Saturday, October 08, 2005

Black hole entropy, special geometry, and dynamical systems

I ended my presentation at the JPS on Septermber 14 and then I heard the talk of Y. Yasui (Osaka City Univ.) on the relation between Kerr black holes (black holes with several charges) and Sasaki-Einstein real 5-fold. I asked whether we can explicitly write "many" metrics for the compact G_2 holonomy manifolds of Joyce and Kovalev after the development of Bryant-Salamon and he said it is like asking whether we can write metrics of K3 surfaces. (which I heard that Eguchi and Yau once tried.) Although string theorists are ordinary interested in non-compact G_2 holonomy manifolds, works like Atiyah-Witten of cone over CP^3 after Kovalev should be an example that we should take care of. One month later, Choissi-Fino(math.DG/0510087) seem to construct an exactly G_2 holonomy metric possibly related to the Hitchin flow equation.

OK, I will write somehow on the story after I went back to the University of Tokyo. Masaki Shigemori performed two seminars (formal and informal) on "Massless black holes and black rings as effective geometries of the D1-D5 system" (hep-th/0508110) and black rings. In that place, we heard something about the Poincare recurrence. He said that the unitarity (preservation of probability) of blackhole systems is apparently broken at the long period, but it will be restored in the longer period. However, Eguchi said we should think about infinite-degree systems because it is a (quantum) field theory.

After the lecture, I reviewed some 19th century physics of statistics from modern ergodic theory point of view, something like Wiener recurrence theorem and infinite Markov process by books of Bunimovich and Aaronson. And then I looked up some papers on the relation between the Banach space theory and entropies, which include "Dynamical entropy in Banach spaces" by David Kerr and Hanfeng Li (Invent. math. (2005)) or "Schreier sets in Ramsey theory" by V. Farmaki AND S. Negrepontis (math.CO/0510102). I hope something like non-separable Banach space theory and the theory of Gowers will find some realistic applications.