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...ack hole in three dimensional theory of gravity with negative cosmological constant [109] and often appears as a factor in the near horizon geometry of higher dimensional black holes in string theory =-=[110, 111, 112]-=-. For this reason it has provided us with a useful tool for relating black hole entropy to the degeneracy of microstates of the black hole, both in three dimensional theories of gravity and also in st...

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...CFT correspondence has been explored. Finally, we would like to mention here a surprising motivation to study three-dimensional gravity. It has been shown in [49] and [50] (see also the recent review =-=[51]-=-) that there exists duality transformations relating fivedimensional black holes with three-dimensional ones. This means that everything we can learn about three-dimensional quantum gravity can be use...

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...hat three-dimensional gravity has no local degree of freedoms, and counts microstates in a two-dimensional conformal field theory living on the boundary of space-time. This is for example reviewed in =-=[32]-=-. Finally the developement of the AdS-CFT correspondence is closely interrelated with various aspects of black hole physics [33]. All these approaches are not tied to BPS states. This opens the perspe...

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...f Strominger’s [10] proposal for the statistical mechanical origin of the three-dimensional black hole entropy. We refer the reader to [11–14] for the stringy aspects of Strominger’s result. See also =-=[15]-=- for a recent review. We shall concentrate here on the gravitational aspects. For a detailed and complete treatment of three dimensional gravity we refer to the recent book by Carlip [16]. In Sec. II ...

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...s N=(4,4) σ-model. We summarize some related facts of IIB supergravity here. IIB supergravity on S1 × K3 whose radius and volume are R and (2π) 4α ′ 2v has the BPS solitonic solution (see for example =-=[13]-=- and references therein): ds 2 10 = f −1 2 1 f −1 2 +f 1 2 1 f 1 2 5 {−dt 2 + dx 2 5 + fN(dt + dx5) 2 } 5 (dx 2 1 + dx22 + dx23 + dx2 1 2 4 ) + f 1 f −1 2 5 ds 2 K3 , (2.15) with periodic identificati...

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...er, the central charge is a robust quantity which can often be determined quite easily by anomaly considerations. This makes it easy to carry out a microscopic calculation of the entropy, [27], [28], =-=[29]-=-. In addition, when the condition, eq.(1.1) is valid subleading corrections to the entropy can also often be easily calculated. These continue to have the form, eq.(1.2). The subleading corrections ar...

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...er, the central charge is a robust quantity which can often be determined quite easily by anomaly considerations. This makes it easy to carry out a microscopic calculation of the entropy, [27], [28], =-=[29]-=-. In addition, when the condition, eq.(1.1) is valid subleading corrections to the entropy can also often be easily calculated. These continue to have the form, eq.(1.2). The subleading corrections ar...

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...evance of the brane solutions to black hole physics. A nice introductory review to black holes is[442]. Branes in the context of black hole physics are reviewed e.g. in[337, 338], [46], [464], [362], =-=[428]-=-, [151], [344, 345], [348]. 6.2.2 Research Papers The elementary string solution was found in[119]. The five brane solution has been considered e.g. in [433], [154], [97]. The general p-brane solution...