Conifold

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In mathematics, 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.

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 $\mathbb{CP}^4$ . The space $\mathbb{CP}^4$ 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 so that it is 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 $S^2 \times S^3$ .

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, as originally pointed out by Andrew Strominger. Andrew Strominger, together with Dave Morrison and Brian Greene, found that the topology near the conifold singularity can undergo a topological transition. It is believed that nearly all Calabi-Yau manifolds can be connected via these "critical transitions".