Gravity and Strings

5
N = 1, 2, d = 4 supergravities
In the previous chapter, we introduced increasingly complex theories of gravity, starting
from GR, to accommodate fermions and we saw that the generalizations of GR that we had
to use could be thought of as gauge theories of the symmetries of flat spacetime.
Avery important development of the last few decades has been the discovery of supersymmetry
and its application to the theory of fundamental particles and interactions.
This symmetry relating bosons and fermions can be understood as the generalization of
the Poincaré orAdS groups which are the symmetries of our background spacetime to the
super-Poincaré or super-AdS (super-)groups which are the symmetries of our background
superspacetime, ageneralization of standard spacetime that has fermionic coordinates.
It is natural to construct generalizations of the standard gravity theories that can be understood
as gauge theories of the (super-)symmetries of the background (vacuum) superspacetime.
These generalizations are the supergravity (SUGRA) theories. Given that the kind
of fermions that one can have depends critically on the spacetime dimension, the SUGRA
theories that one can construct also depend critically on the spacetime dimension. Furthermore,
one can extend the standard bosonic spacetime in different ways by including more
than one (N) set of fermionic coordinates. This gives rise to additional supersymmetries
relating them and, therefore, to supersymmetric field theories and SUGRA theories with N
supersymmetries. The latter are also known as extended SUGRAs (SUEGRAs).
There is, thus, a large variety of supergravities, but not infinitely large, because the gauging
of supersymmetries with N > 8ind = 4 dimensions or N = 1ind = 11 needs the
inclusion of more than one graviton and/or fields of spin higher than 2, which we do not
know how to couple consistently.
We are going to study SUGRA theories because they provide an interesting extension of
the ideas we have reviewed so far and because the effective-field theories that describe the
behavior of superstrings at low energies are SUGRA theories.
Supersymmetry and SUGRA have been developed over the last several years and are
currently the object of extensive work, so we cannot give here a complete review of any of
these subjects. There are excellent books and reviews that cover most of the basic aspects,
though, for instance [150, 404, 912, 915, 916, 946, 948]. Reference [828] contains reprints
of many of the original articles on SUGRA.
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