~gmoore/Strings2011FinalPDF.pdf))

* {#Moore12} [[Greg Moore]], _Applications of the six-dimensional (2,0) theories to Physical Mathematics_, [Felix Klein lectures Bonn (2012)](http://www.hcm.uni-bonn.de/events/eventpages/felix-klein-lectures/applications-of-the-six-dimensional-20-theories-to-physical-mathematics/) ([pdf](http://www.physics.rutgers.edu/~gmoore/FelixKleinLectureNotes.pdf), [[MooreKleinLectures12.pdf:file]])

* {#Workshop14}  [[Qiaochu Yuan]]: _[lecture notes](https://math.berkeley.edu/~qchu/Notes/6d/)_ for _[Mathematical Aspects of Six-Dimensional Quantum Field Theories](http://www.math.northwestern.edu/~celliott/workshop/)_, Berkeley, December 8th-12th, 2014 at the University of California, Berkeley

Discussion of [[anomaly cancellation]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], Kazuya Yonekura, _Anomaly polynomial of general 6d SCFTs_, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 ([arXiv:1408.5572](https://arxiv.org/abs/1408.5572))

* [[Hiroyuki Shimizu]], _Aspects of anomalies in 6d superconformal field theories_, Tokyo 2018 ([spire:1802462](https://inspirehep.net/literature/1802462), [[ShimizuAnomaliesIn6dSCFT.pdf:file]])

Review of construction from [[F-theory]] [[KK-compactification]]:

* {#HeckmanRudelius18} [[Jonathan Heckman]], [[Tom Rudelius]], _Top Down Approach to 6D SCFTs_, J. Phys. A: Math. Theor. 52 093001 2018 ([arXiv:1805.06467](https://arxiv.org/abs/1805.06467), [doi:10.1088/1751-8121/aafc81](https://doi.org/10.1088/1751-8121/aafc81))

New approach to construction of candidate [[Lagrangian densities]] for [[D=6 N=(2,0) SCFTs]]:

* [[Neil Lambert]], _$(2,0)$ Lagrangian Structures_, Physics Letters B Volume 798, 10 November 2019, 134948 ([arXiv:1908.10752](https://arxiv.org/abs/1908.10752))

See also the references and discussion at _[[M5-brane]]_.

### Compactification to 5d super-Yang-Mills

[[KK-compactification]] on [[circle]] [[fibers]] to [[D=5 super Yang-Mills theory]] is discussed in (see also at [[Perry-Schwarz Lagrangian]]):

* [[Nathan Seiberg]], Sec. 7 of _Notes on Theories with 16 Supercharges_, Nucl. Phys. Proc. Suppl. 67:158-171, 1998 ([arXiv:hep-th/9705117](https://arxiv.org/abs/hep-th/9705117))

* {#Douglas11} [[Michael Douglas]], _On D=5 super Yang-Mills theory and (2,0) theory_, JHEP 1102:011, 2011 ([arXiv:1012.2880](https://arxiv.org/abs/1012.2880))

* [[Neil Lambert]], Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, _M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills_, JHEP 1101:083, 2011 ([arXiv:1012.2882](https://arxiv.org/abs/1012.2882))

* {#Witten11} [[Edward Witten]], Sections 4 and 5 of _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))

* {#Hu13} Shan Hu, _6D $(2,0)$ theory and M5 branes: A KK mode approach_, 2013 ([hdl:1969.1/151094](http://hdl.handle.net/1969.1/151094))

* [[Chris Hull]], [[Neil Lambert]], _Emergent Time and the M5-Brane_, JHEP06(2014)016 ([arXiv:1403.4532](https://arxiv.org/abs/1403.4532))

* [[Andreas Gustavsson]], _Five-dimensional Super-Yang-Mills and its Kaluza-Klein tower_. JHEP01(2019)222 ([arXiv:1812.01897](https://arxiv.org/abs/1812.01897))

