String Theory and M-Theory

7.3 Toroidal compactification 271
to fixed points of O(n, n; ) transformations. At these special values of
(GIJ, BIJ) the spectrum has additional massless gauge bosons, and there is
unbroken nonabelian gauge symmetry.
Enhanced gauge symmetry
Nonabelian gauge symmetries can arise from toroidal compactifications.
From a Kaluza–Klein viewpoint this is very surprising. In a point-particle
theory the gauge symmetries one would expect are those that correspond
to isometries of the compact dimensions. The isometry of T n is simply
U(1) n and this is abelian. So the feature in question is a purely stringy one
involving winding modes in addition to Kaluza–Klein excitations.
This section considers the bosonic string theory compactified on a T n
as before. The extension to the heterotic string is given in the following
section. The basic idea in both cases is that for generic values of the moduli
the gauge symmetry is abelian. In the case of the bosonic string theory
it is actually U(1) 2n , so there are 2n massless U(1) gauge bosons in the
spectrum. Half of them arise from reduction of the 26-dimensional metric
(namely, components of the form gµI) and half of them arise from reduction
of the 26-dimensional two-form (namely, components of the form BµI).
At specific loci in the moduli space there appear additional massless particles
including massless gauge bosons. When this happens there is symmetry
enhancement resulting in nonabelian gauge symmetry. For example, in the
n = 1 case, the symmetry is enhanced from U(1) × U(1) to SU(2) × SU(2)
at the self-dual radius. Let us explore how this happens.
The self-dual radius
In order to consider enhanced gauge symmetry of the bosonic string theory
compactified on a circle of radius R, let us assume that the coordinate X 25
is compact and the remaining coordinates are noncompact. The spectrum
is described by the mass formula
M 2 = K2
R 2 + 4R2 W 2 + 4(NL + NR − 2), (7.80)
as well as the level-matching condition
NR − NL = KW. (7.81)
As before, W describes the number of times the string winds around the
circle. Let us now explore some of the low-mass states in the spectrum of
this theory.