String Theory and M-Theory

7.3 Toroidal compactification 283
EXERCISE 7.6
Show that GIJ + BIJ has the right number of components to parametrize
the coset space M 0 n,n.
SOLUTION
The moduli space M 0 n,n is given in terms of a lattice spanned by the leftmoving
and right-moving momenta (pL, pR) under the restriction that
p 2 L − p 2 R ∈ 2
This condition is left invariant by the group of O(n, n, ; ¡ ) transformations,
but the mass formula
M 2 = 2(p 2 L + p 2 R) − 8 + oscillators
is not. The invariance of the mass formula is rather given by O(n, ¡ ) ×
O(n, ¡ ). As a result, the moduli space is given by the quotient space
O(n, n; ¡ )/O(n; ¡ ) × O(n, ¡ ).
Taking into account that O(n, ¡ ) has dimension n(n − 1)/2 and O(n, n, ; ¡ )
has dimension n(2n − 1), we see that the dimension of the moduli space is
n 2 .
On the other hand, the metric G is a symmetric tensor with n(n + 1)/2
parameters while the antisymmetric B field has n(n − 1)/2 independent
components. In total, this gives n 2 components, as we wanted to show. ✷
EXERCISE 7.7
Compute the matrix G for the special case of compactification on a circle
and compare with the results derived in Chapter 6.
SOLUTION
In the special case of n = 1 one simply has a circle of radius R, and G11 = R2 .
Then G reduces to the 2 × 2 matrix
1/(2R2 ) 0
G =
0 2R2
,
so that
M 2 0 = (2W R) 2 + (K/R) 2 ,
in agreement with the result obtained in Chapter 6 (for α ′ = 1/2). The first
term is the winding contribution and the second term is the Kaluza–Klein
.