Black holes, entropy functionals, and topological strings

This thesis is devoted to study of the connection between extremal black holes and topological strings. Important ingredient of this connection is the relation between Hartle-Hawking wave function associated to black holes and topological string partition function. This leads to a natural entropy functional defined on the moduli space of string compactifications. We discuss several examples of such entropy functionals.; We start by proposing a wave function for scalar metric fluctuations on S3 embedded in a Calabi-Yau. This problem maps to a study of non-critical bosonic string propagating on a circle at the self-dual radius. This can be viewed as a stringy toy model for a quantum cosmology. Then we formulate an entropy functional on the moduli space of Calabi-Yau compactifications. We find that the maximization of the entropy is correlated with the appearance of asymptotic freedom in the effective field theory. The points where the entropy is maximized correspond to points on the moduli which are maximal intersection points of walls of marginal stability for BPS states. We then turn to study of the entropy functional on the moduli space of two dimensional conformal field theories captured by the gauged WZW model whose target space is an abelian variety. This gives rise to the effective action on the moduli space of Riemann surfaces, whose critical points are attractive and correspond to Jacobian varieties admitting complex multiplication. The partition function is a generating function for the number of conformal blocks in rational conformal field theories. Finally, we study non-supersymmetric, extremal 4 dimensional black holes which arise upon compactification of type II superstrings on Calabi-Yau threefolds. We propose a generalization of the OSV conjecture for higher derivative corrections to the non-supersymmetric black hole entropy, in terms of the one parameter refinement of topological string introduced by Nekrasov. We also study the attractor mechanism for non-supersymmetric black holes and show how the inverse problem of fixing charges in terms of the attractor value of Calabi-Yau moduli can be explicitly solved.