Topic Suggestions - Kaehler-Einstein metrics in physics

Question

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

Answer 1

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.

Answer 2

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).

Comments

nice references!