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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.
10-dimensional type II supergravity, heterotic supergravity
4-dimensional supergravity
Minmal 4d Supergravity was the first supergravity theory to be constructed, in
See also at supergravity – History.
Discussion in the D'Auria-Fré formulation of supergravity includes
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
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
The maximal $N=8$ supergravity in 4d was obtained by KK-reduction of 11-dimensional supergravity on a 7-torus in
Eugene Cremmer, Bernard Julia, The $SO(8)$ Supergravity, Nucl. Phys. B 159 (1979) 141 (spire)
Eugene Cremmer, Bernard Julia, Phys. Lett. 80B (1978) 48; Nucl. Phys. B159 (1979) 141.
Its $SO(8)$-gauged version was obtained in
Bernard de Wit, Hermann Nicolai, Phys. Lett. 108 B (1982) 285;
Bernard de Wit. Hermann Nicolai, Nucl. Phys. B208 (1982) 323
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)
Evidence for perturbative finiteness of $N=8$ 4d SuGra 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)
and via KLT relationsin
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)
Arguments for finiteness from E7 U-duality is discussed in
Arguments against finiteness to all orders include
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)
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, 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
See also
Textbook accounts include
Survey includes
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
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.
Descent of 4d $\mathcal{N} = 2$ Sugra from type IIA string theory is reviewed for instance in
Discussion of lifts of gauged 4d supergravity to string theory/M-theory includes