subgroup is the type IIB T-duality group. Clearly, E3 ( ) is the smallest

group containing both of these.

Toroidally compacti¬ed M-theory (or type II superstring theory) has a

moduli space analogous to the Narain moduli space of the toroidally com-

pacti¬ed heterotic string described in Chapter 7. Let Hn denote the maximal

compact subgroup of En,n . For example, H6 = U Sp(8), H7 = SU (8) and

H8 = Spin(16). Then one can de¬ne a homogeneous space

M0 = En,n /Hn . (8.148)

n

This is directly relevant to the physics in that the scalar ¬elds in the super-

gravity theory are de¬ned by a sigma model on this coset space. Note that

all the coset generators are noncompact. It is essential that they all be the

same so that the kinetic terms of the scalar ¬elds all have the same sign. The

number of scalar ¬elds is the dimension of the coset space dn = dim M0 . n

For example, in three, four and ¬ve dimensions the number of scalars is

d3 = dim E8 ’ dim Spin(16) = 248 ’ 120 = 128, (8.149)

d4 = dim E7 ’ dim SU (8) = 133 ’ 63 = 70, (8.150)

8.4 M-theory dualities 347

d5 = dim E6 ’ dim U Sp(8) = 78 ’ 36 = 42. (8.151)

The discrete duality-group identi¬cations must still be accounted for, and

this gives the moduli space

Mn = M0 /En ( ). (8.152)

n

A nongeometric duality of M-theory

String theory possesses certain features, such as T-duality, that go beyond

ones classical geometric intuition. This section shows that the same is true

for M-theory by constructing an analogous duality transformation. There is

a geometric understanding of the SL(n, ) subgroup of En ( ) that comes

from considering M-theory on 11’n — T n , since it is the modular group

¡

of T n . But what does the rest of En ( ) imply? To address this question

it su¬ces to consider the ¬rst nontrivial case to which it applies, which is

n = 3. In this case the U-duality group is E3 ( ) = SL(3, ) — SL(2, ).

From the M-theory viewpoint the ¬rst factor is geometric and the second

factor is not. So the question boils down to understanding the implication

of the SL(2, ) duality in the M-theory construction. Speci¬cally, we want

to understand the nontrivial „ ’ ’1/„ element of this group.

To keep the story as simple as possible, let us choose the T 3 to be rectan-

2

gular with radii R1 , R2 , R3 , that is, gij ∼ Ri δij , and assume that C123 = 0.

Choosing R3 to correspond to the “eleventh” dimension makes contact with

the type IIA theory on a torus with radii R1 and R2 . In this set-up, the

stringy duality of M-theory corresponds to simultaneous T-duality transfor-

mations of the type IIA theory for both of the circles. This T-duality gives

a mapping to an equivalent point in the moduli space for which

3

2

p

s

Ri ’ Ri = = i = 1, 2, (8.153)

Ri R3 Ri

with s unchanged. The derivation of this formula has used 3 = R3 2 , which

p s

relates the 11-dimensional Planck scale p to the ten-dimensional string scale

s . Under a T-duality the string coupling constant also transforms. The rule

is that the coupling of the e¬ective theory, which is eight-dimensional in this

case, is invariant:

1 R1 R2 R R2

= 4π 2 2 = 4π 2 1 2 . (8.154)

2 gs (gs )

g8

Thus

gs 2s

gs = . (8.155)

R1 R2

348 M-theory and string duality

What does this imply for the radius of the eleventh dimension R3 ? Using

R3 = gs s ’ R3 = gs s ,

3

gs 3 p

s

R3 = = . (8.156)

R1 R2 R1 R2

However, the 11-dimensional Planck length also transforms, because

3 3

’ ( p )3 = g s 3

= gs (8.157)

p s s

implies that

6

gs 5 p

s

3

( p) = = . (8.158)

R1 R2 R1 R2 R3

The perturbative type IIA description is only applicable for R3 R1 , R 2 .

However, even though T-duality was originally discovered in perturbation

theory, it is supposed to be an exact nonperturbative property. Therefore,

this duality mapping should be valid as an exact symmetry of M-theory

without any restriction on the radii. Another duality is an interchange

of circles, such as R3 ” R1 . This corresponds to the nonperturbative S-

duality of the type IIB superstring theory. Combining these dualities gives

the desired stringy duality of M-theory on T 3 , namely

3

p

R1 ’ , (8.159)

R2 R3

and cyclic permutations, accompanied by

6

p

3

’ . (8.160)

p

R1 R2 R3