* {#Lambert19} [[Neil Lambert]], Sec. 3.1 and 3.4.3. of _Lessons from M2's and Hopes for M5's_, _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html), August 2018_ Fortschritte der Physik Volume 67, Issue 8-9 2019 ([arXiv:1903.02825](https://arxiv.org/abs/1903.02825), [doi:10.1002/prop.201910011]( https://doi.org/10.1002/prop.201910011), [slides pdf](http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4))

### Compactification to 4d super-Yang-Mills
 {#CompactificationTo4DSYM}

[[KK-compactification]] of the $D=6$ $\mathcal{N} = (2,0)$-CFT on [[torus]] [[fibers]] to [[D=4 super Yang-Mills theory]] and the related [[electric-magnetic duality]]/[[S-duality]] in 4-dimensions:

* {#Witten07} [[Edward Witten]], _[[Conformal field theory in four and six dimensions]]_ in [[Ulrike Tillmann]] (ed.) _Topology, geometry and quantum field theory_ LMS Lecture Note Series (2004) ([arXiv:0712.0157](http://arxiv.org/abs/0712.0157))

and the resulting relation to the [[geometric Langlands correspondence]]:

* {#Witten09} [[Edward Witten]], *Geometric Langlands From Six Dimensions*, in Peter Kotiuga (ed.) _A Celebration of the Mathematical Legacy of Raoul Bott_, CRM Proceedings & Lecture Notes **50** AMS (2010) [[arXiv:0905.2720](http://arxiv.org/abs/0905.2720), [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/crmp-50)]

For more references on this see at _[[N=2 D=4 super Yang-Mills theory]]_ the section _[References - Constructions from 5-branes](N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes)_.

Relation to [[BFSS matrix model]] on [[tori]]:

* [[Micha Berkooz]], [[Moshe Rozali]], [[Nathan Seiberg]], _Matrix Description of M-theory on $T^3$ and $T^5$_ ([arXiv:hep-th/9704089](http://arxiv.org/abs/hep-th/9704089))

The [[KK-compactification]] specifically of the [[D=6 N=(1,0) SCFT]] to [[D=4 N=1 super Yang-Mills]]:

* [[Ibrahima Bah]], [[Christopher Beem]], Nikolay Bobev, Brian Wecht, _Four-Dimensional SCFTs from M5-Branes_ ([arXiv:1203.0303](http://arxiv.org/abs/1203.0303))

* Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _$4d$ $\mathcal{N} = 1$ from $6d (1,0)$_, J. High Energ. Phys. (2017) 2017: 64 ([arXiv:1610.09178](https://arxiv.org/abs/1610.09178))

* [[Ibrahima Bah]], [[Amihay Hanany]], Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, _$4d$ $\mathcal{N}=1$ from $6d$ $\mathcal{N}=(1,0)$ on a torus with fluxes_ ([arXiv:1702.04740](https://arxiv.org/abs/1702.04740))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _E-String Theory on Riemann Surfaces_, Fortsch. Phys. ([arXiv:1709.02496](https://arxiv.org/abs/1709.02496))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _D-type Conformal Matter and SU/USp Quivers_, JHEP06(2018)058 ([arXiv:1802.00620](https://arxiv.org/abs/1802.00620))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _Compactifications of ADE conformal matter on a torus_, JHEP09(2018)110 ([arXiv:1806.07620](https://arxiv.org/abs/1806.07620))

* Shlomo S. Razamat, Gabi Zafrir, _Compactification of 6d minimal SCFTs on Riemann surfaces_, Phys. Rev. D 98, 066006 (2018) ([arXiv:1806.09196](https://arxiv.org/abs/1806.09196))

* [[Jin Chen]], [[Babak Haghighat]], [[Shuwei Liu]], [[Marcus Sperling]], _4d $\mathcal{N}=1$ from 6d D-type $\mathcal{N}=(1,0)$_,  J. High Energ. Phys. **2020** 152 (2020) [[arXiv:1907.00536](https://arxiv.org/abs/1907.00536), doi:10.1007/JHEP01(2020)152]

