If I am not mistaken, no explicit metric on a compact Calabi-Yau manifold is known. I guess part of the difficulty is due to the fact that compact Calabi-Yau manifolds do not admit continuous isometries and therefore there is no adapted system of coordinates that can be used to write down the metric in a simpler, explicit way. In fact, in the non-compact case there are explicit examples of Calabi-Yau metrics.

Now, compact Calabi-Yau manifolds are extremely important in String/M/F-theory: they are for example the compactification spaces of fluxless String-Theory compactifications and they are the general compactification spaces of F-theory compactifications, which are very relevant from the phenomenological point of view. Knowing an explicit Calabi-Yau metric would be interesting, because among many other things then one could explicitly know the non-linear sigma model of the string propagating in that background.

My question is then: is this actually an active field of research? What is the actual status of the problem? I don't recall seeing many papers (or any) oriented towards solving this problem and finding an explicit metric.


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    $\begingroup$ Funny, I just saw Andrew Neitzke give a talk about exactly this. He has an idea for a construction using saddle connections, and his initial motivation comes from string theory. I wish I had taken better notes! Should be a paper coming out soon though. $\endgroup$ – Alex Zorn Feb 15 '15 at 19:54
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    $\begingroup$ Maybe it is worth mentionning that some work has been done to approximate numerically Ricci flat metrics on compact Calabi-Yau manifolds: see for example arxiv.org/pdf/math/0512625v1.pdf and arxiv.org/pdf/hep-th/0612075.pdf The first paragraph of the second paper contains the phrase: "it is widely thought that for compact Calabi-Yau manifolds no closed form expression exists, except in trivial cases", which makes me pessimistic on the existence of a positive answer to the question. $\endgroup$ – user25309 Feb 15 '15 at 20:14
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    $\begingroup$ A reply to Alex's comment: The construction I described concerns some special examples of Calabi-Yau manifolds, namely hyperkahler integrable systems, more specifically Hitchin's integrable system. In particular they are not compact. An analogue of the method may apply to compact CY which are also hyperkahler integrable systems, such as K3 surfaces, but for this to work, some serious issues of convergence will have to be overcome. Finally, this construction is not mine alone, it is joint with Davide Gaiotto and Greg Moore. A very brief summary is at arxiv.org/abs/1308.2198 $\endgroup$ – Andy Neitzke Feb 16 '15 at 17:55

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