Paper about Sasaki-Einstein manifolds

Hi, can you give me a good paper (in the sense of a simple introduction) about Sasaki-Einstein manifolds?

Thank you and best regards Florian M.

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Have you tried the references at the Wikipedia article? scholar.google.com? – Qiaochu Yuan Jul 31 '10 at 8:19
This is not yet a real question, as the concept of a "good paper" is vague and subjective. Please add more information about what you're looking for. – Pete L. Clark Jul 31 '10 at 8:26
Okay, I have added "in the sense of a simple introduction" :) Thank you and best regars Florian M. – Differentialgeometer Jul 31 '10 at 8:31

2 Answers

Well, now there is a great textbook for Sasaki-Einstein geometry, by Boyer-Galicki: Here is a link to the book at Oxford University Press

I would definitely start there.

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+1 for mentioning this book! It's the bible on Sasakian geometry. – José Figueroa-O'Farrill Jul 31 '10 at 13:19
Thank you very much! :-) – Differentialgeometer Jul 31 '10 at 13:34

I agree with Spiro's recommendation to pick up the book by Boyer and Galicki, but I would like to add the following.

There are at least two ways to get to Sasaki-Einstein manifolds. The traditional approach is to consider them as special cases of contact manifolds and I believe that this is the point of departure, or at least the main thrust of the above book. But another way, and in my case the way I first met them, is via their characterisation as those manifolds whose metric cone is Calabi-Yau, which is particularly transparent in the context of spin geometry. Said differently, a Calabi-Yau manifold admits parallel spinor fields, and this means that a Sasaki-Einstein manifold admits real Killing spinor fields. The relation between the two, as well as the uniform way of recovering the contact structure and associated Sasakian paraphernalia from the Killing spinors, is explained in the beautiful and short paper of Christian Bär's Real Killing spinors and holonomy, which itself can be thought of as a lucid introduction to this aspect of Sasaki-Einstein manifolds.

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Thanks Pete, for the edit and sorry, Spiro, for adding the "s". – José Figueroa-O'Farrill Jul 31 '10 at 14:18
No problem. The extra 's' is actually on my birth certificate, but I never use it professionally. – Spiro Karigiannis Jul 31 '10 at 16:00