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The exact degeneracies of quarter-BPS dyons in Type II string theory on $K3\times T^2$ are given by Fourier coefficients of the inverse of the Igusa cuspform. For a fixed magnetic charge invariant $m$, the generating function ofthese degeneracies naturally decomposes as a sum of two parts, which aresupposed to account for single-centered black holes, and two-centered blackhole bound states, respectively. The decomposition is such that each part isseparately modular covariant but neither is holomorphic, calling for a physicalinterpretation of the non-holomorphy. We resolve this puzzle by computing thesupersymmetric index of the quantum mechanics of two-centered half-BPSblack-holes, which we model by geodesic motion on Taub-NUT space subject to acertain potential. We compute a suitable index using localization methods, andfind that it includes both a temperature-independent contribution from BPSbound states, as well as a temperature-dependent contribution due to a spectralasymmetry in the continuum of scattering states. The continuum contributionagrees precisely with the non-holomorphic completion term required for themodularity of the generating function of two-centered black hole bound states.

Mock modularity from black hole scattering states