Editing Conifold (section)

==Overview==
Conifolds are important objects in [[string theory]]: [[Brian Greene]] explains the [[physics]] of conifolds in Chapter 13 of his book ''[[The Elegant Universe]]''—including the fact that the space can tear near the cone, and its [[topology]] can change. This possibility was first noticed by {{harvtxt|Candelas|Dale|Lutken|Schimmrigk|1988}} and employed by {{harvtxt|Green|Hübsch|1988}} to prove that conifolds provide a connection between all (then) known Calabi–Yau compactifications in string theory; this partially supports a conjecture by {{harvtxt|Reid|1987}} whereby conifolds connect all possible Calabi–Yau complex 3-dimensional spaces.

A well-known example of a conifold is obtained as a deformation limit of a quintic - i.e. a [[quintic hypersurface]] in the [[complex projective plane|projective space]] <math>\mathbb{CP}^4</math>. The space <math>\mathbb{CP}^4</math> has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations:

<div align=center><math>z_1^5+z_2^5+z_3^5+z_4^5+z_5^5-5\psi z_1z_2z_3z_4z_5 = 0</math></div>

in terms of homogeneous coordinates <math>z_i</math> on <math>\mathbb{CP}^4</math>, for any fixed complex <math>\psi</math>, has complex dimension three. This family of [[quintic hypersurface]]s is the most famous example of [[Calabi–Yau manifold]]s. If the [[Complex manifold|complex structure]] [[parameter]] <math>\psi</math> is chosen to become equal to one, the manifold described above becomes singular since the [[derivative]]s of the quintic [[polynomial]] in the equation vanish when all coordinates <math>z_i</math> are equal or their [[ratio]]s are certain fifth roots of unity. The neighbourhood of this singular point looks like a [[cone (geometry)|cone]] whose base is [[topology|topologically]] just.

<div align=center><math>S^2 \times S^3</math></div>

In the context of [[string theory]], the geometrically singular conifolds can be shown to lead to completely smooth physics of strings. The divergences are "smeared out" by [[D-branes|D3-branes]] wrapped on the shrinking three-sphere in [[Type IIB string theory]] and by [[D-branes|D2-branes]] wrapped on the shrinking two-sphere in [[Type IIA string theory]], as originally pointed out by {{harvtxt|Strominger|1995}}. As shown by {{harvtxt|Greene|Morrison|Strominger|1995}}, this provides the string-theoretic description of the [[topology]]-change via the conifold transition originally described by {{harvtxt|Candelas|Green|Hübsch|1990}}, who also invented the term "conifold" and the diagram
[[Image:3FoldConifoldTransition.pdf|600px|center]]
for the purpose. The two topologically distinct ways of smoothing a conifold are thus shown to involve replacing the singular vertex (node) by either a 3-sphere (by way of deforming the complex structure) or a 2-sphere (by way of a "small resolution"). It is believed that nearly all [[Calabi–Yau manifold]]s can be connected via these "critical transitions", resonating with Reid's conjecture.