String Theory and M-Theory

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 infinite discrete symmetry groups by quantum
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 infinite 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 fields. 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 diffeomorphism group. The subgroup SO(n − 1, n − 1; ) ⊂ En( ) is
the T-duality group of type IIB superstring theory compactified 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)
subgroup is the type IIB T-duality group. Clearly, E3( ) is the smallest
group containing both of these.
Toroidally compactified M-theory (or type II superstring theory) has a
moduli space analogous to the Narain moduli space of the toroidally compactified
heterotic string described in Chapter 7. Let Hn denote the maximal
compact subgroup of En,n. For example, H6 = USp(8), H7 = SU(8) and
H8 = Spin(16). Then one can define a homogeneous space
M 0 n = En,n/Hn. (8.148)
This is directly relevant to the physics in that the scalar fields in the supergravity
theory are defined 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 fields all have the same sign. The
number of scalar fields is the dimension of the coset space dn = dim M 0 n.
For example, in three, four and five 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)