Types of quantum field thories
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)
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)
Its -gauged version was obtained in
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.
Evidence for perturbative finiteness of 4d SuGra is discussed in
and via KLT relationsin
Arguments against finiteness to all orders include
There are two different off-shell formulations, the “old minimal”
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).
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
Textbook accounts include
Discussion of reduction from string theory includes
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 .
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.
Descent of 4d Sugra from type IIA string theory is reviewed for instance in