String Theory and M-Theory

344 M-theory and string duality
dimensions. In each case, the issue is simply whether or not one cycle of the
brane wraps around the spatial circle.
Now let us find the corresponding nine-dimensional p-branes from the Mtheory
viewpoint and explore what can be learned from matching tensions.
The E8 × E8 string arises in ten dimensions from wrapping the M2-brane
on I. Subsequent reduction on a circle can give a 0-brane or a 1-brane. The
story for the M5-brane is just the reverse. Whereas the M2-brane must wrap
the I dimension, the M5-brane must not do so. As a result, it gives a 5-brane
or a 4-brane in nine dimensions according to whether or not it wraps around
the S 1 dimension. So, altogether, both pictures give the electric–magnetic
dual pairs (0, 5) and (1, 4) in nine dimensions.
From the p-brane matching one learns that the SO(32) heterotic string
coupling constant is
g (HO)
s
= L1
. (8.140)
L2
Thus, the SO(32) heterotic string is weakly coupled when the spatial cylinder
of the M-theory compactification is a thin ribbon (L1 ≪ L2). This
is consistent with the earlier conclusion that the E8 × E8 heterotic string
is weakly coupled when L1 is small. Conversely, the type I superstring is
weakly coupled for L2 ≪ L1, in which case the spatial cylinder is long and
thin. The 2 transformation that inverts the modulus of the cylinder, L1/L2,
corresponds to the type I/heterotic S duality of the SO(32) theory. Since it
is not a symmetry of the cylinder it implies that two different-looking string
theories are S dual. This is to be contrasted with the SL(2, ) modular
group symmetry of the torus, which accounts for the self-duality of the type
IIB theory.
The p-brane matching in nine dimensions also gives the relation
L1L 2 2TM2 =
T (HO)
1
2π
L 2 O
−1
, (8.141)
which is the analog of Eq. (8.132). As in that case, it tells us that, for
fixed modulus L1/L2, one has the scaling law LO ∼ A −3/4
, where AC =
C
L1L2 is the area of the cylinder. Equation (8.139) relating TM2 and TM5 is
reobtained, and one also learns that
T (HO)
5 = 1
(2π) 2
L2
In the heterotic string-frame metric, where T (HO)
1
L1
2
(T (HO)
1 ) 3 . (8.142)
is a constant, this implies