nLab D=4 supergravity

Idea

supergravity in dimension 4.

The maximally supersymmetric $N = 8$-version arises from type II supergravity in 10 dimension by 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

• 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)

See also at supergravity – History.

and see:

• Yuri Manin, §5.7 in: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer (1988) [doi:10.1007/978-3-662-07386-5]

Geometric construction

Discussion in the D'Auria-Fré formulation of supergravity:

• Riccardo D'Auria, Pietro Fré, About bosonic rheonomic symmetry and the generation of a spin-1 field in $D=5$ supergravity, Nuclear Physics B

  173 3 (1980) 456-476 [doi:10.1016/0550-3213(80)90013-9]

  (on D=4 N=1 supergravity)

• Pietro Fré, Extended supergravity on the supergroup manifold: $N=3$ and $N=2$ theories, Nuclear Physics B

  186 1 (1981) 44-60 [doi:10.1016/0550-3213(81)90092-4]

  (on D=4 N=2 and D=4 N=3 supergravity)

• 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)

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)

• 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)

• 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)

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)

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)

• Lucrezia Ravera, Hidden Role of Maxwell Superalgebras in the Free Differential Algebras of D=4 and D=11 Supergravity (arXiv:1801.08860)

See also

• Salih Kibaroğlu, Oktay Cebecioğlu, $D=4$ supergravity from the Maxwell-Weyl superalgebra (arXiv: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)

• 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)

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)

Textbook accounts:

• 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)

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]

See also:

• Wikipedia, Supergravity – 4d N=1

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

More on the case 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)

• Edoardo Lauria, Antoine Van Proeyen, $\mathcal{N}=2$ Supergravity in $D=4,5,6$ Dimensions (arXiv: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]

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

Construction

Maximal $N=8$ supergravity in 4d was obtained by KK-reduction of 11-dimensional supergravity on a 7-torus:

• Eugene Cremmer, Bernard Julia: The $N=8$ supergravity theory. I. The Lagrangian, Phys. Lett. B 80 1-2 (1978) 48-51 [doi:10.1016/0370-2693(78)90303-9]

• Eugene Cremmer, Bernard Julia, The $SO(8)$ Supergravity, Nucl. Phys. B 159 (1979) 141 [doi:10.1016/0550-3213(79)90331-6, spire:140465]

Its $SO(8)$-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)

Review:

• Hermann Nicolai: $\mathcal{N}=8$ Supergravity, and beyond [arXiv:2409.18656]

$N = 8$ Perturbative quantum supergravity

For early results on 2-loop finiteness of perturbative quantum supergravity see there.

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)

• 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)

• 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, doi:10.1002/prop.201100037]

and via KLT relations:

• Zvi Bern, John Joseph Carrasco, Lance Dixon, Henrik Johansson, Radu Roiban, Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity (arXiv:1103.1848)

surveyed in

• Radu Roiban, Is Perturbative $\mathcal{N}= 8$ Supergravity Finite? (arXiv:hep-th/0702112)

• Lance Dixon, Ultraviolet Behavior of $N=8$ Supergravity (arXiv:1005.2703)

• Sergio Ferrara, Alessio Marrani, Quantum Gravity Needs Supersymmetry (arXiv:1201.4328)

• Renata Kallosh, An Update on Perturbative $N=8$ Supergravity (arXiv: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”, 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)

• Jacques Distler, Decoupling $N = 8$ supergravity (blog post)

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

• Freddy Cachazo, David Skinner, Gravity from Rational Curves, Phys. Rev. Lett. 110, 161301, 2012 (arXiv:1207.0741, arXiv:10.1103/PhysRevLett.110.161301)

• Tim Adamo, Gravity with a cosmological constant from rational curves, JHEP 1511: 098, 2015 (arXiv: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:

• Krzysztof A. Meissner, Hermann Nicolai, Standard Model Fermions and Infinite-Dimensional R-Symmetries, Phys. Rev. Lett. 121, 091601 (2018) (arXiv:1804.09606)

• Krzysztof A. Meissner, Hermann Nicolai, Planck Mass Charged Gravitino Dark Matter, Phys. Rev. D 100, 035001 (2019) (arXiv:1809.01441)

following the proposal towards the end of

• Murray Gell-Mann, introductory talk at Shelter Island II, 1983 (pdf)

  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

Discussion of gauged supergravity in 4d originates around (Cremmer-Julia 79 (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)

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

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:

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)

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

• Walter H. Baron, Uplifting Maximal Gauged Supergravities (arXiv:1512.05567)