We propose a program for counting microstates of four-dimensional BPS black holes in $\mathcal{N}\ge 2$ supergravities with symmetric-space valued scalars by exploiting the symmetries of timelike reduction to three dimensions. Inspired by the equivalence between the four-dimensional attractor flow and geodesic flow on the three-dimensional scalar manifold, we radially quantize stationary, spherically symmetric BPS geometries. Connections between the topological string amplitude, attractor wave function, the Ooguri-Strominger-Vafa conjecture and the theory of automorphic forms suggest that black hole degeneracies are counted by Fourier coefficients of modular forms for the three-dimensional $U$-duality group, associated to special unipotent representations which appear in the supersymmetric Hilbert space of the quantum attractor flow.

• Received 7 February 2006

DOI:https://doi.org/10.1103/PhysRevD.73.084019