{
  "abstract": "We first construct general saddles of the five-dimensional gravitational supersymmetric index with $S^1\\times\\mathbb{R}^4$ 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 $S^1\\times \\mathbb{R}^4\\times S^1\\times {\\rm K}3$, carrying D1-D5-P charges. Upon taking a decoupling limit, these solutions yield asymptotically Euclidean AdS$_3\\times S^3\\times {\\rm K}3$ 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 $({\\rm AdS}_3\\times S^3)/\\mathbb{Z}_k$ orbifolds. The black hole saddles are modular images of the horizonless ones. They generalize the well-known ${\\rm SL}(2,\\mathbb{Z})$ family of BTZ black holes by placing it in the background of the $\\mathbb{Z}_k$ orbifold. We also consider the asymptotically AdS$_3\\times S^3$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25438",
  "author_total": 3,
  "authors": [
    {
      "name": "Davide Cassani"
    },
    {
      "name": "Alejandro Ruipérez"
    },
    {
      "name": "Enrico Turetta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/872qa6q9scgdv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25438",
  "fulltext_url": null,
  "id": "872qa6q9scgdv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25438",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7204478828",
  "pdf_url": "https://arxiv.org/pdf/2608.25438",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/872qa6q9scgdv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2608.25438",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25438"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Decoupling saddles of the gravitational index and the Farey tail",
  "title_text": "Decoupling saddles of the gravitational index and the Farey tail",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:34Z",
  "url": "https://research.draconitic.com/paper/872qa6q9scgdv",
  "venue": "arXiv",
  "versions": []
}