M2-brane is not understood well enough to check this, though this must

surely be possible.

Matching BPS brane tensions in nine dimensions

We can carry out additional tests of the proposed duality, and learn inter-

esting new relations at the same time, by matching BPS p-branes with p > 0

in nine dimensions. Only some of the simpler cases are described here. Let

us start with strings in nine dimensions. Trivial reduction of the type IIB

strings, that is, not wrapped on the circular dimension gives strings with the

same charges (p, q) and tensions T(p,q) in nine dimensions. The interesting

question is how these should be interpreted in M-theory. The way to answer

this is to start with an M2-brane of toroidal topology in M-theory and to

wrap one of its cycles on a (p, q) homology cycle of the spatial torus. The

minimal length of such a cycle is13

L(p,q) = 2πR11 |p ’ q„M |. (8.133)

13 This is understood most easily by considering the covering space of the torus, which is the

plane tiled by parallelograms. A closed geodesic curve on the torus is represented by a straight

line between equivalent points in the covering space, as shown in Fig. 8.3.

342 M-theory and string duality

Thus, this wrapping gives a string in nine dimensions, whose tension is

(11)

T(p,q) = L(p,q) TM2 . (8.134)

The superscript 11 emphasizes that this is measured in the 11-dimensional

metric. To compare with the type IIB string tensions, we use Eqs (8.130)

and (8.131) to deduce that

(11)

T(p,q) = β ’2 T(p,q) . (8.135)

This agrees with the result given earlier, showing that this is a correct in-

terpretation.

Fig. 8.3. In the covering space of the torus, which is the plane tiled by parallelo-

grams, a closed geodesic curve on the torus is represented by a straight line between

equivalent points.

To match 2-branes in nine dimensions requires wrapping the type IIB

D3-brane on the circle and comparing to the unwrapped M2-brane. The

wrapped D3-brane gives a 2-brane with tension 2πRB TD3 . Including the

metric conversion factor, the matching gives

TM2 = 2πRB β 3 TD3 . (8.136)

Combining this with Eqs (8.131) and (8.132) gives the identity

(TF1 )2

TD3 = , (8.137)

2πgs

in agreement with the tension formulas in Chapter 6. It is remarkable that

the M-theory/type IIB superstring theory duality not only relates M-theory

tensions to type IIB superstring theory tensions, but it even implies a rela-

tion involving only type IIB tensions.

Wrapping the M5-brane on the spatial torus gives a 3-brane in nine di-

mensions, which can be identi¬ed with the unwrapped type IIB D3-brane

in nine dimensions. This gives

TM5 AM = β 4 TD3 , (8.138)

8.4 M-theory dualities 343

which combined with Eqs (8.131) and (8.137) implies that

1

(TM2 )2 .

TM5 = (8.139)

2π

This corresponds to satisfying the Dirac quantization condition with the

minimum allowed product of charges. It also provides a check of the tensions

in (8.22).

An M-theory/SO(32) superstring duality

There is a duality that is closely related to the one just considered that

relates M-theory compacti¬ed on (S 1 / 2 ) — S 1 to the SO(32) theory com-

pacti¬ed on S 1 . Because of the similarity of the two problems, fewer details

are provided this time.

M-theory on a cylinder SO(32) on a circle

RO

R2

L1

Fig. 8.4. Duality between M-theory on a cylinder and SO(32) on a circle.

Since S 1 / 2 can be regarded as a line interval I, (S 1 / 2 ) — S 1 can be re-

garded as a cylinder. Let us choose its height to be L1 and its circumference

to be L2 = 2πR2 . The circumference of the circle on which the dual SO(32)

theory is compacti¬ed is LO = 2πRO as measured in the ten-dimensional

Einstein metric. This is illustrated in Fig. 8.4.

The SO(32) theory in ten dimensions has both a type I and a heterotic

description, which are S dual. As before, the duality can be explored by

matching supersymmetry-preserving (BPS) branes in nine dimensions. Re-

call that in the SO(32) theory, there is just one two-form ¬eld, and the

p-branes that couple to it are the SO(32) heterotic string and its magnetic

dual, which is a solitonic 5-brane. (From the type I viewpoint, both of these

are D-branes.) The heterotic string can give a 0-brane or a 1-brane in nine

dimensions, and the dual 5-brane can give a 5-brane or a 4-brane in nine

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 ¬nd the corresponding nine-dimensional p-branes from the M-

theory 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

L1

(HO)

gs = . (8.140)

L2

Thus, the SO(32) heterotic string is weakly coupled when the spatial cylin-

der of the M-theory compacti¬cation 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 di¬erent-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.