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The degeneracies of $1/4$ BPS states with unit torsion in heterotic string theory compactified on a six-torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form $\Phi_{10}$ of weight $10$. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of $1/\Phi_{10}$. The construction uses two distinct ${\rm SL}(2, \mathbb{Z})$ subgroups of ${\rm Sp}(2, \mathbb{Z})$ which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of $1/\eta^{24}$ by means of a continued fraction structure., Comment: 50 pages, 2 figures

In this note, we model the Bekenstein-Hawking entropy of a four dimensional extremal black hole in terms of classifying particles moving in its near horizon AdS_2 geometry. We use the framework of SLE curves in AdS_2 to classify these particle trajectories in terms of their boundary conditions. These turn out to be related to singular vectors in two-dimensional conformal quantum gravity theory in AdS_2 and the dynamics of these particles are governed by the Hamiltonians of the integrable Calogero-like models, for these boundary conditions. We use this classification to count the leading order Bekenstein-Hawking entropy of the black hole and arrive at a first principle microscopic computation of black hole degeneracy., 31 pages, 3 figures

We construct the on-shell double copy for linearised four-dimensional ${\cal N}=2$ supergravity coupled to one vector multiplet with a quadratic prepotential. We apply this dictionary to the weak-field approximation of dyonic BPS black holes in this theory., Comment: 27 pages

We present the on-shell double copy dictionary for linearised ${\mathcal N}=2$ supergravity coupled to an arbitrary number of vector multiplets in four dimensions. Subsequently, we use it to construct a double copy description of multi-centered BPS black hole solutions in these theories in the weak-field approximation., Comment: 22 pages, comments added, replaced to match published version

The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions., 28 pages, 4 figures, LaTeX; v.2: references added, coffee stains removed; v.3: footnote added, minor typos corrected

In these notes we describe recent progress in understanding finite size corrections to the black hole entropy. Much of the earlier work concerning quantum black holes has been in the limit of large charges when the area of the even horizon is also large. In recent years there has been substantial progress in understanding the entropy of supersymmetric black holes within string theory going well beyond the large charge limit. It has now become possible to begin exploring finite size effects in perturbation theory in inverse size and even nonperturbatively, with highly nontrivial agreements between thermodynamics and statistical mechanics. Unlike the leading Bekenstein-Hawking entropy which follows from the two-derivative Einstein-Hilbert action, these finite size corrections depend sensitively on the phase under consideration and contain a wealth of information about the details of compactification as well as the spectrum of nonperturbative states in the theory. Finite-size corrections are therefore very interesting as a valuable window into the microscopic degrees of freedom of the quantum theory., Comment: 79 pages, based on lectures given at the summer school 'Strings and Fundamental. Physics 2010' in Munich

We explain why multi-centered black hole configurations where at least one of the centers is a large black hole do not contribute to the indexed degeneracies in theories with N=4 supersymmetry. This is a consequence of the fact that such configurations, although supersymmetric, belong to long supermultiplets. As a result, there is no entropy enigma in N=4 theories, unlike in N=2 theories., 14 pages

In this note, we use the first order formalism for attractor flows in N=2 SUGRA extremal black hole backgrounds to establish a formal correspondence between the RG flow of moduli in the underlying N=(2,2) SCFT and bulk attractor flows. Starting from a study of moduli flow trajectories in the CFT, we derive a potential which generates the aforesaid flow. This potential is shown to be a symplectic invariant with the same form as the black hole potential which drives the attractor flows in the bulk. We use these results to make comments on the non-renormalization of extremal black hole entropy and indicate a similar correspondence for two-centered forked flows in CFT., Comment: 28 pages, 3 figures, References added and clarifications

The Kerr/CFT correspondence employs the Cardy formula to compute the entropy of the left moving CFT states. This computation, which correctly reproduces the Bekenstein--Hawking entropy of the four-dimensional extremal Kerr black hole, is performed in a regime where the temperature is of order unity rather than in a high-temperature regime. We show that the comparison of the entropy of the extreme Kerr black hole and the entropy in the CFT can be understood within the Cardy regime by considering a D0-D6 system with the same entropic properties., Comment: 20 pages; LaTeX; JHEP format; v.2 references added, v.3 Section 4 added