String Theory and M-Theory

292 The heterotic string
SO(32) current algebra. Making the same GSO projection as in the leftmoving
sector of the heterotic string, find the ground state and the massless
states of this theory.
PROBLEM 7.3
Exercise 7.1 introduced free-fermion representations of current algebras and
showed that fermions in the fundamental representation of SO(n) give a
level-one current algebra.
(i) Find the level of the current algebra for fermions in the adjoint representation
of SO(n).
(ii) Find the level of the current algebra for fermions in a spinor representation
of SO(16).
PROBLEM 7.4
Generalize the analysis of Exercise 7.6 to the heterotic string. In particular,
verify that the Wilson lines, together with the B and G fields, have the right
number of parameters to describe the moduli space M0 16+n,n in Eq. (7.121).
PROBLEM 7.5
In addition to the SO(32) and E8 × E8 heterotic string theories, there is
a third tachyon-free ten-dimensional heterotic string theory that has an
SO(16) × SO(16) gauge group. This theory is not supersymmetric. Invent
a plausible set of GSO projection rules for the fermionic formulation of
this theory that gives an SO(16) × SO(16) gauge group and does not give
any gravitinos. Find the complete massless spectrum.
PROBLEM 7.6
The SO(16) × SO(16) heterotic string theory, constructed in the previous
problem, is a chiral theory. Using the rules described in Chapter 5, construct
the anomaly 12-form. Show that anomaly cancellation is possible by showing
that this 12-form factorizes into the product of a four-form and an eightform.
PROBLEM 7.7
The ten-dimensional SO(32) and E8 × E8 string theories have the same
number of states at the massless level. Construct the spectrum at the first
excited level explicitly in each case using the formulation with 32 left-moving
fermions. What is the number of left-moving states at the first excited level