String Theory and M-Theory

274 The heterotic string
In the case of type II superstrings, compactification on a T n again gives
rise to 2n abelian gauge fields. However, unlike the bosonic string theory,
there is no possibility of symmetry enhancement. One way of understanding
this is to note that all 2n of the gauge fields belong to the supergravity
multiplet in 10 − n dimensions, and this cannot be extended to include
additional gauge fields. Another way of understanding this is to observe that
symmetry enhancement in the bosonic string utilized winding and Kaluza–
Klein excitations so that NL = NR ± 1. The same relations in the case of
type II superstrings imply that the mass is strictly positive.
Toroidal compactification of the heterotic string is studied in Section 7.4.
It is shown that compactification to 10 − n dimensions gives n right-moving
U(1) currents and 16 + n left-moving U(1) currents. Moreover, there can be
no symmetry enhancement for the right-moving current algebra, but there
can be symmetry enhancement for the left-moving current algebra. In fact,
in the special case n = 0, the U(1) 16 is necessarily enhanced, to either
SO(32) or E8 × E8.
One-loop modular invariance
Chapter 3 showed that one-loop amplitudes are given by integrals over
the moduli space of genus-one (toroidal) Riemann surfaces. This space
is parametrized by a modular parameter τ whose imaginary part is positive.
An important consistency requirement is that the integral should have
modular invariance. In other words, it should be of the form
F
d2τ I(τ, . . .), (7.91)
(Imτ) 2
where F denotes a fundamental region of the modular group. Modular
invariance requires that I is invariant under the P SL(2, ) modular transformations
τ → τ ′ aτ + b
= , (7.92)
cτ + d
where a, b, c, d ∈ and ad − bc = 1, since the measure d 2 τ/(Imτ) 2 is invariant.
Two examples of modular transformations are shown in Fig. 7.1. This
ensures that it is equivalent to define the integral over the region F or any
of its images under a modular transformation. In other words, the value of
the integral is independent of the particular choice of a fundamental region.
This property is satisfied by the bosonic string theory in 26-dimensional
Minkowski space-time. Accepting that result, we propose to examine here