String Theory and M-Theory

490 Flux compactifications
be true for any choice of the five-dimensional base space other than a fivesphere
of unit radius. As was already mentioned, in the case of the conifold
there are two ways of smoothing out the singularity at the tip of the cone,
called deformation and resolution.
The deformed conifold
The deformation consists in replacing Eq. (10.117) by
4
A=1
(w A ) 2 = z, (10.125)
where z is a nonzero complex constant. Since w A ∈ £ 4 we can rescale
these coordinates and assume that z is real and nonnegative. This defines
a Calabi–Yau three-fold for any value of z. As a result, z spans a onedimensional
moduli space. At the singularity of the moduli space (z = 0)
the manifold becomes singular (at ρ = 0).
For large r the deformed conifold geometry reduces to the singular conifold
with z = 0, that is, it is a cone with an S 2 × S 3 base. Moving from ∞
towards the origin, the S 2 and S 3 both shrink. Decomposing w A into real
and imaginary parts, as before, yields
and using the definition
shows that the range of r is
z = x · x − y · y, (10.126)
ρ 2 = x · x + y · y, (10.127)
z ≤ ρ 2 < ∞. (10.128)
As a result, the singularity at the origin is avoided for z > 0. This shows
that as ρ 2 gets close to z the S 2 disappears leaving just an S 3 with finite
radius.
The resolved conifold
The second way of smoothing out the conifold singularity is called resolution.
In this case as the apex of the cone is approached, it is the S 3 which shrinks
to zero size, while the size of the S 2 remains nonvanishing. This is also
called a small resolution, and the nonsingular space is called the resolved
conifold.
In order to describe how this works, let us make a linear change of variables