What is a Conifold and How is it Related to Orbifold Singularities?

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Discussion Overview

The discussion revolves around the concepts of conifolds and orbifold singularities, exploring their definitions, relationships, and mathematical properties. Participants examine specific manifolds, their embeddings, and the implications of these structures in higher-dimensional spaces.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents a parameterization of a Clifford torus and inquires about a similar manifold resulting from the product of two spheres, suggesting a straightforward parameterization in six dimensions.
  • Another participant questions the existence of a specific name for the embedding of S² x S² into R⁶, paralleling the naming of the Clifford torus.
  • A different participant introduces the idea that S² x S² can be viewed as a Lagrangian submanifold of C³ and discusses associated structures, including U(1) bundles and specific examples like T^{0,1} and T^{1,1}.
  • One participant raises a question about whether a conifold is simply defined as an orbifold with one singular point, elaborating on the local behavior of the metric near singularities.
  • Another participant echoes this inquiry, emphasizing the conifold as a local model for isolated singularities and its relation to orbifold singularities.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of conifolds and orbifold singularities, with no consensus reached on specific terminologies or the nature of these mathematical structures.

Contextual Notes

Some participants note potential technical details that may be overlooked, particularly from a physicist's perspective, regarding the definitions and behaviors of conifolds and orbifold singularities.

Sorento7
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We know that a Clifford torus is parameterized in 4D euclidean space by:
(x1,x2,x3,x4) = (Sin(theta1), Cos(theta1), Sin(theta2), Cos(theta2))
{0<=theta1 and theta2<2pi}

Consider that a clifford torus is the immediate result of Circle * Circle

Now, have you encountered a similar manifold which is a result of Sphere * Sphere?
The parameterization is quite straight-forward in 6 dimensions:
(x1,x2,x3,x4,x5,x6)=(Sin(theta1)Cos(phi1), Cos(theta1)Cos(phi1), Sin(phi1), Sin(theta2)Cos(phi2), Cos(theta2)Cos(phi2), Sin(phi2))
{0<= All angles< 2pi}

Does there exist any name for this special 4D manifold?
 
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I don't think so.
 
If you view the 2-sphere as a complex projective space, then \mathbb{P}^1\times \mathbb{P^1} is the 0th Hirzebruch surface, as well as the exceptional del Pezzo surface.
 
I think the question is not so much "what are other names for S² x S²?" as it is "is there a name for the obvious embedding of S² x S² into R^6=R³ x R³?".

Because that is what the Clifford torus is: it is just a name for the obvious embedding of S^1 x S^1 into R^4 = R² x R².
 
fred.
 
quasar987 said:
I think the question is not so much "what are other names for S² x S²?" as it is "is there a name for the obvious embedding of S² x S² into R^6=R³ x R³?".

Because that is what the Clifford torus is: it is just a name for the obvious embedding of S^1 x S^1 into R^4 = R² x R².

Ah, got it. As an aside, the range of angles quoted is wrong to get 2-spheres, but this is easily corrected. In fact there is a very beautiful structure associated with this type of embedding. I will mention a few highlights.

This S^2\times S^2 is a Lagrangian submanifold of \mathbb{C}^3. He hasn't specified the other angle, but it can be chosen to give a U(1) bundle over S^2\times S^2 with a connection 1-form that is

A_\psi = p \cos\theta_1 d\phi_1 + q \cos\theta_2 d\phi_2.

These bundles are known as T^{p,q}. Two special examples are T^{0,1} = S^2\times S^3, which uses the Hopf fibration, while

T^{1,1} = SU(2)\times SU(2)/U(1).

It turns out that T^{1,1}, viewed as the base of the conical metric on \mathbb{C}^3, is compatible with the Kahler structure on \mathbb{C}^3 The metric on T^{1,1} can be chosen to be Einstein, which makes it a nontrivial example of an Einstein-Sasaki manifold.

In fact, an explicit metric can be written down that describes the small resolution of the singularity in the orbifold \mathbb{C^3/Z_3} viewed as a Calabi-Yau manifold (this is also known as the conifold singularity). The resolved conifold can also be viewed as the total space of the bundle \mathcal{O}(-1)\oplus \mathcal{O}(-1) \rightarrow \mathbb{P^1}.
 
Interesting! Thanks.

Is a conifold defined as just an orbifold with 1 singular pt?
 
quasar987 said:
Interesting! Thanks.

Is a conifold defined as just an orbifold with 1 singular pt?

In this case, it is any singularity such that the metric in a small neighborhood of the singular point behaves like

ds^2 \sim dr^2 + r^2 d\Omega^2.

There are probably some more technical mathematical details that, as a physicist, I will get wrong. The conifold is a local model for the behavior near isolated singularities of some larger manifold. I'm only familiar with the case where these singularities are describable as orbifold singularities of some neighborhood of the singular point.
 

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