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The Gravitational Path Integral for \( N=4\) BPS Black Holes from Black Hole Microstate Counting

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Abstract

We use the exact degeneracy formula of single-centred \(\frac{1}{4}\) BPS dyonic black holes with unit torsion in 4D \(N=4\) toroidally compactified heterotic string theory to improve on the existing formulation of the corresponding quantum entropy function obtained using supersymmetric localization. The result takes the form of a sum over Euclidean backgrounds including orbifolds of the Euclidean \({\textrm{AdS}}_2 \times S^2\) attractor geometry. Using an \(N=2\) formalism, we determine the explicit form of the Abelian gauge potentials supporting these backgrounds. We further show how a rewriting of the degeneracy formula is amenable, at a semi-classical level, to a gravitational interpretation involving 2D Euclidean wormholes. This alternative picture is useful to elucidate different aspects of the gravitational path integral capturing the microstate degeneracies. We also comment on the relation between the associated 1D holographic models.

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Notes

  1. A similar contour was also used in [27, 28].

  2. Under an S-duality transformation \(\begin{pmatrix} a &{} b\\ c &{} d \end{pmatrix} \in SL(2, \mathbb {Z}) \), the vector \(\begin{pmatrix} v \\ \sigma \\ \rho \end{pmatrix}\) transforms as \(\begin{pmatrix} v \\ \sigma \\ \rho \end{pmatrix} \rightarrow M \, \begin{pmatrix} v \\ \sigma \\ \rho \end{pmatrix}\), where \( M = \begin{pmatrix} ad + bc &{} \quad - b d &{} \quad - a c \\ - 2 c d &{} \quad d^2 &{}\quad c^2 \\ - 2 ab &{}\quad b^2 &{} \quad a^2 \end{pmatrix} \).

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Acknowledgements

We would like to thank Guillaume Bossard, Kiril Hristov, Luca Iliesiu, Emil Martinec, Sameer Murthy, Boris Pioline and Ashoke Sen for valuable discussions. This work was partially supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020, through the LisMath PhD fellowship PD/BD/135527/2018 (MR), by the Riemann Fellowship (AK), and by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH (VR). AK thanks the Leibniz Universität Hannover (Institute for Theoretical Physics & the Riemann Center for Geometry and Physics) and the Abdus Salam ICTP for gracious and extended hospitality during key stages of this work, and the Simons Center for Geometry and Physics for hospitality during the final stages of this work. MR thanks KU Leuven for its hospitality during part of this work. GC, SN, VR and MR would also like to thank CERN for its hospitality during the final stages of this work.

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Authors and Affiliations

  1. Department of Mathematics, Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisbon, Portugal

    Gabriel Lopes Cardoso & Suresh Nampuri

  2. Kavli IPMU, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba, 277-8583, Japan

    Abhiram Kidambi

  3. CNRS, CEA, Institut de physique théorique, Université Paris-Saclay, 91191, Gif-sur-Yvette, France

    Valentin Reys

  4. King’s College London, Strand, London, WC2R 2LS, UK

    Martí Rosselló

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  1. Gabriel Lopes Cardoso
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  2. Abhiram Kidambi
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  3. Suresh Nampuri
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Correspondence to Valentin Reys.

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Communicated by Ruben Minasian.

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Cardoso, G.L., Kidambi, A., Nampuri, S. et al. The Gravitational Path Integral for \( N=4\) BPS Black Holes from Black Hole Microstate Counting. Ann. Henri Poincaré 24, 3305–3346 (2023). https://doi.org/10.1007/s00023-023-01297-y

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