Ordinary gravity – as a classical field theory – is determined by an action functional on a space of Riemannian metrics on a given manifold: the Einstein-Hilbert action?.
In supergravity the manifold in question is equipped with a Spin structure and the Einstein-Hilbert action functional is extended to one on the space of Riemannian metrics and sections of a vector bundle associated to the spinor bundle?.
To qualify as a supersymmetric theory this action functional is required to be invariant under certain operations on the space of metrics and spinor sections that may mix both.
Roughly, an action functional of supergravity must be locally invariant under the action of a super Poincare Lie algebra in the way that ordinary gravity is locally invariant under the action of the ordinary Poincare Lie algebra.
In general this desired invariance is achieved only if the action functional is extended still a bit further to a space of metrics, spinor? sections and certain differential forms on the manifold.
The extra degress of freedom in supergravity theories, the spinor sections and possibly the differential forms, have physically the interpretation of fermionic and bosonic matter fields coupled to ordinary gravity. Therefore theories of supergravity may be regarded as ordinary theories of gravity and matter, subject to peculiar constraints on the nature of the matter fields.
The desired supersymmetry constraint in supergravity theories turns out to be rather restrictive. For instance there is, under mild assumptions, a unique maximally supersymmetric supergravity extension of the ordinary Einsten-Hilbert action on a 4-dimensional manifold. This in turn is obtained from the unique (under mild assumptions) maximally supersymmetric supergravity action functional on an 11-dimensional manifold by thinking of the 4-dimensional action function as being a dimensional reduction? of the 11-dimensional one.
This uniqueness (under mild conditions) is one reason for interest in supergravity theories. Another important reason is that supergravity theories tend to remove some of the problmes that are encountered when trying to realize gravity as a quantum field theory. Originally there had been high hopes that the maximally supersymmetric supergravity theory in 4-dimensions is fully renormalizable?. This couldn’t be shown computationally – until recently: triggered by new insights recently there there is lots of renewed activity on the renormalizability of maximal supergravity.
etc. pp.
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effectively realizes supergravity theories as higher gauge theories. See the discussion there.