## Abstract

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

Original language | English |
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Article number | 119 |

Journal | Journal of High Energy Physics |

Volume | 2018 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

## Keywords

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