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In order to connect superstring theories to the observed 4dimensional spacetime
physics,
we have to reduce the critical dimension to four dimensions. To preserve
certain supersymmetry consistent with chirality in four dimensions
it is necessary to require some properties to the ten
dimensional spacetime . Perhaps the simplest ansatz is to assume that
the fourdimensional Minkowski spacetime and a sixdimensional internal space factorizes
as
, where has tiny dimensions and
unobservable in our present experiments. It is worth to say that this factorization ansatz is not
unique and other
possibility is the warped compactification of the celebrated RandallSundrum
scenarios, which are nicely reviewed in Ref. [9].
It is useful to classify the compactifications according to how much
supersymmetries
is broken, because this number is related with the quantum corrections that
we shall consider. We choose to be a
manifold with the property that a
certain
number of supersymmetries are preserved^{3}.
We are now looking for conditions in the background which leave some
supersymmetry
unbroken. These conditions are given by null variations of the Fermi fields.
Consider the diagonal metric for tendimensional spacetime given by
where denotes the compactified coordinates and ,
, . For , heterotic string theory the Fermi fields
variations are:
 gravitino:
,
 dilatino:
 gaugino:
where is a WeylMajorana spinor in ten dimensions,
is the
internal component of
, are the
Dirac matrices.
The compactification ansatz
breaks the Lorentz group SO(9,1) into
SO
. In the spinor representation 16
the WeylMajorana supersymmetry parameter
decomposes as
under
. The general form of
is
with an
arbitrary Weyl spinor. When we put the condition that Fermi fields variations vanish, then
each internal spinor
gives the minimal ()
supersymmetric algebra.
Now, by the null fermi fields variations we can find conditions in the background
fields assuming that . These are:
The last equation tell us that is covariantly
constant on the internal space , and implies that is Ricciflat.
This is because
. For this
reason, in general
do not belong to SO(6) but to SU(3), which is a subgroup that
leaves one component of the spinor invariant. Thus the
compact
manifold must have SU(3) holonomy. The second unbroken susy condition
implies that the warped factor in metric is 1 and the metric is unwarped.
Finally, the first condition implies that the dilation is constant.
This is a CalabiYau threefold. A CalabiYau threefold is also a Kähler manifold
in which the first Chern
class zero i.e.
Any CalabiYau manifold possesses a unique Ricciflat metric.
When we
consider heterotic string theory on CalabiYau threefold we obtain a
fourdimensional chiral theory with spacetime supersymmetry .
In fact, compactification on manifolds of SU(3) holonomy preserves of
supersymmetry. If we consider theories (for example, type II
superstrings) in
dimensions, after compactification on a CalabiYau threefold we obtain
theories in .
In addition to the CYthreefold structure for the unbroken susy condition
leads to the equations in complex
coordinates

(28) 
These equations require to specify a gauge subbundle of a
gauge bundle
over and a gauge connection on with curvature . The condition
tell us that the subbundle as well as the
corresponding connection should be holomorphic. The second condition
is the celebrated DonaldsonUhlenbeckYau equation for . This equation has a
unique solution if the bundle is stable and if it is satisfied the integrability condition
where is the Kähler form of
. There is a further condition to be satisfied by the connection , the Bianchi
identity for and , it is given by

(29) 
The only solution is
which implies that
. This situation is usually known as the standard embedding of the spin connection
in the gauge connection ant it is a method to determine the connection on .
Thus in the compactification of phenomenological interest of the heterotic theory with the
ansatz
, the internal space has to be a CalabiYau threefold and one
has to specify a stable, holomorphic vector bundle over (or ) satisfying
and
.
If is a SU vector bundle over the subgroups of
that
commutes are , SO(10) and SU(5) for respectively. This leads to
GUTs in four dimensions justly with the gauge groups , SO(10) or SU(5).
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Massless Spectrum
In order to describe the impact of the characteristics of and on the
properties of the spectrum of the four dimensional theory we start by decomposing
the tendimensional Dirac operator under
into

(30) 
where
and
. Dirac equation in ten dimensions is

(31) 
Thus the spectrum of the Dirac operator
on determines
the massive spectrum of fermions in four dimensions.
In ten dimensions the Lorentz group only has real spinor representations and
the Clifford modules decomposes as:
. Positive and negative chirality are distinguised by
. theorem implies that we must take
only one chirality

(32) 
Decompose the spinor representation of SO(1,9) under
with
where
and
.
One solution with
is given by
and then the spin
bundle decomposes
under
as

(33) 
Now solve the Dirac equation with the ansatz
and
. It leads to

(34) 
where
and
.
is an
elliptic operator on the
compact manifold , this implies that that operator has a finite number of
fermion zero modes. Massless fermions in four dimensions originate as zero
modes of the Dirac operators
of the internal manifold . By
the
AtiyahSinger theorem, a topological invariant of containing the information of
the chiral fermions on is given by the index of the Dirac operator

(35) 
for chiral fermions on with
Here
are the number of positive and negative chiral zero modes. In dimensions
this index is vanishing. We need to couple gauge fields coming from the heterotic string
theory. Recall that they are
valued gauge fields.
The standard embedding of the
spin connection in the gauge connection leads to the chain of maximal subgroups:
. This breaks SO(16) to SO(10). The computation of the
for this case yields
where are irreps of
SO(6) and are complex irreps of SO(10). These latter determine the irreps where are
distributed the massless fermions of the fourdimensional theory. The former irreps determine
the number of fermionic chiral zero modes described by the topological index
. This is given by
and from the solution
it determines the chiral fermion families in four
dimensions

(36) 
Thus the theory in four dimensions has
where

(37) 
where
with
is the Euler number of
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