# Topic Suggestions - Kaehler-Einstein metrics in physics

+ 3 like - 0 dislike
376 views

I am hoping to get some topic suggestions for a presentation I have to give in a couple of weeks. The course the presentation is for is called Kahler-Einstein metrics. I would really like the presentation to be geared towards applying the material we've learned to concepts in physics.

We've learned about Kahler manifolds and their properties, and have spent a bunch of time learning about PDEs. Topics have included Sobolev inequalities, elliptic regularity, Schauder theory and other elliptic PDE stuff. We are also studying the Calabi-Yau theorem.

Any topic suggestions and reference requests are appreciated.

This post imported from StackExchange Physics at 2015-03-21 18:21 (UTC), posted by SE-user JonHerman
asked Mar 21, 2015

+ 2 like - 0 dislike

Not really my field of expertise but exact space-times in General relativity are often written as a direct product of a Kähler-Einstein and Einstein manifold. A canonical example would be Minkowski with $${\rm d} s^2 = {\rm d}u {\rm d}v + {\rm d}\zeta {\rm d}\bar{\zeta}, u,v= x \pm t, \zeta = y + iz$$
You can also do a "Wigner cheat" by taking $t \equiv i \tau$ (compensated for elsewhere) and you then have $${\rm d} s^2 = {\rm d}u {\rm d}\bar{u} + {\rm d}\zeta {\rm d}\bar{\zeta}$$ i.e. you have Minkowski as a $\sim \mathbb{C}^2$ Einstein-Kähler manifold and this can be done for all Einstein manifolds. But this is just a trick sometimes useful in the study of exact space-times, I hear that Calabi-Yau spaces bear much importance in string theory and there is surely someone on this forum who can talk on that.

answered Mar 22, 2015 by (1,620 points)
+ 2 like - 0 dislike

An interesting special case of Kaehler-Einstein manifolds are the Calabi-Yau manifolds, which like all other Kaehler-Einstein manifolds (and in general, Kaehler manifolds), have $2k$ real dimensions.

Calabi-Yau manifolds are of interest in string theory, where extra dimensions in 10-dimensional superstring theories are compatified on Calabi-Yau manifolds with 3 complex dimensions (= 6 real dimensions) and holonomy group $SU(3)$.

(A similiar anlog exists for the 11-dimensional M-theory, which can be compactified on a G2-manifold. A G2 manifold is an Einistein manifold, because it's Ricci-flat (and all Ricci-flat manifolds are Einstein manifolds), but not a Kaehler manifold, because it's 7-dimensional (while Kaehler manifolds have an even number of real dimensions). However, one may say that it is "analogous" to Calabi-Yau manifolds, see e.g. [1], [2].)

Note that [1] is an updated version of [2], while [2] has some more interesting answers, so I reference it too.

1. Spiro Karigiannis, What is a G2 manifold?. AMS Notices - 58 (4), 2011.
2. Spiro Karigiannis, Answer to "$G_2$ and geometry" (MathOverflow, 2010).
3. Kaehler manifold (nLab wiki)
4. Robert J Berman, Kahler-Einstein metrics emerging from free fermions and statistical mechanics. ArXiV (2010).
answered Mar 23, 2015 by (1,955 points)
edited Mar 23, 2015

nice references!

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.