The school was billed for beginners, and that is why I am here. Yesterday and today, we heard from Per Kraus and Finn Larsen. Kraus talked under the title "Black Hole Entropy and the AdS/CFT Correspondence" and I hope there will eventually be a video of the talk available online, but we shall see... For the impatient you can alreadywatch/listen to Kraus giving a talk based around his papers Microscopic Black Hole Entropy in Theories with Higher Derivatives and Holographic Gravitational Anomalies both with Fin Larsen. But I think the video of the three hour talk from our school will be much more elementary and welcoming. Hopefully I can make some comments about Fin Larsen's complementary talks in a later post.
Kraus began by telling us about the BTZ black hole (so-called for Banados, Teitelboim and Zanelli), emphasising the point that only for the BTZ black hole does a precise agreement occur between the microscopic and macroscopic counts of black hole entropy . The BTZ black hole is a 3-dimensional black hole similar to the Kerr solution, for a review see Carlip's The (2+1)-Dimensional Black Hole. It lives in three dimensional anti de Sitter space, AdS_3, a space with negative cosmological constant. Identifications in AdS_3 give rise to the BTZ black hole.
AdS_3 can be realised as a hyperboloid in a signature (+,+,-,-), i.e.. This is the Sl(2,R) group manifold. The BTZ black hole can be analysed by looking at the conjugacy classes of Sl(2,R). There are three conjugacy classes: hyperbolic, elliptic and parabolic, with the BTZ black hole sitting in the hyperbolic conjugacy class. Kraus, Samuli Hemming and Esko Keski-Vakkuri have written about this in Strings in the Extended BTZ Spacetime, see section 2. An identification is made with the conjugating elements and the left and right moving temperature, and we move into a thermodynamic setting. Mass, angular momentum, entropy formulae follow, and the equivalence of a thermal AdS_3 background with a BTZ upto various modular transformations in each case.
Kraus considered spacetimes whose near horizon geometry (when r approaches the event horizon, and considering only the dominant terms) factorises into AdS_3 x X x S^p, where X is an unspecified geometry (see Strominger's Black Hole Entropy from Near-Horizon Microstates for the motivation for looking at this geometry). Kraus demonstrated the equivalence of the Wald formula, for finding the entropy from a Lagrangian which includes higher-derivative corrections, with the Cardy density of states formula for a CFT for theories which have a general diffeomorphism invariance. Through this equivalence the exact entropy (i.e. including corrections) is derived solely from knowing the central charges of the theory. Furthermore Kraus presented a variational principle to give the central charge for some Lagrangian with higher derivative terms. In his final talk he looked at the use of gravitational anomalies for learning about the pictures on either side of AdS/CFT.
Kraus used two main examples to illustrate his talk:
He demonstrated how the BTZ black hole appears in each case and compared the entropy calculations in each case. The D1-D5-P entropy (Strominger and Vafa) is in exact agreement with the macroscopic Bekenstein-Hawking entropy, while the M5 branes' microscopic entropy (Maldacena, Strominger and Witten) gives a central charge consisting of two parts, the highest order part agreeing with the macroscopic count and the remainder being due to the presence of higher derivative terms in M-theory. I refer you to the two papers with Larsen linked to earlier to see the full application of the method with these examples in mind.
D1-D5-P on T^4 x S^1 or K3 x S^1 M5 branes wrapped on 4-cycles in M-theory on CY_3 x S^1