supergravity in dimension 4.
The maximally supersymmetric -version arises from type II supergravity in 10 dimension by compactification on a 6-torus.
The -version arises from KK-reduction via M-theory on G2-manifolds. This hosts the super 2-brane in 4d.
Discussion in the D'Auria-Fré formulation of supergravity includes
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
Laura Andrianopoli, Riccardo D'Auria, Luca Sommovigo, , 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, , Gauged Supergravity coupled to Vector-Tensor Multiplets, Nucl.Phys.B851:1-29,2011 (arXiv:1103.4813)
- Murat Gunaydin, S. McReynolds, M. Zagermann, Unified Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions, JHEP 0509:026,2005 (arXiv:hep-th/0507227)
Bernard de Wit, Hermann Nicolai, Phys. Lett. 108 B (1982) 285;
Bernard de Wit. Hermann Nicolai, Nucl. Phys. B208 (1982) 323.
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.
Eugene Cremmer, Bernard Julia, Phys. Lett. 80B (1978) 48; Nucl. Phys. B159 (1979) 141.
Chris Hull, New Gauged , Supergravities, Class.Quant.Grav.20:5407-5424,2003 (arXiv:hep-th/0204156)
Evidence for perturbative finiteness of 4d SuGra is discussed in
Zvi Bern, Lance Dixon, Radu Roiban, Is 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)
and via KLT relationsin
Arguments for finiteness from E7 U-duality is discussed in
- N. Beisert, H. Elvang, D. Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, 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 the Workshop “Supergravity versus Superstring Theory in the Ultraviolet”, PennState Univ, PA USA, August 27-30 2009.
On , supergravity
- L. Andrianopoli, M. Bertolini, A. Ceresole, Riccardo D'Auria, S. Ferrara, Pietro Fré, T. Magri. supergravity and 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)
There are two different off-shell formulations, the “old minimal”
and the “new minimal” supergravity
V. Akulov, D. Volkov and V. Soroka, Theor. Math. Phys. 31 (1977) 285; 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, M. 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, SG Limit of Superstring/M-Theory Please Stand Up?, Phys.Lett.B492:178-186,2000 (arXiv:hep-th/0008206)
- Nicolas Boulanger, Mboyo Esole, A Note on the uniqueness of , supergravity, Class.Quant.Grav. 19 (2002) 2107-2124 (arXiv:gr-qc/0110072)
Textbook accounts include
Gauged 4d supergravity
Discussion of gauged supergravity in 4d originates around
(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 , JHEP 0303:044,2003 (arXiv:hep-th/0302174)
Perturbative finiteness properties of gauged 4d supergravity from ungauged 4d supergravity is discussed in BCDJR 11, p. 24:
Another question is whether 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 supergravity , which are known to have improved ultraviolet behavior at one loop .
and Ferrara-Marrani 12, p. 12:
Another interesting aspect  which should be implied by UV finiteness of supergravity in 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 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