An arXiv preprint reports a sharp split in a calculation of gravitational-index contributions. An infinite family of two-center saddles, candidate geometries used to organize the calculation, matches terms in the D1-D5 elliptic genus Farey-tail expansion. The more-than-two-center configurations examined in the same work, including black rings and black lenses, do not match Farey-tail saddles. The comparison is made by matching on-shell actions, the calculated actions of the candidate geometries, with the actions assigned to terms in the CFT expansion.
Following the geometry
The study constructs five-dimensional gravitational-index saddles analytically and uplifts the three-charge configurations to type IIB supergravity. It then takes those solutions through a decoupling limit on S1 x K3, producing AdS3 x S3 x K3 asymptotics. The resulting geometry and action are compared with the D1-D5 two-dimensional conformal field theory, or CFT2, elliptic genus and its Farey-tail expansion. In plain terms, the test asks whether the gravity saddles line up with the separate contributions used to represent the CFT quantity.
Two families, one dictionary
The clearest correspondence appears in the two-center family. The authors report a precise match between an infinite family of two-center saddle contributions and the D1-D5 Farey-tail expansion. Among them, horizonless orbifold saddles are matched with polar-state contributions. The above-threshold family is matched with supersymmetric orbifolds of Euclidean BTZ x S3, the black-hole side of the comparison.
The paper's CFT dictionary assigns meaning to the discrete labels of these solutions. It treats eta as an integral spectral-flow parameter and p-prime as a parameter for fractional spectral flow in the CFT. In the authors' interpretation, horizonless geometries represent fractional-spectral-flow states, while black-hole geometries represent their modular images. That interpretation is the bridge between the five-dimensional saddles and the polar or above-threshold terms.
The black-hole construction also has a specific geometric classification. When n and q3 are coprime, it gives the SL(2, Z) family of BTZ black holes. When the defining integers share common factors, the orbifold may be freely acting or singular, with singularities at one or both poles of the horizon S3. These cases are part of the geometric classification behind the reported action match.
Where the correspondence ends
The negative result is narrower. The detailed on-shell-action discussion focuses on solutions with two centers, while the multi-center analysis examines a relevant subclass of three-center solutions. In that tested sector, the explicit actions do not match Farey-tail saddles. The authors present this as evidence that supersymmetric black-ring and black-lens configurations do not contribute to the D1-D5 elliptic genus.
That wording matters: the paper presents a no-match result for the analyzed setup and subclass, not a general theorem covering every configuration with three or more centers. The conclusion is therefore a constraint on the proposed saddle spectrum, with its reach set by which geometries were constructed and compared.
The limits of the claim
One further obstacle comes from moduli space, the space of allowed theory parameters. The paper states that horizonless geometries with more than one compact cycle, after uplift to type IIB on K3, do not exist as supersymmetric configurations at a generic point in moduli space and are not expected to provide protected index states. This gives a separate reason to question whether some multi-center candidates can represent protected contributions.
The three-center action also carries a technical caveat. Its chemical-potential-independent phase is inferred from a smooth limit as n approaches 0, and the authors say they did not perform an explicit rigorous derivation of that phase. The multi-center mismatch is therefore reported with a qualification attached to the action analysis.
Because the work is an analytic construction, the reported correspondence has no statistical uncertainty; its scope is set by the five-dimensional supergravity model, the type IIB uplift, the decoupling limit and the D1-D5 comparison. The document is arXiv version 1, posted on 26 August 2026.
Paper data and sources
Original title: Decoupling saddles of the gravitational index and the Farey tail
Authors: Davide Cassani, Alejandro Ruipérez, Enrico Turetta
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text