Journal of High Energy Physics

, 2018:119 | Cite as

Mock modularity from black hole scattering states

  • Sameer MurthyEmail author
  • Boris Pioline
Open Access
Regular Article - Theoretical Physics


The exact degeneracies of quarter-BPS dyons in Type II string theory on K3 × T2 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.


Black Holes in String Theory Supersymmetry and Duality Solitons Monopoles and Instantons 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College LondonLondonU.K.
  2. 2.Laboratoire de Physique Théorique et Hautes Energies (LPTHE), Sorbonne Université et CNRS UMR 7589ParisFrance

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