Conifold

In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.

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 (Candelas Dale) and employed by (Green Hübsch) to prove that conifolds provide a connection between all (then) known Calabi–Yau compactifications in string theory; this partially supports a conjecture by (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 projective space [math]\displaystyle{ \mathbb{CP}^4 }[/math]. The space [math]\displaystyle{ \mathbb{CP}^4 }[/math] has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations:

[math]\displaystyle{ 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]\displaystyle{ z_i }[/math] on [math]\displaystyle{ \mathbb{CP}^4 }[/math], for any fixed complex [math]\displaystyle{ \psi }[/math], has complex dimension three. This family of quintic hypersurfaces is the most famous example of Calabi–Yau manifolds. If the complex structure parameter [math]\displaystyle{ \psi }[/math] is chosen to become equal to one, the manifold described above becomes singular since the derivatives of the quintic polynomial in the equation vanish when all coordinates [math]\displaystyle{ z_i }[/math] are equal or their ratios are certain fifth roots of unity. The neighbourhood of this singular point looks like a cone whose base is topologically just.

[math]\displaystyle{ 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 D3-branes wrapped on the shrinking three-sphere in Type IIB string theory and by D2-branes wrapped on the shrinking two-sphere in Type IIA string theory, as originally pointed out by (Strominger 1995). As shown by (Greene Morrison), this provides the string-theoretic description of the topology-change via the conifold transition originally described by (Candelas Green), who also invented the term "conifold" and the diagram

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

Candelas, Philip; Dale, A.M.; Lutken, Andrew; Schimmrigk, Rolf (1988), "Complete intersection Calabi-Yau manifolds", Nuclear Physics B 298 (3): 493–525, doi:10.1016/0550-3213(88)90352-5, Bibcode: 1988NuPhB.298..493C, https://cds.cern.ch/record/178648
Reid, Miles (1987), "The moduli space of 3-folds with K=0 may nevertheless be irreducible", Mathematische Annalen 278 (1–4): 329–334, doi:10.1007/bf01458074
Green, Paul; Hübsch, Tristan (1988), "Connecting Moduli Spaces of Calabi-Yau Threefolds", Communications in Mathematical Physics 119 (3): 431–441, doi:10.1007/BF01218081, Bibcode: 1988CMaPh.119..431G, http://projecteuclid.org/euclid.cmp/1104162497
Candelas, Philip; Green, Paul; Hübsch, Tristan (1990), "Rolling Among Calabi-Yau Vacua", Nuclear Physics B 330 (1): 49–102, doi:10.1016/0550-3213(90)90302-T, Bibcode: 1990NuPhB.330...49C
Strominger, Andrew (1995), "Massless black holes and conifolds in string theory", Nuclear Physics B 451 (1–2): 96–108, doi:10.1016/0550-3213(95)00287-3, Bibcode: 1995NuPhB.451...96S
Greene, Brian; Morrison, David; Strominger, Andrew (1995), "Black hole condensation and the unification of string vacua", Nuclear Physics B 451 (1–2): 109–120, doi:10.1016/0550-3213(95)00371-X, Bibcode: 1995NuPhB.451..109G

v t e String theory
Background	Strings History of string theory First superstring revolution Second superstring revolution String theory landscape
Theory	Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism GS formalism
String duality	T-duality S-duality U-duality Montonen–Olive duality
Particles and fields	Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon
Branes	D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition
Conformal field theory	Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model
Gauge theory	Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G₂, F₄, E₆, E₇, E₈) ADE classification Dirac string p-form electrodynamics
Geometry	Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G₂ manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory Twisted K-theory
Supersymmetry	Supergravity Superspace Lie superalgebra Lie supergroup
Holography	Holographic principle AdS/CFT correspondence
M-theory	Matrix theory Introduction to M-theory
String theorists	Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Cleaver Curtright Dijkgraaf Distler Douglas Duff Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Naqvi Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind 't Hooft Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach

Conifold

Overview

References

Further reading