time can be viewed as a slab of thickness πR11 . The two ten-dimensional

boundaries are the orbifold singularities where the super Yang“Mills ¬elds

are localized. The two boundaries are sometimes called end-of-the-world

9-branes. Each of them carries the gauge ¬elds for an E8 group. This is a

very intuitive way of understanding why this theory has a gauge group that

is a product of two identical factors. The fact that the boundaries carry

E8 gauge supermultiplets is required for anomaly cancellation. There are

no anomalies in odd dimensions, except at a boundary. In this case the

boundary anomaly cancels only for the gauge group E8 . No other choice

works, as was explained in Chapter 5.

There is an alternative route by which one can deduce that M-theory com-

pacti¬ed on S 1 / 2 is dual to the E8 — E8 heterotic string in ten dimensions.

It uses the following sequence of dualities that have been introduced previ-

ously: (1) T-duality between the E8 — E8 heterotic string and the SO(32)

heterotic string; (2) S-duality between the SO(32) heterotic string and the

type I superstring; (3) T-duality between the type I superstring and the

type I superstring; (4) identi¬cation of the type I superstring as a type IIA

orientifold; (5) duality between the type IIA superstring and M-theory on a

circle. Quantitative details of this construction are described in Exercise 8.6.

M2-branes, with the topology of a cylinder, are allowed to terminate on

a boundary of the space-time, so that the boundary of the M2-brane is a

closed loop inside the end-of-the-world 9-brane. In this picture, an E8 — E8

heterotic string is a cylindrical M2-brane suspended between the two space-

time boundaries, with one E8 associated with each boundary. This cylinder

is well approximated by a string living in ten dimensions when the separation

πR11 is small, as indicated in Fig. 8.2. Since perturbation theory in gs is an

expansion about R11 = 0, the fact that there really are 11 dimensions and

that the string is actually a membrane is invisible in that approach. The

336 M-theory and string duality

tension of the heterotic string is therefore

TH = 2πR11 TM2 = (2π 2 )’1 . (8.121)

s

All of these statements are straightforward counterparts of statements con-

cerning the strongly coupled type IIA superstring theory.

There are two possible strong coupling limits of the E8 — E8 heterotic

string theory. One possibility is a limit in which both boundaries go to

in¬nity, so that one ends up with an 11 space-time geometry. This is the

¡

same limit as one obtains by starting with type IIA superstring theory and

letting R11 ’ ∞. The strongly coupled E8 —E8 heterotic string and the type

IIA superstring theory are identical in the 11-dimensional bulk. The only

thing that distinguishes them is the existence of boundaries in the former

case. The second possibility is to hold one boundary ¬xed as R11 ’ ∞.

This limit leads to a semi-in¬nite eleventh dimension. Since there is just

one boundary in this limit, there is just one E8 gauge group. This limit has

received very little attention in the literature. It is also possible to consider

11-dimensional geometries with more than two boundaries, and therefore

more than two E8 groups.

In studies of possible phenomenological applications of the strongly cou-

pled E8 —E8 heterotic string, a subject sometimes called heterotic M-theory,

one considers compacti¬cation of six more spatial dimensions (usually on

a Calabi“Yau space). An interesting possibility that does not arise in

Fig. 8.2. A cylindrical M2-brane suspended between two end-of-the-world 9-branes

is approximated by an E8 — E8 heterotic string as R11 ’ 0.

8.3 M-theory 337

the weakly coupled ten-dimensional description is that moduli of this six-

dimensional space, as well as other moduli (such as the vev of the dilaton),

can vary along the length of the compact eleventh dimension. Thus, for

example, one E8 theory can be more strongly coupled than the other one.

This is explored further in Chapter 10.

EXERCISES

EXERCISE 8.5

Use T-duality to deduce the tension of the type IIB Kaluza“Klein 5-brane.

SOLUTION

The type IIB KK5-brane is T-dual to the NS5-brane in the type IIA theory.

In the type IIA theory one can form the dimensionless combination

TNS5 1

= .

2 2π

TD2

Since this is a dimensionless number, it is preserved under T-duality ir-

respective of the coordinate frame used to measure distances. Under the

T-duality

TNS5 ’ TKK5 TD2 ’ 2πR9 TD3 .

and

Therefore, in the type IIB string frame

2

1 R9

2

TKK5 = (2πR9 TD3 ) = 2 . (8.122)

gs (2π)5 8

2π s

It is an interesting fact that this is proportional to the square of the radius.

2

EXERCISE 8.6

Show that the duality between M-theory on S 1 / 2 and the E8 —E8 heterotic

string is a consequence of previously discussed dualities.

SOLUTION

Consider the E8 — E8 heterotic string with string coupling gs and x9 coor-

dinate compacti¬ed on a circle of radius R9 . This is T-dual to the SO(32)

338 M-theory and string duality

heterotic string on a circle of radius

2

R9 = s /R9

and coupling

gs = s gs /R9 .

As discussed in Chapter 7, Wilson lines need to be turned on to give the

desired E8 — E8 gauge symmetry. Applying an S-duality transformation

then gives the type I string with

1 R9

gs = =

gs s gs

and

gs = ( s )3/2 /

R9 = R9 gs = R9 / R9 g s .

Another T-duality then gives the type I theory with

2

R9 = s /R9 = R9 s g s

and

’1/2

= (R9 / s )3/2 gs

gs = s gs /R9 .

In the bulk this is the type IIA theory, so we can use the type IIA/M-theory

duality to introduce R11 = gs s and p = (gs )1/3 s . A little algebra then