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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.08551 |
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| _version_ | 1866910120042037248 |
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| author | Pioline, Boris Raj, Rishi |
| author_facet | Pioline, Boris Raj, Rishi |
| contents | In Type II Calabi-Yau string compactifications, S-duality predicts that suitable generating series of BPS indices counting microstates of D4-D2-D0 black holes are in general mock modular forms of higher depth. The non-holomorphic contributions needed to cancel the anomaly under modular transformations involve certain indefinite theta series with kernels constructed from generalized error functions. Physically, these contributions are expected to arise from a spectral asymmetry in the continuum of scattering states of $n$ BPS dyons with mutually non-local charges. For $n=2$, the (standard, depth one) error function completion was derived long ago by explicitly computing the bosonic and fermionic density of states in the two-body supersymmetric quantum mechanics. Here we derive the general non-holomorphic completion for an arbitrary number of centers by evaluating the refined Witten index of the supersymmetric quantum mechanics using localization. In a nutshell, the index reduces to an integral over $\mathbb{R}^{3n-3}$ (the relative location of the centers), and splits into an integral over the $2n-2$ dimensional phase space of BPS ground states times an integral over $n-1$ transverse directions, which ultimately produces the expected generalized error functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Black Hole Quantum Mechanics and Generalized Error Functions Pioline, Boris Raj, Rishi High Energy Physics - Theory In Type II Calabi-Yau string compactifications, S-duality predicts that suitable generating series of BPS indices counting microstates of D4-D2-D0 black holes are in general mock modular forms of higher depth. The non-holomorphic contributions needed to cancel the anomaly under modular transformations involve certain indefinite theta series with kernels constructed from generalized error functions. Physically, these contributions are expected to arise from a spectral asymmetry in the continuum of scattering states of $n$ BPS dyons with mutually non-local charges. For $n=2$, the (standard, depth one) error function completion was derived long ago by explicitly computing the bosonic and fermionic density of states in the two-body supersymmetric quantum mechanics. Here we derive the general non-holomorphic completion for an arbitrary number of centers by evaluating the refined Witten index of the supersymmetric quantum mechanics using localization. In a nutshell, the index reduces to an integral over $\mathbb{R}^{3n-3}$ (the relative location of the centers), and splits into an integral over the $2n-2$ dimensional phase space of BPS ground states times an integral over $n-1$ transverse directions, which ultimately produces the expected generalized error functions. |
| title | Black Hole Quantum Mechanics and Generalized Error Functions |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.08551 |