String Theory and M-Theory

334 M-theory and string duality
Since the integral gives 2πR,
TD6 = (2πR)2
16πG11
= 2πR
16πG10
=
2π
(2πℓs) 7 gs
, (8.115)
where we have used R = gsℓs. This agrees with the value obtained in
Chapter 6.
There is a simple generalization of the above, the multi-center Taub–NUT
metric, that describes a system of N parallel D6-branes. The metric in this
case is
where
ds 2 = V (x)dx · dx + 1
dy +
V (x)
2 A · dx , (8.116)
B = − ∇V = ∇ × A and V (x) = 1 + R
2
N
α=1
1
. (8.117)
|x − xα|
Since this system is BPS, the tension and magnetic charge are just N times
the single D6-brane values.
A similar construction applies to other string theories compactified on
circles. Indeed, the type IIB superstring theory compactified on a circle
contains a Kaluza–Klein 5-brane, constructed in the same way as the D6brane,
which is the magnetic dual of the Kaluza–Klein 0-brane. A T-duality
transformation along the circular dimension transforms the type IIB theory
into the type IIA theory compactified on the dual circle. The Kaluza–Klein
0-brane is dual to a fundamental type IIA string wound on the dual circle.
Therefore, the Kaluza–Klein 5-brane must map to the magnetic dual of the
fundamental IIA string, which is the type IIA NS5-brane.
E8 × E8 heterotic string theory at strong coupling
Let us briefly review the Hoˇrava–Witten picture of the strongly coupled
E8 × E8 heterotic string theory. One starts with the strongly coupled type
IIA superstring theory, or equivalently M-theory on ¡ 9,1 × S 1 , and mods
out by a certain 2 symmetry, much like one does in deriving the type I
superstring theory from the type IIB superstring theory. The appropriate
2 symmetry in this case includes the following reversals:
x 11 → −x 11
and A3 → −A3. (8.118)
In particular, modding out by this 2 action implies that the zero mode of
the Fourier expansion of Aµνρ in the x 11 direction is eliminated from the