Conifold

Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base.

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.

has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations: in terms of homogeneous coordinates

is chosen to become equal to one, the manifold described above becomes singular since the derivatives of the quintic polynomial in the equation vanish when all coordinates

The divergences are "smeared out" by D3-branes wrapped on the shrinking three-sphere in Type IIB string theory and by D2-branes wrapped on the shrinking two-sphere in Type IIA string theory, as originally pointed out by Strominger (1995).

As shown by Greene, Morrison & Strominger (1995), this provides the string-theoretic description of the topology-change via the conifold transition originally described by Candelas, Green & Hübsch (1990), who also invented the term "conifold" and the diagram for the purpose.

It is believed that nearly all Calabi–Yau manifolds can be connected via these "critical transitions", resonating with Reid's conjecture.