Quantum Black Holes, Wall Crossing, and Mock Modular Forms

By A. Dabholkar, S. Murthy, Don Zagier. 2012

362 citations recorded by the source corpus.

We show that the meromorphic Jacobi form that counts the quarter-BPS states in N=4 string theories can be canonically decomposed as a sum of a mock Jacobi form and an Appell-Lerch sum. The quantum degeneracies of single-centered black holes are Fourier coefficients of this mock Jacobi form, while the Appell-Lerch sum captures the degeneracies of multi-centered black holes which decay upon wall-crossing. The completion of the mock Jacobi form restores the modular symmetries expected from $AdS_3/CFT_2$ holography but has a holomorphic anomaly reflecting the non-compactness of the microscopic CFT. For every positive integral value m of the magnetic charge invariant of the black hole, our analysis leads to a special mock Jacobi form of weight two and index m, which we characterize uniquely up to a Jacobi cusp form. This family of special forms and another closely related family of weight-one forms contain almost all the known mock modular forms including the mock theta functions of Ramanujan, the generating function of Hurwitz-Kronecker class numbers, the mock modular forms appearing in the Mathieu and Umbral moonshine, as well as an infinite number of new examples.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/aa2e33c8037571743db43ccce8425ee65935ea31

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Citations in the corpus

Curve Counting on K3 × E, The Igusa Cusp Form χ10, and Descendent Integration (2016, background): “ Fourier expansion of Hd is independent of the moduli τ. Hence, all wall-crossings are related to φd. The splitting of ψd into a ﬁnite and polar part has been studied by Dabholkar, Murthy, and Zagier [7] and has a direct interpretation in a physical model of quantum black holes. In fact, up to the E”
Umbral Moonshine (2012, methodology): “The relation between the zeros of Φ((2)) and the wall-crossing phenomenon in this theory has been studied in [124, 125, 126, 127] and this relation leads to a connection between the counting of BPS black hole states in string theory and mock modular forms [45].”