Gravity and Strings

524 The extended objects of string theory
The D8-brane has with be associated with a d = 11 8- or 9-brane, but none of these is
explicitly known. As we explained in Section 16.2, the natural candidate would be the
KK9M-brane, but no associated solution of d = 11 supergravity corresponding to it is
known, and possibly no such solution exists. However, it may exist as a solution of a modified
theory [123, 142, 666]. Its inclusion is necessary for the consistency of the diagrams in
Figures 19.4.1 and 19.4.1 on pages 552 and 553.
19.1.1 The masses of string- and M-theory extended objects from duality
We can immediately use our knowledge of the duality relations between the extended objects
of string and M theories to find their masses when they are compactified on tori. All
we need to know is the duality transformation rules of compactification radii, the string
coupling constant and the string masses, and the mass of one of the extended objects. Let
us first recall the duality transformation rules.
In T duality, the relation between the compactification radii RA(B) and coupling constants
gA(B) of the type-IIA(B) theories (or any other pair of string theories) is given by
Eqs. (14.61) and (14.62), which we rewrite here for convenience:
RA,B = ℓ 2 s /RB,A, gA,B = gB,Aℓs/RB,A. (19.1)
In type-IIB S duality (with S = η and vanishing Ĉ (0) ), the transformation rules for the
string coupling constant and radii can be deduced froms Eqs. (17.21) and (17.24), and take
the forms
g ′ B = 1/gB, R ′ i = Ri/g 1 2
B .
(19.2)
These rules have to be supplemented by the following transformation rule for the masses,
which follows from Eq. (17.24) and the definition of mass, as we will explain in Section
19.3:
M ′ = g 1 2
B M.
(19.3)
In type-IIA/M-theory duality we have to use the relations Eqs. (16.48) (rewritten below)
between the string length and the 11-dimensional Planck length and between the string
coupling constant and the compactification radius of the 11th coordinate that we call here
R10 for convenience:
ℓs = −(11) 2
ℓ
Planck /[(2π)2 R10], gA = (2π) 2 R 2 10 /− ℓ
(11) 2
Planck
. (19.4)