100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Space Time in String Theory 325an entropy formula, which agreed parametrically (as a function of variouscharges) with the degeneracies of the states computed in weakly coupledstring theory. Strominger and Vafa 23 found the first example of an extremalblack hole with non-zero classical horizon area where the comparison couldbe made. The near horizon geometry of these black holes was AdS 3 × K,with K a compact manifold. They found that the entropy formula agreedwith that derived from weakly coupled string theory. In the string theorycalculation the entropy is calculated from a 1 + 1 dimensional conformalfield theory describing the world volume of Dirichlet strings (D-1 branes).Even the coefficient in the entropy works. This can be traced to a specialproperty of 1 + 1 CFT’s : the entropy is the same everywhere along a lineof fixed points. Strominger and Vafa were able to compute in a solubleCFT, at regions in moduli space where the black hole horizon is tiny andgravitational effects unimportant, but get the precise extremal black holeentropy because of the BPS property and the invariance of the entropyalong fixed lines.The Strominger Vafa paper set off a flurry of activity. Entropy calculationswere generalized to near extremal black branes, and calculationsof gray body factors for scattering off a black hole at low energy wereperformed 24 . The agreement, particularly in the latter calculation, wherean inclusive cross section is reproduced over a whole energy range, wasspectacular. Furthermore, the successes could no longer be explained byinvoking the BPS property. Maldacena 25 , pondering the reason for thesesuccesses, realized that they could all be explained by a remarkable conjecture.In the perturbative string theory calculations, the black brane wasdescribed by a world volume CFT on a set of D-branes. This theory had theisometry group of AdS d × K (with K some compact manifold) as a quantumsymmetry group - it was a conformal field theory (CFT). This was thenear horizon geometry of the black brane and Maldacena conjectured thatthe CFT was the correct quantum theory of the AdS d × K space-time.Maldacena’s AdS/CFT conjecture was clarified some months later inwork of Gubser, Klebanov and Polyakov 42 and of Witten 27 . These authorsconsidered classical solutions of the supergravity field equations on AdS×K,with boundary conditions on its conformal boundary. They calculated theaction as a functional of the boundary conditions S[φ(b)]. They showed ina few examples that this could be viewed as the generating functional ofconnected Green functions in a CFT living on the boundary of AdS space.The coordinates of the compact manifold were realized as fields in the CFT,much as in Matrix Theory. Thus, classical supergravity was shown to be an