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"). a quintic hypersurface in the projective space C P 4 has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations:įor the purpose. (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.Ī well-known example of a conifold is obtained as a deformation limit of a quintic - i.e. This possibility was first noticed by Candelas et al. 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.Ĭonifolds 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. points whose neighbourhoods look like cones over a certain base. Unlike manifolds, conifolds can contain conical singularities, i.e. In mathematics and string theory, a conifold is a generalization of a manifold.
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