Gravity and Strings

14 String theory 407
generically called T dualities, and the archetype of them (known long before as T duality,
Tfrom “target space,” see e.g. [456]) relates string theories compactified on circles of dual
radii. This is an exact symmetry at all orders in string perturbation theory [33].
On top of these, there are the so-called S dualities which are non-perturbative in the string
coupling constant and cannot be studied using the standard worldsheet approach. In fact,
their existence in four-dimensional heterotic string theory was suggested by the existence
of the corresponding global symmetry in the effective action [397, 803, 842] which is nothing
but that of N = 4, d = 4 SUGRA. As we saw in Section 12.2, the equations of motion
of this theory are invariant under global SL(2, R) transformations, some of which invert
the dilaton field which can be interpreted as the string coupling constant. In general S dualities
are associated with this group 2 [583] and involve the inversion of the dilaton (string
coupling constant) and the interchange or electric and magnetic fields. Another interesting
example that we will study in Chapter 17 is N = 2B, d = 10 SUGRA, the effective-field
theory of the type-IIB superstring: the equations of motion of the supergravity theory are invariant
under global SL(2, R) rotations that invert the dilaton, which suggests the existence
of an S duality between type-IIB superstring theories.
The T- and S-duality groups are sometimes (in type-II theories) part of a bigger duality
group called in [583] the U duality group.
The interpretation as string dualities of the supergravity symmetries (the upper ← arrow)
is possible if the string spectra reflect the same duality properties. For T dualities, this is
easy to see in the perturbative spectrum, but S dualities necessarily imply the existence
of new non-perturbative states. Thus, using the upper ← relation, we start learning new
things about string theory and its spectrum. Needless to say, the worldvolume actions for
the corresponding states must also be related by the same dualities.
The symmetries of the supergravity theories also relate different solitonic solutions. In
some cases this has been used to generate new solitonic solutions out of old ones. In the
end one should be able to obtain complete duality-invariant families of solutions. Now,
the relation between effective-field theories and string theories can be used to relate solitonic
supergravity solutions to perturbative and non-perturbative string states (the lower ←
arrow). Whereas in the supergravity theories the duality groups are continuous, in string
theory quantum effects generically break them to the discrete subgroups that result from
the restriction from real numbers to integers. In particular, the S-duality group is broken
by charge quantization of the string states to SL(2, Z) [843]. The full spectrum is invariant
under that group [844].
To develop these relations, we must find a dictionary to interpret, in terms of string theory,
supergravity field configurations and symmetries, because only the fields associated
with massless string-theory modes appear in the effective field theories, but string dualities
often involve massive modes. However, these massive modes are usually charged and couple
to the massless long-range potentials that appear in supergravity, 3 and we know their
transformations under dualities, which help us to know how the massive and charged string
2 But not the other way around: the T-duality group can contain SL(2, R) subgroups.
3 The only bosonic fields apart from the potentials and the metric are the scalars, whose interpretation is
different: they represent coupling constants or geometrical data of the compactification space, moduli.