Ivancevic_Applied-Diff-Geom

1226 Applied Differential Geometry: A Modern Introductionsquare of the partition function for a gas of topological strings on a Calabi–Yau whose moduli are those associated to the charges/potentials (p Λ , φ Λ )via the attractor equations [Ooguri et. al. (2004)]. Both sides of (6.336)are defined in a perturbation expansion in 1/Q, where Q is the graviphotoncharge carried by the black hole. 29 The non–perturbative completion ofeither side of (6.336) might in principle be defined as the partition functionof the holographic CFT dual to the black hole, as in [Strominger and Vafa(1996)]. Then we have the triple equality,Z CF T = Z BH = |Z top | 2 .The existence of fundamental connection between 4D black holes andthe topological string might have been anticipated from the following observation.Calabi–Yau spaces have two types of moduli: Kähler and complexstructure. The world–sheet twisting which produces the A (B) model topologicalstring from the critical superstring eliminates all dependence on thecomplex structure (Kähler) moduli at the perturbative level. Hence theperturbative topological string depends on only half the moduli. Blackhole entropy on the other hand, insofar as it is an intrinsic property of theblack hole, cannot depend on any externally specified moduli. What happensat leading order is that the moduli in vector multiplets are driven toattractor values at the horizon which depend only on the black hole chargesand not on their asymptotically specified values. Hypermultiplet vevs onthe other hand are not fixed by an attractor mechanism but simply dropout of the entropy formula. It is natural to assume this is valid to all ordersin a 1/Q expansion. Hence the perturbative topological string and the largeblack hole partition functions depend on only half the Calabi–Yau moduli.It would be surprising if string theory produced two functions on the samespace that were not simply related. Indeed [Ooguri et. al. (2004)] arguedthat they were simply related as in (6.336).Supergravity Area–Entropy FormulaRecall that a well–known hypothesis by J. Bekenstein and S. Hawking statesthat the entropy of a black hole is proportional to the area of its horizon(see [Hawking and Israel (1979)]). This area is a function of the black holemass, or in the extremal case, of its charges. Here we review the leadingsemiclassical area–entropy formula for a general N = 2, d = 4 extremalblack hole characterized by magnetic and electric charges (p Λ , q Λ ), recently29 The string coupling g s is in a hypermultiplet and decouples from the computation.