’1

(HO) 2

T1 LO

L1 L2 TM2 = , (8.141)

2

2π

which is the analog of Eq. (8.132). As in that case, it tells us that, for

’3/4

¬xed modulus L1 /L2 , one has the scaling law LO ∼ AC , where AC =

L1 L2 is the area of the cylinder. Equation (8.139) relating TM2 and TM5 is

reobtained, and one also learns that

2

1 L2

(HO) (HO) 3

T5 = (T1 ). (8.142)

(2π)2 L1

(HO)

In the heterotic string-frame metric, where T1 is a constant, this implies

8.4 M-theory dualities 345

that

(HO) (HO) ’2

∼ (gs

T5 ), (8.143)

as is typical of a soliton. In the type I string-frame metric, on the other

hand, it implies that

(I) (I)

TD1 ∼ 1/gs TD5 ∼ 1/gs ,

and (8.144)

consistent with the fact that both are D-branes from the type I viewpoint.

U-duality

It is natural to seek type II counterparts of the O(n, n; ) and O(16+n, n; )

duality groups that were found in Chapter 7 for toroidal compacti¬cation

of the bosonic and heterotic string theories, respectively. A clue is provided

by the fact that the massless sector of type II superstring theories are max-

imal supergravity theories (ones with 32 conserved supercharges), with a

noncompact global symmetry group.

In the case of type IIB supergravity in ten dimensions the noncompact

global symmetry group is SL(2, ), as was shown earlier in this chapter.

¡

Toroidal compacti¬cation leads to theories with maximal supersymmetry in

lower dimensions.14 So, for example, toroidal compacti¬cation of the type

IIB theory to four dimensions and truncation to zero modes (dimensional

reduction) leads to N = 8 supergravity. N = 8 supergravity has a noncom-

pact E7 symmetry. More generally, for d = 11 ’ n, 3 ¤ n ¤ 8, one ¬nds a

maximally noncompact form of En , denoted En,n . The maximally noncom-

pact form of a Lie group of rank n has n more noncompact generators than

compact generators. Thus, for example, E7,7 has 133 generators of which

63 are compact and 70 are noncompact. A compact generator generates a

circle, whereas a noncompact generator generates an in¬nite line. En are

standard exceptional groups that appear in Cartan™s classi¬cation of simple

Lie algebras for n = 6, 7, 8. The de¬nition for n < 6 can be obtained by

extrapolation of the Dynkin diagrams. This gives the identi¬cations (listing

the maximally noncompact forms)15

) — SL(2,

E5,5 = SO(5, 5), E4,4 = SL(5, ), E3,3 = SL(3, ). (8.145)

¡ ¡ ¡

These noncompact Lie groups describe global symmetries of the classical

low-energy supergravity theories. However, as was discussed already for the

14 Chapter 9 describes compacti¬cation spaces that (unlike tori) break some or all of the super-

symmetries.

15 The compact forms of the same sequence of exceptional groups was encountered in the study

of type I superstrings in Chapter 6.

346 M-theory and string duality

case of the E1,1 = SL(2, ) symmetry of type IIB superstring theory in ten

¡

dimensions, they are broken to in¬nite discrete symmetry groups by quan-

tum and string-theoretic corrections. The correct statement for superstring

theory/M-theory is that, for M-theory on d — T n or (equivalently) type

¡

IIB superstring theory on d — T n’1 , the resulting moduli space is invariant

¡

under an in¬nite discrete U-duality group. The group, denoted En ( ), is a

maximal discrete subgroup of the noncompact En,n symmetry group of the

corresponding supergravity theory.

The U-duality groups are generated by the Weyl subgroup of En,n plus

discrete shifts of axion-like ¬elds. The subgroup SL(n, ) ‚ En ( ) can be

understood as the geometric duality (modular group) of T n in the M-theory

picture. In other words, they correspond to disconnected components of

the di¬eomorphism group. The subgroup SO(n ’ 1, n ’ 1; ) ‚ En ( ) is

the T-duality group of type IIB superstring theory compacti¬ed on T n’1 .

These two subgroups intertwine nontrivially to generate the entire En ( )

U-duality group. For example, in the n = 3 case the duality group is

E3 ( ) = SL(3, ) — SL(2, ). (8.146)

The SL(3, ) factor is geometric from the M-theory viewpoint, and an

SO(2, 2; ) = SL(2, ) — SL(2, ) (8.147)