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In [[mathematics]] and [[string theory]], a '''conifold''' is a generalization of a [[manifold]]. Unlike manifolds, conifolds can contain [[gravitational singularity#Conical|conical singularities]], i.e. points whose neighbourhoods look like [[cone (geometry)|cones]] over a certain base. In [[physics]], in particular in [[Compactification (physics)#Flux compactification|flux compactification]]s of [[string theory]], the base is usually a five-[[dimension]]al real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.


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
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:<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>
 
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 structure]]{{dn|date=September 2012}} [[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 <math>S^2 \times S^3</math>.
 
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]]
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]] manifolds can be connected via these "critical transitions", resonating with Reid's conjecture.
 
==References and external links==
*{{citation| last1=Candelas|first1=Philip|last2=Dale|first2=A.M.|last3=Lutken|first3=Andrew|last4=Schimmrigk|first4=Rolf|year = 1988| title = Complete intersection Calabi-Yau manifolds | journal = Nuclear Physics B | volume = 298 | pages = 493 | doi = 10.1016/0550-3213(88)90352-5 |url=http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198706385|bibcode = 1988NuPhB.298..493C }}
*{{citation| last1=Reid|first1=Miles|year = 1987 | title = The moduli space of 3-folds with K = 0 may nevertheless be irreducible | journal = Math. Ann.| volume = 278 | pages = 329–334}}
*{{citation| last1=Green|first1=Paul|last2=Hübsch|first2=Tristan|year = 1988 | title = Connecting Moduli Spaces of Calabi-Yau Threefolds | journal = Communication in Mathematical Physics | volume = 119 | pages = 431–441 | doi = 10.1007/BF01218081|url=http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198804429|bibcode = 1988CMaPh.119..431G }}
*{{citation| last1=Candelas|first1=Philip|last2=Green|first2=Paul|last3=Hübsch|first3=Tristan|year = 1990 | title = Rolling Among Calabi-Yau Vacua | journal = Nuclear Physics B | volume = 330 | pages = 49–102 | doi = 10.1016/0550-3213(90)90302-T|url=http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198905563|bibcode = 1990NuPhB.330...49C }}
* {{Citation | last1=Hübsch | first1=Tristan | title=Calabi&ndash;Yau Manifolds: a Bestiary for Physicists | publisher=[[World Scientific]] | location=Singapore, New York | isbn=981-02-1927-X | year=1994 | oclc=34989218|url=http://www.worldscibooks.com/physics/1410.html}}
*{{citation| last1=Strominger|first1=Andrew|year = 1995 | title = Massless black holes and conifolds in string theory | journal = Nuclear Physics B | volume = 451 | pages = 96–108| doi=10.1016/0550-3213(95)00287-3|arxiv=hep-th/9504090|bibcode = 1995NuPhB.451...96S }}
*{{citation| last1=Greene|first1=Brian|last2=Morrison|first2=David|last3=Strominger|first3=Andrew|year = 1995 | title =Black hole condensation and the unification of string vacua | journal = Nuclear Physics B | volume = 451 | pages = 109–120| doi=10.1016/0550-3213(95)00371-X|arxiv=hep-th/9504145|bibcode = 1995NuPhB.451..109G }}
*{{citation| last1=Gross|first1=Mark|year = 1997 | title = Primitive Calabi-Yau threefolds | journal = Journal of Differential Geometry | volume = 45 | pages = 288–318|arxiv=alg-geom/9512002|bibcode = 1995alg.geom.12002G }}
*Greene, Brian "[http://arxiv.org/abs/hep-th/9702155v1 String Theory On Calabi&ndash;Yau Manifolds]" (1997)
* {{Citation | last1=Greene | first1=Brian | title=The Elegant Universe | publisher=W.W. Norton & Co. | isbn= 0-393-05858-1 | year=2003}}
*Hübsch, Tristan "[http://homepage.mac.com/thubsch/Research/Conifolds.html Conifolds and 'The (Real Worlds-Wide-)Web']" (2009)
 
[[Category:Algebraic geometry]]
[[Category:Generalized manifolds]]
[[Category:Singularity theory]]
[[Category:String theory]]

Latest revision as of 03:44, 10 October 2014


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