Conifold
From Academic Kids
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. The base is usually a five-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 <math>CP^4<math>. The space <math>CP^4<math> has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equation
- <math>z_1^5+z_2^5+z_3^5+z_4^5+z_5^5-5\psi z_1z_2z_3z_4z_5 = 0<math>
for the homogeneous coordinate <math>z_i<math> has complex dimension three; it is the most famous example of a Calabi-Yau manifold. If the complex structure parameter <math>\psi<math> 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 <math>z_i<math> 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 <math>S^2 \times S^3<math>.
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".Template:Geometry-stub