### Compactification to 2d CFT
 {#CompactificationTo2dCFT}

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:

* [[Abhijit Gadde]], [[Sergei Gukov]], [[Pavel Putrov]], *Fivebranes and 4-manifolds*, in: *Arbeitstagung Bonn 2013*, Progress in Mathematics **319**, Birkhäuser (2016) [[arXiv:1306.4320](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/10.1007/978-3-319-43648-7_7)]

* [[Mykola Dedushenko]], [[Sergei Gukov]], [[Pavel Putrov]], *Vertex algebras and 4-manifold invariants*, Chapter 11 in: *Geometry and Physics: Volume I*  (2018) 249–318 [[arXiv:1705.01645](https://arxiv.org/abs/1705.01645), [doi:10.1093/oso/9780198802013.003.0011](https://doi.org/10.1093/oso/9780198802013.003.0011)]

* [[Boris Feigin]], [[Sergei Gukov]], $VOA[M_4]$, J. Math. Phys. **61** 012302 (2020) [[arXiv:1806.02470](https://arxiv.org/abs/1806.02470), [doi:10.1063/1.5100059](https://doi.org/10.1063/1.5100059)]

In relation to [[M5-brane elliptic genus]]:

* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*,  J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), doi:10.1007/JHEP05(2021)084, [spire:1704312](https://inspirehep.net/literature/1704312)]

and in relation to [[QFT with defects|defects]]:

* [[Jin Chen]], [[Wei Cui]], [[Babak Haghighat]], [[Yi-Nan Wang]], *SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds*, JHEP **2023** 208 (2023) [[arXiv:2305.09734](https://arxiv.org/abs/2305.09734), doi:10.1007/JHEP11(2023)208]

### ADE classification

Discussion of the [[ADE classification]] of the 6d theories includes, after ([Witten95](#Witten95))

* Julie D. Blum, [[Kenneth Intriligator]], _New Phases of String Theory and 6d RG Fixed Points via Branes at Orbifold Singularities_, Nucl.Phys.B506:199-222,1997 ([arXiv:hep-th/9705044](http://arxiv.org/abs/hep-th/9705044))

* {#HeckmannMorrisonVafa13} [[Jonathan Heckman]], [[David Morrison]], [[Cumrun Vafa]], _On the Classification of 6D SCFTs and Generalized ADE Orbifolds_ ([arXiv:1312.5746](http://arxiv.org/abs/1312.5746))

### Models and special properties
 {#ModelsAndSpecialProperties}

Realization of the 6d theory in [[F-theory]] is discussed in ([Heckmann-Morrison-Vafa 13](#HeckmannMorrisonVafa13)).

A proposal for related higher nonabelian differential form data is made in

* {#SSW11} [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, _(1,0) superconformal models in six dimensions_,  J. High Energ. Phys. **2011** 62 (2011) [[arXiv:1108.4060](http://arxiv.org/abs/1108.4060), doi:10.1007/JHEP12(2011)062]

Since by [[transgression]] every nonabelian [[principal 2-bundle]]/[[gerbe]] gives rise to some kind of nonabelian 1-bundle on [[loop space]] it is clear that some parts (but not all) of the nonabelian gerbe theory on the 5-brane has an equivalent reformulation in terms of ordinary gauge theory on the [[loop space]] of the 5-brane and possibly for gauge group the [[loop group]] of the original gauge group.

Comments along these lines have been made in

* [[Andreas Gustavsson]], _Selfdual strings and loop space Nahm equations_ ([arXiv:0802.3456](http://arxiv.org/abs/0802.3456)).

