Conifold

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.

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 et al. (1988) and employed by 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 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 \mathbb{CP}^4. The space \mathbb{CP}^4 has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations

z_1^5+z_2^5+z_3^5+z_4^5+z_5^5-5\psi z_1z_2z_3z_4z_5 = 0

in terms of homogeneous coordinates zi on \mathbb{CP}^4, for any fixed complex ψ, has complex dimension three. This family of quintic hypersurfaces is the most famous example of Calabi-Yau manifolds. If the complex structure parameter ψ 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 zi 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 S^2 \times S^3.

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 & Strominger (1995), this provides the string-theoretic description of the topology-change via the conifold transition originally described by Candelas, Green & Hübsch (1990), who also invented the term "conifold" and the diagram

3FoldConifoldTransition.pdf

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


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • conifold — noun A certain generalization of a manifold …   Wiktionary

  • Transition de conifold — La transition de conifold n est pas très différente de la transition de flop seulement, les accrocs et les déchirures sont beaucoup plus cataclysmiques. C est aussi une évolution de la composante d un espace de Calabi Yau où le tissu même de l… …   Wikipédia en Français

  • Topological string theory — In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry.… …   Wikipedia

  • Brian Greene — Infobox Scientist name = Brian Greene image width = 170px caption = At the launch of the World Science Festival, April 2008 birth date = Birth date and age|mf=yes|1963|2|9 birth place = New York City, U.S. residence = United States nationality =… …   Wikipedia

  • Transition geometrique — Transition géométrique Une transition géométrique est un changement de l espace de compactification d une théorie des cordes qui peut faire intervenir un changement de topologie mais sous lequel la physique de la théorie est inchangée. Il existe… …   Wikipédia en Français

  • Transition géométrique — Une transition géométrique est un changement de l espace de compactification d une théorie des cordes qui peut faire intervenir un changement de topologie mais sous lequel la physique de la théorie est inchangée. Il existe entre autres : la… …   Wikipédia en Français

  • Calabi-Yau — Variété de Calabi Yau Un exemple de variété de Calabi Yau Une variété de Calabi Yau, ou espace de Calabi Yau est un type particulier de variété en mathématiques intervenant dans des domaines comme la géométrie algébrique mais également en… …   Wikipédia en Français

  • Espace de Calabi-Yau — Variété de Calabi Yau Un exemple de variété de Calabi Yau Une variété de Calabi Yau, ou espace de Calabi Yau est un type particulier de variété en mathématiques intervenant dans des domaines comme la géométrie algébrique mais également en… …   Wikipédia en Français

  • Histoire de la théorie des cordes — Cet article résume l histoire de la théorie des cordes. La théorie des cordes est une théorie de la physique moderne qui tente d unifier la mécanique quantique (physique aux petites échelles) et la théorie de la relativité générale (nécessaire… …   Wikipédia en Français

  • Theorie des cordes — Théorie des cordes Les niveaux de grossissements : monde macroscopique, monde moléculaire, monde atomique, monde subatomique, monde des cordes. La théorie des cordes est l une des voies envisagées pour régler une des questions majeures de la …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”