212 Supergravity and cosmology
includes a theory of branes of various dimensions. The fieldsxμ(σ)have their
own dynamics. The zero modes of the excitations of such extended objects are
coordinates of spacetime,xμ(σ)=xconstantμ +···. Thus the concept of spacetime
is an approximation to a full quantum theory of gravity!
Supergravity (gravity+supersymmetry) may be viewed as an approximate
effective description of a fundamental theory when the dependence on coordinates
of the world-volume is ignored. The smallest theory of supergravity includes
two types of fields, the graviton and the gravitino. Supergravity interacting with
matter multiplets includes also scalars, spinors and vectors. All these fields are
functions of the usual spacetime coordinatest,xin a four-dimensional spacetime.
The fundamental M-theory, which should encompass both supergravity and string
theory, at present experiences rapid changes. Over the last few years M-theory
and string theory focused its main attention on the superconformal theories
and adS/CFT (anti-de Sitter/conformal field theory) correspondence [1]. It has
been discovered that IIB string theory onadS 5 ×S^5 is related toSU( 2 , 2 | 4 )
superconformal symmetry. In particular, one finds theSU( 2 , 2 | 1 )superconformal
algebra from the anti-de Sitter compactification of the string theory with one-
quarter of the unbroken supersymmetry. These recent developments in M-theory
and non-perturbative string theory suggest that we should take afresh look at the
superconformal formulation underlying the supergravity.
The ‘phenomenological supergravity’ based on the most generalN = 1
supergravity [2] has an underlying superconformal structure. This has been
known for a long time but only recently the complete most generalN = 1
gauge theory superconformally coupled to supergravity was introduced [4]. The
theory has localSU( 2 , 2 | 1 )symmetry and no dimensional parameters. The phase
of this theory with spontaneously broken conformal symmetry gives various
formulations ofN=1 supergravity interacting with matter, depending on the
choice of theR-symmetry fixing.
The relevance of supergravity to cosmology is that it gives a framework of
an effective field theory in the background of the expanding universe and time-
dependent scalar fields. Let us remind here that the early universe is described by
an FRW metric which can be written in a form which is conformal to a flat metric:
ds^2 =a^2 (η)[−dη^2 +γijdxidxj]. (6.1)This fact leads to an interest in the superconformal properties of supergravity.
6.2 Superconformal symmetry, supergravity and cosmology
The most general four-dimensional N = 1 supergravity [2] describes a
supersymmetric theory of gravity interacting with scalars, spinors and vectors
of a supersymmetric gauge theory. It is completely defined by the choice of the
three functions: the superpotentialW[φ]and the vector couplingfab[φ]which
are holomorphic functions of the scalar fields (depend onφiand do not depend