In fact, via the [[strict 2-group]] version of the [[string 2-group]] there is a local gauge in which the loop group variables appear already before transgression of the 5-brane gerbe to loop space. This is discussed from a [[holographic principle|holographic]] point of view in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_

### On the holographic dual
 {#ReferencesOnTheHolographicDual}

The basics of the relation of the 6d theory to a 7d theory under [[AdS-CFT]] is reviewed [[holographic principle|holographic duality]]

* {#Maldacena} [[Juan Maldacena]], _The Large N limit of superconformal field theories and supergravity_, Adv. Theor. Math. Phys. 2:231, 1998, [hep-th/9711200](http://arxiv.org/abs/hep-th/9711200); _Wilson loops in Large N field theories_, Phys. Rev. Lett. __80__ (1998) 4859, [hep-th/9803002](http://arxiv.org/abs/hep-th/9803002)

The argument that the abelian [[7d Chern-Simons theory]] of a [[circle n-bundle with connection|3-connection]] yields this way the [[conformal blocks]] of the abelian [[self-dual higher gauge theory]] of the 6d theory on a _single_ brane is due to

* {#WittenI} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ J. Geom. Phys.22:103-133,1997 ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))

* {#Witten98} [[Edward Witten]], _AdS/CFT Correspondence And Topological Field Theory_ JHEP 9812:012,1998 ([arXiv:hep-th/9812012](http://arxiv.org/abs/hep-th/9812012))

The nonabelian generalization of this 7d action functional that follows from taking the quantum corrections ([[Green-Schwarz mechanism]] and flux quantization) of the [[supergravity C-field]] into account is discussed in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_

See also

* {#HEGK} [[Eric D'Hoker]], John Estes, Michael Gutperle, Darya Krym,

  _Exact Half-BPS Flux Solutions in M-theory I
Local Solutions_ ([arXiv:0806.0605](http://arxiv.org/abs/0806.0605))

   _Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to $AdS_7 \times S^4$ ([arXiv:0810.4647](http://arxiv.org/abs/0810.4647))

### Solitonic 1-brane excitations

* {#GGT} [[Jerome Gauntlett]], [[Joaquim Gomis]], [[Paul Townsend]], _BPS Bounds for Worldvolume Branes_ ([arXiv:hep-th/9711205](http://arxiv.org/abs/hep-th/9711205))

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Threebrane Soliton of the M-Fivebrane_ ([arXiv:hep-th/9710033](http://arxiv.org/abs/hep-th/9710033))

[[!include D=6 N=(2,0) SCFT as an extended functorial field theory -- references]]

[[!redirects 6d (2,0)-susy QFT]]
[[!redirects 6d (2,0)-superconformal QFT]]

[[!redirects 6d (2,0)-superconformal field theory]]
[[!redirects 6d (2,0)-superconformal field theories]]

[[!redirects 6d (2,0)-superconformal SCFT]]
[[!redirects 6d (2,0)-superconformal SCFTs]]

[[!redirects 6d superconformal gauge field theory]]
[[!redirects 6d superconformal gauge field theories]]

[[!redirects D=6 N=(2,0) SCFT]]
[[!redirects D=6 N=(2,0) SCFTs]]

[[!redirects D=6 N=(1,1) SCFT]]

[[!redirects 6d (2,0)-supersymmetric QFT]]

[[!redirects 6dN20SCFT]]


D=6 N=(2,0) SCFT as an extended functorial field theory -- references

### The $D=6$, $\mathcal{N}=(2,0)$ SCFT as an extended functorial field theory
 {#TheoryXAsAnFQFTReferences}

On the (conjectural) suggestion to view at least some aspects of the [[D=6 N=(2,0) SCFT]] (such as its [[quantum anomaly]] or its image as a [[2d TQFT]] under the [[AGT correspondence]]) as a [[functorial field theory]]   given by a [[functor]] on a suitable [[cobordism category]], or rather as an [[extended TQFT|extended]] such [[FQFT]], given by an [[n-functor]] (at least a [[2-functor]] on a [[(∞,2)-category of cobordisms|2-category of cobordisms]]):

