Gravity and Strings

19.4 Duality of string-theory solutions 551
19.4 Duality of string-theory solutions
In Section 19.1 we used string dualities to find and relate all the extended objects of string
and M theories. In the subsequent sections we have established a relation between those
objects and certain classical solutions of the string effective actions and d = 11 supergravity
using arguments based on the symmetries of the solutions which determine the dimensions
of the worldvolumes of the objects they describe, on the basis of the charges they carry and
the matching with p-brane sources.
On the other hand, in Chapters 15–17 we learned how string dualities manifest themselves
in string effective actions and, to close the loop, here we are going to see how the
duality relations between string states are realized as relations between solutions of the
effective actions. These relations are represented in Figures 19.4.1 and 19.4.1.
The three main types of duality relations that we are going to study are (i) those between
the solutions of d = 11 supergravity and solutions of N = 2A, d = 10 supergravity, via the
dimensional-reduction formulae Eqs. (16.35); (ii) those between solutions of N = 2A, d =
10 and N = 2B, d = 10 supergravity, via the type-II Buscher T-duality rules Eqs. (17.36)
and (17.37); and (iii) those between solutions of N = 2B, d = 10 supergravity, via SL(2, Z)
transformations Eqs. (17.21) or (17.23) and (17.24). We are also going to need the results
of Section 11.3.1 in order to perform reductions on transverse directions.
The supergravity duality transformations can also be used to construct new solutions.
We will study two families of solutions constructed in this way: pq strings and pq
5-branes.
19.4.1 N = 2A, d = 10 SUEGRA solutions from d = 11 SUGRA solutions
There are two basic p-brane solutions of d = 11 SUGRA: the M2- and M5-brane solutions
Eqs. (19.45) and (19.50). If they really describe the M2- and M5-brane states of M theory,
their reduction must give rise to the F1A, D2, D4, and S5A solutions Eqs. (19.56), (19.64)
and (19.60) of N = 2A, d = 10 SUEGRA [14, 335, 899] under double and direct dimensional
reductions, i.e. in a worldvolume direction (corresponding to branes wrapped in the
compact dimension) or in a transverse direction.
Double dimensional reductions are, by definition, made in a direction none of the fields
depends on, and one just has to rewrite the solution in d = 10 variables using Eqs. (16.35)
in a straightforward manner. The only subtlety is that, in order to have a non-trivial value
for ˆg = e ˆφ0, one must first rescale the compact worldvolume coordinate (that we call here
z) z → e 2 3 ˆφ0z in Eqs. (19.45) and (19.50).
Direct dimensional reductions are made precisely in one of the directions on which the
p-brane metric depends. We could substitute the harmonic function for another one independent
of the compact direction but, in that case, we would lose the relation to the
quantum object it represents. The right procedure is, as we explained in Section 11.3.1, to
construct first the correct solution that describes the p-brane in a transverse space with a
compact coordinate, which amounts to solving the Laplace equation in such a space, and
then Fourier-expand the solution, keeping only the zero mode. The solution is the same
harmonic function as that which describes an infinite periodic array of parallel p-branes
separated by a distance equal to the length of the compact direction. This harmonic function