Quantum Black Holes, Wall Crossing, and Mock Modular Forms

We show that the quantum degeneracies of single-centered black holes in N = 4 theories are coefficients of a mock modular form. The failure of modularity of such a form is of a very special type and is governed by a holomorphic modular form called the shadow, such that the sum of the mock modular form and a simple non-holomorphic, real-analytic function obtained from the shadow transforms like a modular form and is called its completion. The shadow can be viewed as a holomorphic anomaly associated with the completion. The spectral-flow invariant partition function of these black holes is a mock Jacobi form.

Mock modularity is a consequence of the meromorphy of the generalized elliptic genus which is closely related to the wall-crossing phenomenon. The completion makes manifest the modular symmetries expected from holography and provides a starting point for a Rademacher-type expansion of the degeneracies with implications for the exact quantum entropy and the Poincare series. Mock modular forms are thus expected to provide a proper framework for AdS2/CFT1 and AdS3/CFT2 holography in the context of the MSW string for N = 2 black holes, and more generally have applications in other physical problems involving noncompact conformal field theories and meromorphic Jacobi forms.