Black Hole Entropy in String Theory

Abstract

In the first part of the thesis we examine the entropy of black strings in compactifications of F-theory on an elliptically fibered Calabi-Yau threefold or of type IIB string theory on K3. The black strings arise as D3-branes wrapped on a curve in either the base of the Calabi-Yau threefold or in K3 and when compactified on an additional circle, they reduce to black holes. Macroscopically, they are described by solutions of 6d (1,0) supergravity coupled to tensor multiplets. We derive a general local form for supersymmetric solutions of this theory after which we consider several special cases in which the resulting equations can be solved explicitly. In particular, we derive expressions for the black string solutions we are interested in. In F-theory we consider black string solutions that have an asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) space transversely. The black strings are dual to 2d (0,4) superconformal field theories (SCFTs) that arise as the IR fixed point of the compactified worldvolume theory of the branes. The entropy of the black strings is via the Cardy formula expressed in terms of the central charges and levels of the SCFT. We compute this data using the macroscopic black string solution. The central charges and levels correspond to coefficients of Chern-Simons terms in the 3d action resulting from the reduction of the 6d supergravity theory on the compact space surrounding the string. It turns out that to capture degrees of freedom living outside of the horizon one must perform this reduction at asymptotic infinity. In addition, one should include one-loop Chern-Simons terms arising from integrating out massive Kaluza-Klein modes. For the black strings considered we can only match with a microscopic computation when the transverse space is either flat or equal to Taub-NUT. Therefore, we also consider black strings probing ALE/ALF spaces transversely in compactifications of type IIB string theory on K3. We again compute the central charges and levels using the 6d supergravity theory. When the space transverse to the string is either ALE or Taub-NUT there is a complementary microscopic computation. We find a perfect matching to the macroscopic computations. In the last part of the thesis we change gears a little. We study compactifications of the four-dimensional Labc quiver gauge theories on a Riemann surface. We compute central charges and R-charges of baryonic operators in the resulting 2d (0,2) theories in three distinct ways: from the field theory side utilizing c-extremization, its recently discovered geometric dual formulation, and holographically using new AdS3 duals of two-dimensional field theories. The latter solutions can be thought of as the near-horizon geometry of black strings that are asymptotically AdS5 x Labc. Since the central charge is related to the entropy density of the strings, the matching we find between the supergravity and field theory sides can be viewed as a microscopic counting of the degrees of freedom of these strings.