String Theory and M-Theory

that
8.4 M-theory dualities 345
T (HO)
5
∼ (g (HO)
s ) −2 , (8.143)
as is typical of a soliton. In the type I string-frame metric, on the other
hand, it implies that
TD1 ∼ 1/g (I)
s and TD5 ∼ 1/g (I)
s , (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 compactification
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 maximal
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 compactification leads to theories with maximal supersymmetry in
lower dimensions. 14 So, for example, toroidal compactification 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 noncompact
E7 symmetry. More generally, for d = 11 − n, 3 ≤ n ≤ 8, one finds a
maximally noncompact form of En, denoted En,n. The maximally noncompact
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 infinite line. En are
standard exceptional groups that appear in Cartan’s classification of simple
Lie algebras for n = 6, 7, 8. The definition for n < 6 can be obtained by
extrapolation of the Dynkin diagrams. This gives the identifications (listing
the maximally noncompact forms) 15
E5,5 = SO(5, 5), E4,4 = SL(5, ¡ ), E3,3 = SL(3, ¡ ) × SL(2, ¡ ). (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 compactification spaces that (unlike tori) break some or all of the supersymmetries.
15 The compact forms of the same sequence of exceptional groups was encountered in the study
of type I ′ superstrings in Chapter 6.