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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Conifold ： ウィキペディア英語版 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 and employed by to prove that conifolds provide a connection between all (then) known Calabi–Yau compactifications in string theory; this partially supports a conjecture by 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 $\backslash mathbb^4$. The space $\backslash mathbb^4$ has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations :$z\_1^5+z\_2^5+z\_3^5+z\_4^5+z\_5^5-5\backslash psi\; z\_1z\_2z\_3z\_4z\_5\; =\; 0$ in terms of homogeneous coordinates $z\_i$ on $\backslash mathbb^4$, for any fixed complex $\backslash psi$, has complex dimension three. This family of quintic hypersurfaces is the most famous example of Calabi–Yau manifolds. If the complex structure parameter $\backslash psi$ 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 $z\_i$ 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 $S^2\; \backslash times\; S^3$. 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 . As shown by , this provides the string-theoretic description of the topology-change via the conifold transition originally described by , 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 and external links== * * * * * * * * *Greene, Brian "(String Theory On Calabi–Yau Manifolds )" (1997) * *Hübsch, Tristan "(Conifolds and 'The (Real Worlds-Wide-)Web' )" (2009) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Conifold」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.