Decoupling saddles of the gravitational index and the Farey tail
Abstract
We first construct general saddles of the five-dimensional gravitational supersymmetric index with S1×ℝ4 asymptotics, carrying an arbitrary number of electric charges and Gibbons-Hawking centers. We then uplift the three-charge configurations to type IIB supergravity, obtaining solutions asymptotic to S1× ℝ4× S1× K3, carrying D1-D5-P charges. Upon taking a decoupling limit, these solutions yield asymptotically Euclidean AdS3× S3× K3 candidate saddles of the dual CFT2 elliptic genus. We focus on the two-center saddles, which involve three integers specifying an orbifold action. Depending on these integers, the saddles may or may not possess a Euclidean horizon, and may be smooth or exhibit orbifold singularities. We give the on-shell action of these two-center saddles in the appropriate ensemble and establish a precise match with the Farey-tail expansion of the elliptic genus at the level of saddle-point actions. The horizonless saddles are the Euclidean version of fractional spectral-flow solutions previously discussed in the literature, corresponding to (AdS3× S3)/ℤk orbifolds. The black hole saddles are modular images of the horizonless ones. They generalize the well-known SL(2,ℤ) family of BTZ black holes by placing it in the background of the ℤk orbifold. We also consider the asymptotically AdS3× S3 solutions arising as decoupling limit of five-dimensional black ring and black lens saddles, provide their on-shell action and confirm that they do not contribute to the D1-D5 elliptic genus.
Related
- Allowable complex saddles with real charges in the gravitational index
- Regular hairy black holes by gravitational decoupling: de Sitter and Minkowski cores
- Unpolarized Low-Frequency Tail of Causal Chiral Gravitational Waves
- On the Mathematical Foundations of Gravitational Decoupling in General Relativity
- Particle Dynamics and Thermodynamics of a Charged-Like Hairy Black Hole in Extended Gravitational Decoupling
BibTeX
@misc{cassani2026decoupling,
title = {{Decoupling saddles of the gravitational index and the Farey tail}},
author = {Davide Cassani and Alejandro Ruipérez and Enrico Turetta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.25438},
eprint = {2608.25438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25438},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Decoupling saddles of the gravitational index and the Farey tail
AU - Davide Cassani
AU - Alejandro Ruipérez
AU - Enrico Turetta
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.25438
UR - https://arxiv.org/abs/2608.25438
AB - We first construct general saddles of the five-dimensional gravitational supersymmetric index with S^1×ℝ^4 asymptotics, carrying an arbitrary number of electric charges and Gibbons-Hawking centers. We then uplift the three-charge configurations to type IIB supergravity, obtaining solutions asymptotic to S^1× ℝ^4× S^1× K3, carrying D1-D5-P charges. Upon taking a decoupling limit, these solutions yield asymptotically Euclidean AdS_3× S^3× K3 candidate saddles of the dual CFT_2 elliptic genus. We focus on the two-center saddles, which involve three integers specifying an orbifold action. Depending on these integers, the saddles may or may not possess a Euclidean horizon, and may be smooth or exhibit orbifold singularities. We give the on-shell action of these two-center saddles in the appropriate ensemble and establish a precise match with the Farey-tail expansion of the elliptic genus at the level of saddle-point actions. The horizonless saddles are the Euclidean version of fractional spectral-flow solutions previously discussed in the literature, corresponding to (AdS_3× S^3)/ℤ_k orbifolds. The black hole saddles are modular images of the horizonless ones. They generalize the well-known SL(2,ℤ) family of BTZ black holes by placing it in the background of the ℤ_k orbifold. We also consider the asymptotically AdS_3× S^3 solutions arising as decoupling limit of five-dimensional black ring and black lens saddles, provide their on-shell action and confirm that they do not contribute to the D1-D5 elliptic genus.
ER -