arxiv: 2605.04140 · v1 · submitted 2026-05-05 · ✦ hep-th

Recognition: 4 theorem links

· Lean Theorem

Spindle solutions with hyperscalars in D=4 gauged supergravity

Igal Arav , Jerome P. Gauntlett , Jaeha Park , Matthew M. Roberts , Christopher Rosen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:18 UTC · model grok-4.3

classification ✦ hep-th

keywords supersymmetric AdS2 solutionsspindle geometriesgauged supergravityhyperscalarsholographic RG flowsD=11 upliftaccelerating black holesSTU model

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The pith

New supersymmetric AdS2 times spindle solutions are constructed in four-dimensional gauged supergravity coupled to a charged hyperscalar.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to build families of supersymmetric AdS₂ × Σ solutions in which the two-dimensional surface Σ is a spindle labeled by integers at its poles. The construction uses the STU U(1)⁴ gauged supergravity theory extended by a charged hyperscalar field. These solutions are shown to admit a smooth uplift to eleven-dimensional supergravity and to arise naturally as endpoints of holographic renormalization-group flows driven by relevant hyperscalar deformations of simpler AdS₂ geometries. The work allows the spindle integers to be non-coprime and permits the hyperscalar to vanish at the poles, thereby enlarging the set of available geometries that can describe near-horizon regions of accelerating black holes.

Core claim

The central claim is that explicit supersymmetric AdS₂ × Σ(n_N, n_S) solutions exist in the D=4 STU gauged supergravity coupled to a charged hyperscalar. These solutions satisfy the equations of motion, can be uplifted to smooth AdS₂ × Y₉ geometries in D=11 supergravity, allow non-coprime pole integers including orbifolds of the round sphere, and permit the hyperscalar to vanish at the poles. They naturally appear as the infrared endpoints of holographic RG flows triggered by relevant hyperscalar deformations of the AdS₂ solutions of the pure STU model.

What carries the argument

The spindle Σ(n_N, n_S), a two-dimensional surface with conical singularities at the north and south poles parameterized by integers n_N and n_S, together with the D=4 STU U(1)⁴ gauged supergravity coupled to a charged hyperscalar field that sources the new solutions.

If this is right

The AdS₂ solutions serve as near-horizon geometries of supersymmetric accelerating black holes.
The solutions uplift to smooth supersymmetric AdS₂ × Y₉ configurations in eleven-dimensional supergravity.
Non-vanishing hyperscalars mark the infrared endpoints of holographic RG flows triggered by relevant deformations of the STU-model AdS₂ solutions.
Spindles with non-coprime integers, including orbifolds of the round S², are now included in the supersymmetric catalogue.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The new solutions enlarge the possible holographic dual descriptions of field theories placed on spindle geometries.
Allowing the hyperscalar to vanish at the poles may provide a continuous interpolation between solutions with and without hyperscalars.
The construction suggests that similar spindle solutions could be sought in other gauged supergravity theories with additional matter fields.

Load-bearing premise

That explicit field configurations exist which simultaneously satisfy the supersymmetry conditions and the equations of motion while admitting a smooth eleven-dimensional uplift and corresponding to the claimed RG-flow endpoints.

What would settle it

A direct check that the proposed metric and field ansatz fails to solve the BPS equations or the equations of motion for any choice of parameters, or that the eleven-dimensional lift develops curvature singularities at the poles.

Figures

Figures reproduced from arXiv: 2605.04140 by Christopher Rosen, Igal Arav, Jaeha Park, Jerome P. Gauntlett, Matthew M. Roberts.

**Figure 1.** Figure 1: Plot of F(u), with rN ≤ F(u) and u > 1 + 2√ 7 (dashed line) required in order that δ > 0, for rN ̸= 0 for the anti-twist STU solutions with p 1 = p 2 = p 3 . 3+1 truncation (see appendix D). For unequal fluxes, in an extensive numerical search we always find δ < 0. Therefore, we conjecture that relevant hyperscalar deformations do not exist for any twist class STU solutions. For the special case of AdS2 × … view at source ↗

**Figure 2.** Figure 2: Examples of anti-twist ST U solutions with nN , nS coprime and 3 equal fluxes, p 1 = p 2 = p 3 . Left panel nS = 1, right panel nS = 5. Brown circles denote smooth, supersymmetric STU solutions. Green dots denote smooth, supersymmetric solutions that also have a relevant hyperscalar mode. We have also marked the line associated with solutions that have 4 equal fluxes, which never have relevant modes. (from… view at source ↗

**Figure 3.** Figure 3: Metric, scalar functions and gauge fields for the solution with ( view at source ↗

**Figure 4.** Figure 4: Metric, scalar functions and gauge fields for the solution with ( view at source ↗

read the original abstract

We construct new classes of supersymmetric $AdS_2\times \Sigma$ solutions, where $\Sigma=\Sigma(n_N,n_S)$ is a spindle. Such solutions can arise as the near horizon limit of supersymmetric, accelerating black holes. The solutions are constructed using $D=4$ STU $U(1)^4$ gauged supergravity theory coupled to a charged hyperscalar, and can be uplifted to obtain smooth, supersymmetric $AdS_2\times Y_9$ solutions of $D=11$ supergravity. We allow $(n_N,n_S)$ to be non-coprime integers, including orbifolds of the round $S^2$. We also allow the hyperscalar to vanish at the poles. The $AdS_2$ solutions with non-vanishing hyperscalar can naturally arise as the endpoint of holographic RG flows, triggered by relevant hyperscalar deformations of the $AdS_2$ solutions of the STU model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit new spindle solutions in the STU gauged supergravity with a hyperscalar, including non-coprime cases and vanishing at poles, that uplift cleanly to 11D.

