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

Abstract

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.

Figures (6)

• Figure 1: Penrose diagram of the non-extremal (left) and extremal (right) Reissner-Nordström black holes. Dotted lines denote event horizons, dashed lines represent time-like singularities. The diagram on the left should be doubled along the dashed-dotted line.
• Figure 2: Radial flow for the Gaussian one-scalar model, for charges $(p^0,p^1,q_1,q_0)=(4,1,1,2)$. All trajectories are attracted to $z_*=X^1/X^0=(1-3i)/10$ at $r=0$.
• Figure 3: Left: the cylinder amplitude in string theory can be viewed either as a trace over the open string Hilbert space (quantizing along $\tau$) channel) or as an inner product between two wave functions in the closed string Hilbert space (quantizing along $\sigma$). Right: The global geometry of Lorentzian $AdS_2$ has the topology of a strip; its Euclidean continuation at finite temperature becomes a cylinder. $\tau$ and $t$ are the global and Poincaré time, respectively.
• Figure 4: Potential governing the radial evolution of the complex scalar in the same model as in Figure \ref{['su22fig0']} and same charges, at $U=0$. The potential has a global maximum at $z_*=X^1/X^0=(1-3i)/10$.
• Figure 5: Left: Potential governing the motion along the $U$ variable in the universal sector. The horizon is reached at $U\to-\infty$. Right: Root diagram of the $SU(2,1)$ symmetries in the universal sector.