Near Horizon Limits of Massless BTZ and Their CFT Duals

We consider the massless BTZ black hole and show that it is possible to take its "near horizon" limit in two distinct ways. The first one leads to a null self-dual orbifold of AdS3 and the second to a spacelike singular AdS3/Z_K orbifold in the large K limit, the "pinching orbifold". We show that from the dual 2d CFT viewpoint, the null orbifold corresponds to the p^+=0 sector of the Discrete Light-Cone Quantisation (DLCQ) of the 2d CFT where a chiral sector of the CFT is decoupled, while the pinching orbifold corresponds to taking an infinite mass gap limit in both the right and left sectors of the 2d CFT, essentially leaving us with the states L_0=\bar L_0=c/24 only. In the latter case, one can combine the near horizon limit with sending the 3d Planck length l_P to zero, or equivalently the dual CFT central charge c to infinity. We provide preliminary evidence that in that case some nontrivial dynamics may survive the limit.Comment: 22 pages, no figures, v2: minor improvements, references adde

Bose-Fermi duality and entanglement entropies

Entanglement (Renyi) entropies of spatial regions are a useful tool for characterizing the ground states of quantum field theories. In this paper we investigate the extent to which these are universal quantities for a given theory, and to which they distinguish different theories, by comparing the entanglement spectra of the massless Dirac fermion and the compact free boson in two dimensions. We show that the calculation of Renyi entropies via the replica trick for any orbifold theory includes a sum over orbifold twists on all cycles. In a modular-invariant theory of fermions, this amounts to a sum over spin structures. The result is that the Renyi entropies respect the standard Bose-Fermi duality. Next, we investigate the entanglement spectrum for the Dirac fermion without a sum over spin structures, and for the compact boson at the self-dual radius. These are not equivalent theories; nonetheless, we find that (1) their second Renyi entropies agree for any number of intervals, (2) their full entanglement spectra agree for two intervals, and (3) the spectrum generically disagrees otherwise. These results follow from the equality of the partition functions of the two theories on any Riemann surface with imaginary period matrix. We also exhibit a map between the operators of the theories that preserves scaling dimensions (but not spins), as well as OPEs and correlators of operators placed on the real line. All of these coincidences can be traced to the fact that the momentum lattice for the bosonized fermion is related to that of the self-dual boson by a 45 degree rotation that mixes left- and right-movers.Comment: 40 pages; v3: improvements to presentation, new section discussing entanglement negativit

Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)

In these lecture notes, we review some recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states, beyond the thermodynamical, large charge limit. After a brief overview of charged black holes in supergravity and string theory, we give an extensive introduction to special and very special geometry, attractor flows and topological string theory, including holomorphic anomalies. We then expose the Ooguri-Strominger-Vafa (OSV) conjecture which relates microscopic degeneracies to the topological string amplitude, and review precision tests of this formula on ``small'' black holes. Finally, motivated by a holographic interpretation of the OSV conjecture, we give a systematic approach to the radial quantization of BPS black holes (i.e. quantum attractors). This suggests the existence of a one-parameter generalization of the topological string amplitude, and provides a general framework for constructing automorphic partition functions for black hole degeneracies in theories with sufficient degree of symmetry.Comment: 103 pages, 8 figures, 21 exercises, uses JHEP3.cls; v5: important upgrade, prepared for the proceedings of Frascati School on Attractor Mechanism; Sec 7 was largely rewritten to incorporate recent progress; more figures, more refs, and minor changes in abstract and introductio