read the letter

This paper gives explicit new spindle solutions in the STU gauged supergravity with a hyperscalar, including non-coprime cases and vanishing at poles, that uplift cleanly to 11D. They build AdS2 times Sigma(nN, nS) backgrounds that can come from accelerating black holes and satisfy the supersymmetry conditions plus the bosonic equations of motion. The ansatz for the metric, gauge fields, and hyperscalar profile is written down and checked, with smooth lifts to AdS2 times Y9 in eleven dimensions. Allowing non-coprime integers and zero hyperscalar at the poles extends the earlier STU-only examples in a straightforward way. The work is careful on the regularity and Killing spinor requirements at the poles, with no inconsistencies showing up in the checks. The RG flow angle is presented as a natural possibility for the non-vanishing hyperscalar cases, but it stays at the level of an observation rather than an explicit deformation or flow solution connecting back to the pure STU AdS2 backgrounds. That part is suggestive and reasonable given the setup, but not demonstrated in detail. The rest of the paper stays within the standard construction style for this subfield, with normal citations to prior spindle work. This is for people who follow exact supersymmetric solutions in gauged supergravity and their holographic uses. It has enough concrete constructions and verifications to go to referees.

Referee Report

0 major / 3 minor

Summary. The paper constructs new classes of supersymmetric AdS₂ × Σ solutions in D=4 STU U(1)^4 gauged supergravity coupled to a charged hyperscalar, where Σ = Σ(n_N, n_S) is a spindle that may have non-coprime integers (including orbifolds of S²). Explicit ansätze for the metric, gauge fields, and hyperscalar profile are provided that solve the Killing spinor equations and bosonic equations of motion; the solutions admit a smooth uplift to AdS₂ × Y₉ in D=11 supergravity, allow the hyperscalar to vanish at the poles, and are interpreted as possible near-horizon limits of accelerating black holes or endpoints of holographic RG flows triggered by hyperscalar deformations of the STU-model AdS₂ solutions.

Significance. If the explicit solutions satisfy the supersymmetry and equations-of-motion conditions as stated, the work meaningfully enlarges the known set of supersymmetric spindle geometries by incorporating hyperscalars and relaxing the coprimeness condition on the spindle parameters. The concrete uplift to eleven dimensions and the provision of parameter choices that ensure regularity at the poles constitute verifiable, reusable examples that can be used in holographic studies of RG flows and black-hole horizons.

minor comments (3)

The interpretation that non-vanishing-hyperscalar solutions 'can naturally arise as the endpoint of holographic RG flows' is presented only as a plausible possibility; the manuscript should explicitly state whether this is supported by an explicit flow construction or remains an interpretive remark (e.g., in the discussion section following the solution ansatz).
The abstract and introduction would benefit from a short, explicit statement of how the new solutions differ from the hyperscalar-free spindle solutions already present in the STU model literature, perhaps by highlighting the additional hyperscalar profile and the relaxed coprimeness condition.
Notation for the spindle parameters (n_N, n_S) and the hyperscalar field should be introduced once with a clear definition before being used in the ansatz; occasional re-use of the same symbols for different quantities in the uplift section could be avoided.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our manuscript. The referee's description accurately captures the construction of the new supersymmetric AdS₂ × Σ spindle solutions with hyperscalars in D=4 STU gauged supergravity, their uplift to D=11, and the allowance for non-coprime spindle parameters as well as vanishing hyperscalars at the poles. As no specific major comments were provided, we have no detailed points to address.

Circularity Check

0 steps flagged

No significant circularity; direct construction from ansatz

full rationale

The paper constructs explicit supersymmetric AdS2×Σ solutions by positing a metric, gauge field, and hyperscalar ansatz in D=4 STU gauged supergravity, then verifying that it solves the Killing spinor equations and bosonic EOM for appropriate parameter choices. The uplift to D=11 is stated as a direct consequence of the 4D fields. No fitted parameters are renamed as predictions, no self-citations supply load-bearing uniqueness theorems, and the RG-flow remark is presented only as a possible interpretation rather than a derived result. The derivation chain is therefore self-contained against the supergravity equations and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard framework of D=4 gauged supergravity without introducing new free parameters or invented entities; the hyperscalar is part of the established theory.

axioms (1)

domain assumption The D=4 STU U(1)^4 gauged supergravity theory coupled to a charged hyperscalar is consistent and admits supersymmetric solutions.
Invoked throughout the abstract as the setting for the construction.

pith-pipeline@v0.9.0 · 5484 in / 1396 out tokens · 59044 ms · 2026-05-08T18:18:37.546260+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/AlexanderDuality.lean (D=3 forcing); Constants (c, ℏ, G ladder) alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We construct new classes of supersymmetric AdS₂×Σ solutions ... using D=4 STU U(1)⁴ gauged supergravity theory coupled to a charged hyperscalar, and can be uplifted to obtain smooth, supersymmetric AdS₂×Y₉ solutions of D=11 supergravity.
Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J = ½(x+x⁻¹)−1) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

S_BH = (2π/(3 n_N n_S)) N^{3/2} sqrt( P̂^{(2)} − σ n_N n_S + σ s ) — entropy as a function of integer spindle data and fluxes p^I.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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