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I would like to know which explicit metrics on non-compact Calabi-Yau (CY) threefolds are known.

For instance, an important class of such spaces can be constructed algebraically, including local $\mathbb{CP}^1$ (a.k.a. the resolved conifold), local $\mathbb{CP}^2$, local $\mathbb{CP}^1 \times \mathbb{CP}^1$, and the deformed conifold. However, as far as I have searched the math and physics literature, I have found explicit CY metrics only in the case of the resolved and deformed conifold.

Is the CY metric for, e.g., local $\mathbb{CP}^2$ known? What about other cases?

(I have followed terminology from this paper).

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  • $\begingroup$ My understanding is that very few Kaehler-Einstein metrics for Calabi-Yaus are known. In the compact case, Numerical Kaehler-Einstein metric on the third del Pezzo by C. Doran, M. Headrick, C. P. Herzog, J. Kantor, T. Wiseman uses numerical techniques to compute an appropriate metric. $\endgroup$
    – Ursula
    Commented Jul 22, 2015 at 18:58
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    $\begingroup$ Are you OK with metrics on the singular cone, or do you want a metric that extends to the resolution? There was a flurry of activity in finding irregular Sasaki-Einstein metrics on S^2 x S^3 (and maybe other low del Pezzos?) about a decade ago. You might start with hep-th/0411238. $\endgroup$
    – Aaron Bergman
    Commented Jul 22, 2015 at 23:03
  • $\begingroup$ Thank you Aaron for your comment! I am actually OK with singular cone metrics and the Ypq metrics you point out are an excellent example. In the paper you point out they mention that these spaces can be realized as GLSMs in the ultraviolet. Do you know however if the explicit RG flow from the UV to the deep IR has/can be described? (At least in some cases such as the conifold?). $\endgroup$
    – Nuno
    Commented Jul 22, 2015 at 23:31
  • $\begingroup$ I'm not sure exactly what you're looking for, but I've been out of the field long enough that I likely wouldn't remember anyways. Sorry. $\endgroup$
    – Aaron Bergman
    Commented Jul 23, 2015 at 4:00

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The simplest example would be on the, non-small resolution, $\mathbf{K}_{\mathbb{CP}^1 \times\mathbb{CP}^1}$ of $\mathbb{CP}^1 \times\mathbb{CP}^1$. This the C-Y metric is the Calabi Ansazt using the usual homogeneous metric on $\mathbb{CP}^1 \times\mathbb{CP}^1$.

The resolved conifold, i.e. the small resolution you have in mind, has a C-Y metric, but I don't recall if it has an explicit formula.

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