String Theory and M-Theory

280 The heterotic string
the circle. However, all radii R > 0 are allowed, in contrast to the bosonic
string theory where R and R = 1/(2R) are equivalent.
As in the case of the bosonic string theory, one can form a moduli space
Mn,n of inequivalent toroidal compactifications of type II superstrings as
a quotient of M 0 n,n by a suitable duality group. The appropriate duality
group is smaller in this case than in the bosonic theory. It is only a subgroup
of the O(n, n; ) transformation group. Specifically, it is the subgroup that
preserves the chirality of the spinors. This reduces the group to SO(n, n; ).
To summarize, one could say that the distinction between type IIA and
type IIB dissolves after T n compactification, and there is a single moduli
space for the pair constructed in the way indicated here, but this moduli
space is twice as large as in the case of the bosonic string theory. Chapter
8 shows that, when other dualities are taken into account, the duality
group SO(n, n; ) is extended to En+1( ), which is a discrete subgroup of
a noncompact exceptional group.
EXERCISES
EXERCISE 7.3
Consider the bosonic string theory compactified on a circle of radius R =
√ α ′ . Verify that there is SU(2)×SU(2) gauge symmetry by constructing the
conserved currents. Show that the modes of the currents satisfy a level-one
Kac–Moody algebra.
SOLUTION
To do this let us focus on the holomorphic right-moving currents, since the
antiholomorphic left-moving currents work in an identical fashion. Let us
define
and
J ± (z) = e ±2iX25 (z)/ √ α ′
J 3 (z) = i 2/α ′ ∂X 25 (z).
The coefficients in the exponent have been chosen to ensure that J ± (z) have
conformal dimension h = 1. These currents are single valued at the self-dual
radius R = √ α ′ , because X 25 (z) contains the zero mode 1
2 x25 . Note that in
the text we have been setting α ′ = 1/2.