nLab D=4 supergravity





physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics




supergravity in dimension 4.

The maximally supersymmetric N=8N = 8-version arises from type II supergravity in 10 dimension by compactification on a 6-torus.

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


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

See also at supergravity – History.

and see:

Geometric construction

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

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

based on

  • Murat Gunaydin, S. McReynolds, M. Zagermann, Unified N=2N=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

See also

  • Salih Kibaroğlu, Oktay Cebecioğlu, D=4D=4 supergravity from the Maxwell-Weyl superalgebra (arXiv:1812.09861)

On N=1N=1 d=4d = 4 supergravity

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

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=1N=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=4D = 4, N=1N=1 supergravity, Class.Quant.Grav. 19 (2002) 2107-2124 (arXiv:gr-qc/0110072)

Textbook accounts:


  • Robin Ducrocq, Michel Rausch de Traubenberg, Mauricio Valenzuela, A pedagogical discussion of N=1N = 1 four-dimensional supergravity in superspace (arXiv:2104.06671)

Survey of the formulation in superspace:

See also:

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

More on the case N=2:

Review of the superspace formulations:

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


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

Its SO(8)SO(8)-gauged version was obtained in

and further gaugings by non-compact gauge groups in

N=8N = 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=8N=8 4d supergravity as as perturbative quantum field theory (perturbative quantum gravity) is discussed in

and via KLT relations:

surveyed in

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)E_{7(7)} Constraints on Counterterms in N= 8 Supergravity_, Phys. Lett. B694, 265 (2010).

Arguments against finiteness to all orders include

See also

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

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:

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×T 2/ 2K_3 \times T^2 / \mathbb{Z}_2, JHEP 0303:044,2003 (arXiv:hep-th/0302174)

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

Another question is whether N=8N = 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=8N = 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=8,6,5N = 8, 6, 5 supergravity in D=4D = 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 𝒩=2\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

Last revised on April 25, 2024 at 06:01:16. See the history of this page for a list of all contributions to it.