%0 Generic %1 dabholkar2012quantum %A Dabholkar, Atish %A Murthy, Sameer %A Zagier, Don %D 2012 %K black crossing holes wall %T Quantum Black Holes, Wall Crossing, and Mock Modular Forms %U http://arxiv.org/abs/1208.4074 %X 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.

@misc{dabholkar2012quantum, abstract = {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.}, added-at = {2013-12-23T06:55:47.000+0100}, author = {Dabholkar, Atish and Murthy, Sameer and Zagier, Don}, biburl = {https://www.bibsonomy.org/bibtex/2dbf25f4612ca96730bab6f813237355d/aeu_research}, description = {Quantum Black Holes, Wall Crossing, and Mock Modular Forms}, interhash = {b5a20dd4bdb1c544f8722690c67de069}, intrahash = {dbf25f4612ca96730bab6f813237355d}, keywords = {black crossing holes wall}, note = {cite arxiv:1208.4074Comment: 151 pages, 1 figure}, timestamp = {2013-12-23T08:22:34.000+0100}, title = {Quantum Black Holes, Wall Crossing, and Mock Modular Forms}, url = {http://arxiv.org/abs/1208.4074}, year = 2012 }