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In order to construct string models in less than 10D as well as to
understand the heterotic string construction, we need to
consider the simplest compactifications which correspond to the extra
dimensions being circles and their higher dimensional generalization.
Let us first see the case of a circle. This means that the 10D space
is represented by flat 9D spacetime times a circle . We know that a
circle is just the real line identifying all the numbers differing by ,
where is the radius of the circle.
So the only difference with the flat space discussed above are the boundary conditions. The solution of the wave equations are now as in (4).
But now and , is an integer
reflecting the fact that the momentum in the compact
direction has to be quantized in order to get singlevalued
wave function. The integer however refers to the fact that
the string can wind around several times in the compact dimension and is thus
named the `winding number'. The mass formula is then:

(6) 
This shows several interesting facts.
First, for and varying , we obtain an infinite
tower of massive states with masses ; these are the
standard `momentum states' of KaluzaKlein compactifications
in field theory. In particular the massless states with
and one oscillator in the compact direction
are vector fields in the extra dimensions giving rise
to a
KaluzaKlein gauge symmetry. The states with are
the winding states and are purely stringy; they represent
string states winding around the circle, they have
mass .
Second, there are special values of and which can give rise
to extra massless states. In particular for we can see that
at the special radius in units of , there are massless states with a single
oscillator corresponding to massless vectors which in this
case generate
. This means that the special point
in the `moduli space' of the circle is a point of enhanced symmetry.
The original
KaluzaKlein symmetry of compactification on a circle gets enhanced to
. This is a very stringy effect because it depends crucially on the existence of winding modes
().
The third interesting fact about this compactification is that the spectrum
is invariant under the following `duality' transformations [8]:

(7) 
This is also a stringy property. It exchanges small
with large distances but at the same time it
exchanges momentum (KaluzaKlein) states with
winding states. This symmetry can be shown to hold not
only for the spectrum but also for the interactions and
therefore it is an exact symmetry of string perturbation theory.
Let us now extend the compactification to two dimensions, i.e.,
the 26D spacetime is the product of flat 24D spacetime and a
2D generalization of a circle, the torus . Again the only difference with flat space is the boundary
conditions. The two compact dimensions are identified by
vectors of a 2D lattice, defining the torus .
Out of the three independent components of the compactified
metric
and the single component of
namely we can build two complex `moduli' fields:







(8) 
is the standard modular parameter of any geometrical
2D torus and it is usually identified as the `complex
structure' modulus. is the `Kähler structure' modulus
(since is a complex Kähler space) and its imaginary part
measures the overall size of the torus, since is the
determinant of the 2D metric. It plays the same role as
did for the 1D circle. In terms of and we can write the left and
rightmoving momenta as:







(9) 
The mass forula, depending on , again shows that there are enhanced symmetry points for
special values of and . It also shows the following symmetries:

(10) 
Where are integers satisfying .
The first transformation is the standard
`modular' symmetry of 2D
tori and is independent of string theory; it is purely geometric.
The second transformation is a stringy
named
duality and it is a generalization of (7)
for the 2D case. Again this is a symmetry as long as we also transform
momenta with winding .
The third symmetry exchanges the
complex structure with the Kähler structure and
it is called `mirror symmetry'.
If and each parametrize a complex plane
, the duality symmetry implies that they can only live
in the fundamental domain defined by all the points of the product of complex spaces
identified under the
duality group
.
This is the situation that gets generalized to higher dimensions.
In general, compactification on a dimensional torus has the
moduli space
with
points identified under the duality group
.
For the heterotic string with 16 extra left moving coordinates
with
a similar modification to the duality group.
The left and right moving momenta live on an
even, selfdual lattice of signature , which
is usually called the Narain lattice
[9].
This generalizes the lattice defined by the
integers
of eq. (9).
We can easily verify in this case that the dimension of
is corresponding to the number of independent
components of
with
. For we have a
4D string model with a moduli space of dimension . To
this we have to add the dilaton field which, together
with the spacetime components of the antisymmetric
tensor , can be combined into a new modular
parameter:

(11) 
Here the axion field is defined as
.
parametrizes again a coset
. It is then natural to
believe there is also a duality symmetry for the field
of the type ; by analogy with the situation for and .
Such a symmetry was proposed in ref.[10] and it
has received a lot of attention recently. If true it
may have far reaching
consequences since (similar to equation (7)) it relates
strong to weak string coupling.
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