Mock modularity from black hole scattering states

Abstract

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 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.

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Cited by in corpus (5)

• APS $\eta$-invariant, path integrals, and mock modularity
• Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds
• The gravitational path integral for $N=4$ BPS black holes from black hole microstate counting
• Squashed Toric Manifolds and Higher Depth Mock Modular Forms
• Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the ${\mathbb{Z}}_2$ Model