String Theory and M-Theory

7.3 Toroidal compactification 273
Note that this vanishes for R 2 = 1/2 = α ′ , which is precisely the self-dual
radius of the T-duality transformation R → α ′ /R.
In the same way we can consider the states which have K = −W = ±1.
Then there are again two vectors
|V µ
µ
+− 〉 = ˜αµ −1 | + 1, −1〉 and |V −+ 〉 = ˜αµ −1 | − 1, +1〉, (7.89)
and two scalars
|φ+−〉 = ˜α−1| + 1, −1〉 and |φ−+〉 = ˜α−1| − 1, +1〉. (7.90)
The mass of these states is also given by Eq. (7.88). Altogether, at the
self-dual radius there are four additional massless vectors in the spectrum
in addition to the two that are present for any radius. The interpretation
is that there is enhanced gauge symmetry for this particular value of
the radius. The gauge group U(1) × U(1), which is present in general,
is a subgroup of the enhanced symmetry group, which in this case is
SU(2) × SU(2). This is explored in Exercise 7.3. The three vectors that
involve an α µ
−1 excitation are associated with a right-moving SU(2) on
the string world sheet. Similarly, the other three involve a ˜α µ
−1 excitation
and are associated with a left-moving SU(2) on the string world sheet.
The case of SU(3) × SU(3) is studied in Exercise 7.8.
This enhancement of gauge symmetry at the self-dual radius is a “stringy”
effect. For other values of the radius the gauge symmetry is broken to
U(1)L × U(1)R. The four gauge bosons |V µ
±± 〉 eat the four scalars |φ±±〉
as part of a stringy Higgs effect. On the other hand, the U(1)L × U(1)R
gauge bosons, as well as the associated scalar |φ〉, remain massless for all
values of the radius. This neutral scalar has a flat potential (meaning that
the potential function does not depend on it), which corresponds to the
freedom of choosing the radius of the circle to be any value with no cost
in energy. Altogether, the spectrum of the bosonic string compactified on
a circle is characterized by a single parameter R, called the modulus of the
compactification. It is the radius of the circle, whose value is determined by
the vacuum expectation value of the scalar field |φ〉.
As was explained in the previous section, the T-duality symmetry of the
bosonic string theory requires that the moduli space of the theory compactified
on a circle be defined as the quotient space of the positive line R > 0
modulo the identification of R and 1/(2R). Therefore, the point of enhanced
gauge symmetry, which is the fixed point of the T-duality transformation,
is also the singular point of the moduli space.