Conifold

In mathematics and string theory, 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, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.

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. This possibility was first noticed by Candelas et al. (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.

A well-known example of a conifold is obtained as a deformation limit of a quintic - i.e. a quintic hypersurface in the projective space ${\displaystyle \mathbb {CP} ^{4}}$. The space ${\displaystyle \mathbb {CP} ^{4}}$ has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations

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