Black Hole Quantum Mechanics and Generalized Error Functions

[2507.08551] Black Hole Quantum Mechanics and Generalized Error Functions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search High Energy Physics - Theory arXiv:2507.08551 (hep-th) [Submitted on 11 Jul 2025 ( v1 ), last revised 10 Apr 2026 (this version, v3)] Title: Black Hole Quantum Mechanics and Generalized Error Functions Authors: Boris Pioline , Rishi Raj View a PDF of the paper titled Black Hole Quantum Mechanics and Generalized Error Functions, by Boris Pioline and 1 other authors View PDF HTML (experimental) Abstract: 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. Comments: 32 pages, 4 figures; v3: minor changes, final version to appear in JHEP Subjects: High Energy Physics - Theory (hep-th) Cite as: arXiv:2507.08551 [hep-th] (or arXiv:2507.08551v3 [hep-th] for this version) https://doi.org/10.48550/arXiv.2507.08551 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Boris Pioline [ view email ] [v1] Fri, 11 Jul 2025 12:55:42 UTC (58 KB) [v2] Sat, 20 Sep 2025 14:44:23 UTC (66 KB) [v3] Fri, 10 Apr 2026 20:19:44 UTC (66 KB) Full-text links: Access Paper: View a PDF of the paper titled Black Hole Quantum Mechanics and Generalized Error Functions, by Boris Pioline and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: hep-th < prev | next > new | recent | 2025-07 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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