TASI Lectures on Branes, Black Holes and Anti-de Sitter Space
M. J. Duff
TL;DR
Duff reviews the M2, D3 and M5 branes and their near-horizon AdS×S geometries, deriving the corresponding compactifications of D=11 supergravity on S^7, Type IIB supergravity on S^5, and D=11 supergravity on S^4. The text links the bulk AdS physics to boundary singleton theories, and develops non-linear KK truncations that yield multi-charge AdS black holes whose ten-/eleven-dimensional embeddings reinterpret charges as brane angular momenta. It then connects these structures to AdS/CFT, the membrane/supergravity bootstrap, and the rotating-brane perspective on black hole microphysics, while detailing explicit KK Ansätze and brane actions. The work also surveys rotating AdS black holes, magnetic solutions, and the near-horizon limits that reveal enhanced conformal symmetries, with extensive appendices listing the full equations and transformation rules. Overall, the lectures provide a unifying framework for branes, AdS vacua, and holographic dualities, and supply tools to embed lower-dimensional AdS black holes into M-/string theory. The results have lasting impact on holography, consistent truncations, and the interpretation of AdS black holes in higher dimensions.
Abstract
In the light of the duality between physics in the bulk of anti-de Sitter space and a conformal field theory on the boundary, we review the M2, D3 and M5 branes and how their near-horizon geometry yields the compactification of D=11 supergravity on S^{7}, Type IIB supergravity on S^{5} and D=11 supergravity on S^{4}, respectively. We discuss the ``Membrane at the End of the Universe'' idea and its relation to the corresponding superconformal singleton theories that live on the boundary of the AdS_{4}, AdS_{5} and AdS_{7} vacua. The massless sectors of these compactifications are described by the maximally supersymmetric D=4, D=5 and D=7 gauged supergravities. We construct the non-linear Kaluza-Klein ansätze describing the embeddings of the U(1)^4, U(1)^3 and U(1)^2 truncations of these supergravities, which admit 4-charge AdS_{4}, 3-charge AdS_{5} and 2-charge AdS_{7} black hole solutions. These enable us to embed the black hole solutions back in ten and eleven dimensions and reinterpret them as M2, D3 and M5 branes spinning in the transverse dimensions with the black hole charges given by the angular momenta of the branes. A comprehensive Appendix lists the field equations, symmetries and transformation rules of D=11 supergravity, Type IIB supergravity, and the M2, D3 and M5 branes.