Conifold

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In mathematics, a conifold is a generalization of the notion of a manifold. Unlike manifolds, a conifold can (or should) contain conical singularities i.e. points whose neighborhood looks like a cone with a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five real-dimensional manifold.

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.

A well-known example of a conifold is obtained as a deformed limit of the quintic - i.e. the quintic hypersurface in the projective space $C P 4$ . The space $C P 4$ has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equation

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

for the homogeneous coordinate $z i$ has complex dimension three; it is the most famous example of a Calabi-Yau manifold. If the complex structure parameter $ψ$ is chosen equal to one, the manifold described above becomes singular because the derivatives of the quintic polynomial in the equation vanish when all coordinates $z i$ are equal to each other (or their ratios are certain fifth roots of unity). The neighborhood of this singular point then looks like a cone whose base is topologically equivalent to a product of two spheres, namely $S^2 \times S^3$ .

In the context of string theory, the geometrically singular conifolds were shown to lead to completely smooth physics of strings. The divergences are "smeared out" by D3-branes wrapped on the shrinking three-sphere, as originally pointed out by Andrew Strominger. Andrew Strominger together with Dave Morrison and Brian Greene have also found that the topology near the conifold singularity can undergo a topology transition. It is believed that nearly all Calabi-Yau manifolds can be connected via these "critical transitions".