Conifold

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 CP^4. The space 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_5^5-5psi 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 psi 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 imes 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".

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Cone: A cone is a basic geometrical shape: see cone (solid). Several things have also been called "cones" on account of their shape:...

Dimension: Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space....

String theory: String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that are the basis of the Standard Model of particle physics. For this reason, string theories are able to avoid problems associat...