Mock modularity from black hole scattering states

• Published in: The journal of high energy physics Vol. 2018; no. 12; pp. 1 - 33
• Main Authors: Murthy, Sameer, Pioline, Boris
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2018; Springer Nature B.V; SpringerOpen
• Subjects: Black holes; Black Holes in String Theory; Classical and Quantum Gravitation; Decomposition; Elementary Particles; High energy physics; Modularity; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum mechanics; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Scattering; Solitons Monopoles and Instantons; String Theory; Supersymmetry; Supersymmetry and Duality; Temperature dependence; Black Holes in String Theory; Supersymmetry and Duality; Solitons Monopoles and Instantons
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP12(2018)119

A bstract The exact degeneracies of quarter-BPS dyons in Type II string theory on K 3 × T 2 are given by Fourier coefficients of the inverse of the Igusa cusp form. For a fixed magnetic charge invariant m , the generating function of these degeneracies naturally decomposes as a sum of two parts, which are supposed to account for single-centered black holes, and two-centered black hole bound states, respectively. The decomposition is such that each part is separately modular covariant but neither is holomorphic, calling for a physical interpretation of the non-holomorphy. We resolve this puzzle by computing the supersymmetric index of the quantum mechanics of two-centered half-BPS black-holes, which we model by geodesic motion on Taub-NUT space subject to a certain potential. We compute a suitable index using localization methods, and find that it includes both a temperature-independent contribution from BPS bound states, as well as a temperature-dependent contribution due to a spectral asymmetry in the continuum of scattering states. The continuum contribution agrees precisely with the non-holomorphic completion term required for the modularity of the generating function of two-centered black hole bound states.