Black Holes and the AdS/CFT Correspondence

String theory has been acknowledged as the most promising theory for quantising gravity and for unifying all forces of nature, with its development bringing an incredible amount of ideas into mathematics and physics. One its most useful discoveries has been a duality between a particular string theory and a quantum theory very similar to that describing the interactions between the fundamental particles that make out our world. This duality is commonly referred as the AdS/CFT correspondence, and it is referred as a duality, since both theories, even though intrinsically different, describe the same physics. Furthermore, on each side physicists can perform calculations in regimes which are inaccessible by the dual theory. The discovery of the AdS/CFT correspondence, has originated many exciting developments. My research programme intends to use the AdS/CFT duality as a way of learning about phenomena that cannot be accessed in a given picture, but can be studied in the dual picture. In particular, I am interested in black hole physics, as they might lead to a more universal understanding of black hole entropy and its quantum corrections. Black holes were predicted by General Relativity and are now supported by experimental evidence. In string theory, they can be understood in terms of strings and branes, which are multidimensional objects. It has been a long-standing problem to give a microscopic description to the entropy of a black hole, and string theory has been able to provide explicit examples in which this can be achieved. Several black holes can be studied using the AdS/CFT correspondence if their horizon geometry happens to contain an anti-de Sitter (AdS) factor. This is a common feature that arises in black holes built from strings and branes. It is then natural to think of using the AdS/CFT conjecture to compute the entropy for a wider class of black holes, while matching all possible corrections from both sides exactly. The achievement of this goal would give us a fundamental understanding of quantum gravity. For black holes with underlying 3D AdS factors, it is possible to define a 2D dual theory which is said to be defined on the boundary. I am concerned with black hole geometries that contain these AdS factors in their geometries, and with their interpretation from the dual theory. In particular, it is of great interest to study the theory dual to a black hole formed from N strings, for which several proposals have been made, and to give an explicit realisation of the symmetries of the black hole from the dual theory. Black holes with 2D AdS factors are less understood, as there is less information that can be gained from the AdS/CFT correspondence, given that it is not clear which is the dual theory. It was recently determined that 2D AdS gravity in the presence of an electric field, can be related to a 2D conformal theory (CFT), in the same way as 3D AdS is related to a 2D CFT. It would be gratifying to understand the dual theory and to build it explicitly, as it would also establish the connection between the dual theory to 2D AdS and 3D AdS. Another interesting aspect in which black holes and the AdS/CFT duality are relevant, is related to the recently emergent connection between hydrodynamics and black hole physics. Using the AdS/CFT correspondence it has been possible to describe thermal plasmas in terms of black holes in AdS space, and in particular it has been found that the value of the ratio between the shear viscosity and the entropy density is fixed an of order 1/10. However, it is not clear if such behaviour should hold for small black holes, which have vanishing entropy at leading order, and lead to a singular ratio. Given that these black holes have near horizon geometries containing AdS factors, it is of interest to compute the ratio for a small black hole, as it will enlarge our understanding of these objects.