= (gs )2/3

R9 / p

and

2

R9

R11 = .

R9

Now we can decompactify R9 ’ ∞ at ¬xed R9 and p . Note that R11 ’ ∞

at the same time. On the one hand, this gives the ten-dimensional E8 — E8

heterotic string, with coupling constant gs , while on the other hand it gives

a dual M-theory description with a compact eleventh dimension that is an

interval of length πR9 satisfying the expected relation R9 = (gs )2/3 p . 2

8.4 M-theory dualities

The previous section showed that the strongly coupled type IIA superstring

and the strongly coupled E8 — E8 heterotic string have a simple M-theory

interpretation. There are additional dualities involving M-theory that relate

it to the other superstring theories as well as to itself.

8.4 M-theory dualities 339

An M-theory/type IIB superstring duality

M-theory compacti¬ed on a circle gives the type IIA superstring theory,

while type IIA superstring theory on a circle corresponds to type IIB super-

string theory on a dual circle. Putting these two facts together it follows

that there should be a duality between M-theory on a two-torus T 2 and type

IIB superstring theory on a circle S 1 . The M-theory torus is characterized

by an area AM and a modulus „M , while the IIB circle has radius RB . Let

us explore this duality directly without reference to the type IIA theory.

Speci¬cally, the plan is to compare various BPS states and branes in nine

dimensions.

Since all of the (p, q) strings in type IIB superstring theory are related

by SL(2, ) transformations,12 they are all equivalent, and any one of them

can be weakly coupled. However, when one is weakly coupled, all of the

others are necessarily strongly coupled. Let us consider an arbitrary (p, q)

string and write down the spectrum of its nine-dimensional excitations in

the limit of weak coupling using the standard string theory formulas given

in Chapter 6:

2

K

2

+ (2πRB W T(p,q) )2 + 4πT(p,q) (NL + NR ).

MB = (8.123)

RB

As before, K is the Kaluza“Klein excitation number and W is the string

winding number. NL and NR are excitation numbers of left-moving and

right-moving oscillator modes, and the level-matching condition is

NR ’ NL = KW. (8.124)

The plan is to use the formula above for all the (p, q) strings simultane-

ously. However, the formula is completely meaningless at strong coupling,

and at most one of the strings is weakly coupled. The appropriate trick in

this case is to consider only BPS states, that is, ones belonging to short

supersymmetry multiplets, since their mass formulas can be reliably extrap-

olated to strong coupling. They are easy to identify, being given by either

NL = 0 or NR = 0. (Ones with NL = NR = 0 are ultrashort.) In this way,

one obtains exact mass formulas for a very large part of the spectrum “ much

more than appears in any perturbative limit. Of course, the appearance of

this rich spectrum of BPS states depends crucially on the compacti¬cation.

There is a unique correspondence between the three integers W, p, q, where

p and q are coprime, and an arbitrary pair of integers n1 , n2 given by

(n1 , n2 ) = (W p, W q). The integer W is the greatest common divisor of n1

12 It is assumed here that p and q are coprime.

340 M-theory and string duality

and n2 . The only ambiguity is whether to choose W or ’W , but since W

is the (oriented) winding number and the (’p, ’q) string is the orientation-

reversed (p, q) string, the two choices are actually equivalent. Thus BPS

states are characterized by three integers K, n1 , n2 and oscillator excitations

corresponding to NL = |W K|, tensored with a 16-dimensional short mul-

tiplet from the NR = 0 sector (or vice versa). Note that the combination

|W |T(p,q) , which appears in Eq. (8.123), can be rewritten using Eq. (8.97)

in the form

|W |T(p,q) = |n1 ’ n2 „B |TF1 . (8.125)

Let us now consider M-theory compacti¬ed on a torus. If the two periods

in the complex plane, which de¬ne the torus, are 2πR11 and 2πR11 „M , then

AM = (2πR11 )2 Im „M (8.126)

is the area of the torus. In terms of coordinates z = x + iy on the torus, a

single-valued wave function has the form

n2 Re „M ’ n1

i

ψn1 ,n2 ∼ exp n2 x ’ y . (8.127)

R11 Im „M

These characterize Kaluza“Klein excitations. The contribution to the mass-

2 2

squared is given by the eigenvalue of ’‚x ’ ‚y ,

(n2 Re „M ’ n1 )2 |n1 ’ n2 „M |2

1

2 2

MKK = 2 n2 + = . (8.128)

(Im „M )2 (R11 Im „M )2

R11

Clearly, this term has the right structure to match the type IIB string

winding-mode terms, described above, for the identi¬cation

„M = „ B . (8.129)

2 2

The normalization of MKK and the winding-mode contribution to MB is not

the same, but that is because they are measured in di¬erent metrics. The

matching tells us how to relate the two metrics, a formula to be presented

soon.

The identi¬cation in Eq. (8.129) is a pleasant surprise, because it implies

that the nonperturbative SL(2, ) symmetry of type IIB superstring theory,

after compacti¬cation on a circle, has a dual M-theory interpretation as the

modular group of a toroidal compacti¬cation! Modular transformations of

the torus are certainly symmetries, since they correspond to the disconnected

components of the di¬eomorphism group. Once the symmetry is established

for ¬nite RB , it should also persist in the decompacti¬cation limit RB ’ ∞.

To go further requires an M-theory counterpart of the term (K/RB )2 in

8.4 M-theory dualities 341

the type IIB superstring mass formula (8.123). Here there is also a natural

candidate: wrapping M-theory M2-branes so as to cover the torus K times.

If the M2-brane tension is TM2 , this gives a contribution (AM TM2 K)2 to the

mass-squared. Matching the normalization of this term and the Kaluza“

Klein term gives two relations. One learns that the metrics in nine dimen-

sions are related by

g (M) = β 2 g (B) , (8.130)

where

2πR11 TM2

β2 = , (8.131)

TF1

and that the compacti¬cation volumes are related by

3/2

2

gs AM

3

2 = TM2 (2πR11 ) = TM2 . (8.132)

Im „M

TF1 RB

Since all the other factors are constants, this gives (for ¬xed „B = „M ) the

’3/4

scaling law RB ∼ AM .

There still are the oscillator excitations of the type IIB superstring BPS

mass formula to account for. Their M-theory counterparts must be exci-