Gravity and Strings

448 From eleven to four dimensions
string theories which would be different, dual, manifestations of the same unique
theory. 1
Our interest is mainly in four-dimensional string effective-field theories and classical solutions
and their connection to higher-dimensional theories and solutions. It is then natural
to start by introducing 11-dimensional supergravity and then performing the reduction on
a circle to find the type-IIA superstring effective action. We will do this in Section 16.1.
Since we are interested in classical solutions, we will study only the bosonic sectors of
these theories. However, we will also need the supersymmetry transformation rules for the
fermions in order to study their unbroken supersymmetries.
To study T duality between the effective actions of the type-IIA and -IIB theories, following
the philosophy of Section 15.2, we will have to perform dimensional reduction of both
theories to nine dimensions. The reduction of the IIA theory will be done in Section 16.3
whereas the reduction of the IIB theory will be postponed to the next chapter, in which we
will find the type-II Buscher rules.
The E8 × E8 heterotic string theory can be obtained by compactification of M theory on a
segment (the simplest orbifold), each E8 factor group living on one of the ten-dimensional
boundaries. From the point of view of effective actions, we can easily obtain the heterotic
string effective action (without the gauge fields) by compactifying 11-dimensional supergravity
on an orbifold, which amounts to the S 1 compactification which we carry out in
Section 16.1 followed by a truncation that we study in Section 16.4. A similar truncation
of the type-IIB theory that gives the type-I theory (again without the gauge fields) will be
studied in the next chapter and corresponds to the O9 plus 32 D9 construction of the type-I
SO(32) theory.
Further compactification increases the number of dualities: on the one hand, one can perform
T dualities in more directions that can also be rotated into each other. Also, in even
dimensions, new dualities that involve the Hodge-dualization of differential-form potentials
appear: in d = 4, vectors can be dualized, in d = 6 2-forms, and in d = 8 3-forms. Furthermore,
in odd dimensions, Hodge-dualization of differential forms can increase the number
of vector fields that can be rotated into other vector fields, enhancing the duality group.
Usually these dualities are manifest only in the Einstein frame and were known as hidden
symmetries of supergravity theories. In Section 16.5 we are going to study the toroidal
compactification of N = 1, d = 10 supergravity, the effective theory of the heterotic string
down to d = 4, as an example and we will find that, generically, the classical duality group 2
is O(n, n + 16) for compactification on an n-torus, all of it due to T duality, but, in d = 4,
vectors can be dualized into vectors and the symmetry is increased by the S-duality group
SL(2, R). (The duality groups that appear in toroidal compactifications of N = 2, d = 10
theories are given in Table 16.1.)
Finally, we are going to study the preservation of unbroken supersymmetry under duality
transformations in Section 16.6.
1 A 12-dimensional origin for M theory and type-IIB superstring theory by the name of F theory has also been
suggested.
2 Quantum effects such as charge quantization break the classical supergravity duality groups to discrete
subgroups, typically the ones obtained by restricting the matrix entries to taking integer values [583]. On
the other hand, if G is the classical duality group, the scalars parametrize a coset space G/H, where H is the
maximal compact subgroup of H. For the heterotic-string case H = O(n) × O(n + 16).