Abstract
We begin to develop the formalism of localization for the functional integral of supergravity on AdS3 × S2. We show how the condition of supersymmetry in the Euclidean ℍ3/ℤ × S2 geometry naturally leads to a twist of the S2 around the time direction of AdS3. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of (0, 4) SCFT2 at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on ℍ3/ℤ × S2 localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on AdS2 × S2. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of arXiv:1112.5371, which could be of independent interest.
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Acknowledgments
We would like to thank Satoshi Nawata and Valentin Reys for useful discussions and comments. This work is supported by the ERC Consolidator Grant N. 681908, “Quantum black holes: A microscopic window into the microstructure of gravity”, and by the STFC grant ST/P000258/1. A.C. is supported by the STFC studentship ST/S505468/1. IJ is supported by an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government, by the Korean Local Governments — Gyeongsangbuk-do Province and Pohang City, and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A1048531).
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Ciceri, A., Jeon, I. & Murthy, S. Localization on AdS3 × S2. Part I. The 4d/5d connection in off-shell Euclidean supergravity. J. High Energ. Phys. 2023, 218 (2023). https://doi.org/10.1007/JHEP07(2023)218
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DOI: https://doi.org/10.1007/JHEP07(2023)218