* {#Witten09} [[Edward Witten]], Section 1 of: _Geometric Langlands From Six Dimensions_, in Peter Kotiuga (ed.) _A Celebration of the Mathematical Legacy of Raoul Bott_, CRM Proceedings & Lecture Notes Volume: 50, AMS 2010 ([arXiv:0905.2720](http://arxiv.org/abs/0905.2720), [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/crmp-50))

* [[Daniel Freed]], _[[4-3-2 8-7-6]]_, talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_ Dec 2012 ([pdf](https://people.maths.ox.ac.uk/tillmann/ASPECTSfreed.pdf), [[Freed432876.pdf:file]])

* [[Daniel Freed]], p. 32 of: _The cobordism hypothesis_, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, ([arXiv:1210.5100](http://arxiv.org/abs/1210.5100), [doi:10.1090/S0273-0979-2012-01393-9](https://doi.org/10.1090/S0273-0979-2012-01393-9))

* [[Daniel Freed]], [[Constantin Teleman]], _Relative quantum field theory_, Commun. Math. Phys. 326, 459–476 (2014) ([arXiv:1212.1692](https://arxiv.org/abs/1212.1692), [doi:10.1007/s00220-013-1880-1](https://doi.org/10.1007/s00220-013-1880-1))

* [[David Ben-Zvi]], _Theory $\mathcal{X}$ and Geometric Representation Theory_, talks at _[Mathematical Aspects of Six-Dimensional Quantum Field Theories](http://www.ihes.fr/~celliott/workshop/)_ IHES 2014, notes by [[Qiaochu Yuan]] ([pdf I](https://math.berkeley.edu/~qchu/Notes/6d/Day%201,%20Talk%201,%20Ben-Zvi.pdf), [pdf II](https://math.berkeley.edu/~qchu/Notes/6d/Day%202,%20Talk%201,%20Ben-Zvi.pdf), [pdf III](https://math.berkeley.edu/~qchu/Notes/6d/Day%203,%20Talk%201,%20Ben-Zvi.pdf))

* [[David Ben-Zvi]], _Algebraic geometry of topological field theories_, talk at _[Reimagining the Foundations of Algebraic Topology April 07, 2014 - April 11, 2014](https://www.msri.org/workshops/689)_ ([web video](https://www.msri.org/workshops/689/schedules/18216))

* [[Lukas Müller]], _Extended Functorial Field Theories and Anomalies in Quantum Field Theories_ ([arXiv:2003.08217](https://arxiv.org/abs/2003.08217))


D=6 super Yang-Mills theory

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###Context###
#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#### Super-Geometry
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[[!include supergeometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

[[super Yang-Mills theory]] in [[dimension]] 6

## Related entries

* [[D=4 super Yang-Mills theory]]

* [[D=5 super Yang-Mills theory]]

* [[D=10 super Yang-Mills theory]]

* [[D=6 N=(1,0) SCFT]]

* [[D=6 N=(2,0) SCFT]]

## References

* Shuichi Yokoyama, _Supersymmetry Algebra in Super Yang-Mills Theories_, JHEP09(2015)211 ([arXiv:1506.03522](https://arxiv.org/abs/1506.03522))

* Freddy Cachazo, Alfredo Guevara, Matthew Heydeman, Sebastian Mizera, John H. Schwarz, Congkao Wen, _The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps_, JHEP09(2018)125 ([arXiv:1805.11111](https://arxiv.org/abs/1805.11111))

* Evgeny Ivanov, _Classical and quantum superfield invariants in $\mathscr{N}$ = (1, 1), 6D SYM theory_, Journal of Physics: Conference Series, Volume 965, conference 1 ([doi:10.1088/1742-6596/965/1/012021](https://iopscience.iop.org/article/10.1088/1742-6596/965/1/012021))