Lectures on Black Holes and the AdS3/CFT2 Correspondence
We present a detailed discussion of AdS_3 black holes and their connection to two-dimensional conformal field theories via the AdS/CFT correspondence. Our emphasis is on deriving refined versions of black hole partition functions that include the effect o
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Abstract We present a detailed discussion of AdS3 black holes and their connection to two-dimensional conformal field theories via the AdS/CFT correspondence. Our emphasis is on deriving refined versions of black hole partition functions that include the effect of higher derivative terms in the spacetime action as well as nonperturbative effects. We include background material on gravity in AdS3 , in the context of holographic renormalization.
1 Introduction The fact that string theory is able to provide a successful microscopic description of certain black holes provides strong evidence that it is a consistent theory of quantum gravity. Correctly reproducing the Bekenstein-Hawking entropy formula S = A/4G from an explicit sum over states indicates that the right microscopic degrees of freedom have been identified. Since string theory also reduces to conventional general relativity (coupled to matter) at low energy, it seems to provide us with a coherent theory encompassing both the microscopic and macroscopic regimes. Needless to say, however, there is still much to be learned about the full implications of string theory for quantum gravity. One approach to deepening our understanding is to examine the string theory description of black holes with improved precision. This program has been highly fruitful so far. The earliest successful black hole entropy matches, following [1], appeared somewhat miraculous, the emergence of the Bekenstein-Hawking formula from the microscopic side not becoming apparent until all the last numerical factors were accounted for. The precise agreement seemed even more astonishing once additional features like rotation and non-extremality were included. It was eventually understood that the essential ingredients on the two sides are the near horizon AdS region of the black hole geometry, and the low energy CFT describing P. Kraus Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547, USA, [email protected] Kraus, P.: Lectures on Black Holes and the AdS3 /CFT2 Correspondence. Lect. Notes Phys. 755, 193–247 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-79523-0 4
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the underlying branes, and this (among other observations) led to the celebrated AdS/CFT correspondence [2, 3]. This improved understanding largely demystifies the nature of the entropy matching, as we will discuss in these lectures. One of our goals here will be to show how just a few basic features, like the existence of a near horizon AdS region with the appropriate symmetries, is enough to make the agreement manifest, even in rather complicated contexts, and including incorporating subleading corrections to the area law formula. A survey of the examples in which there is a precise microscopic accounting of black entropy reveals the near ubiquitous appearance of a near horizon AdS3 factor (possibly after a suitable duality transformation).1 In these examples, the dual theory is a two-dimensional CFT, for which there are powerful results constraining the spectrum of states
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