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arxiv: 2606.23893 · v1 · pith:ZBRFKMRLnew · submitted 2026-06-22 · ✦ hep-th

Towards OSV in AdS

Nikolay Bobev , Sunjin Choi , Junho Hong , Valentin Reys This is my paper

Pith reviewed 2026-06-26 06:59 UTC · model grok-4.3

classification ✦ hep-th
keywords supersymmetric localizationsuperconformal indexsquashed sphere partition functionAdS4 black holesOSV conjecture3d SCFTsholographyM2-branes
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The pith

The superconformal index of 3d N=2 SCFTs equals the square of the squashed three-sphere partition function in the Cardy-like limit, producing an OSV-like relation for AdS4 black holes.

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

The paper applies supersymmetric localization to general 3d N=2 SCFTs and obtains the relation Z_{S^1 × S^2} ∼ |Z_{S^3_b}|^2 between the superconformal index and the squashed sphere partition function. The derivation uses a saddle point approximation when the S1 radius is small and the squashing parameter is large. In the holographic setting this implies that the partition function of supersymmetric asymptotically AdS4 black holes is determined by the large N limit of Z_{S^3_b} of the dual SCFT, because that quantity encodes the gauged supergravity prepotential. The same logic yields a parallel relation for the topologically twisted index, and the authors verify the AdS4 result explicitly for SCFTs arising from M2-branes.

Core claim

Supersymmetric localization establishes a relation of the form Z_{S^1×S^2} ∼ |Z_{S^3_b}|^2 between the superconformal index and the squashed three-sphere partition function for general 3d N=2 SCFTs, obtained via saddle point approximation in the Cardy-like limit of small S1 radius and large squashing parameter. A similar relation holds for the topologically twisted index. Holographically the result connects the partition function of supersymmetric asymptotically AdS4 black holes to the large N limit of Z_{S^3_b} of the dual SCFT; since the latter encodes the gauged supergravity prepotential, the relation is akin to the OSV conjecture for asymptotically flat black holes. The relation is confi

What carries the argument

Saddle point approximation to supersymmetric localization results in the Cardy-like limit of small S1 radius and large squashing parameter, which produces the index-partition function relation Z_{S^1×S^2} ∼ |Z_{S^3_b}|^2.

If this is right

  • The index-partition function relation holds for every 3d N=2 SCFT.
  • An analogous relation exists between the topologically twisted index and the squashed sphere partition function.
  • The partition function of supersymmetric AdS4 black holes is determined by the large N limit of the dual SCFT's Z_{S^3_b}.
  • The relation is verified at large N for SCFTs dual to M2-branes.
  • A parallel relation holds for 5d SCFTs and asymptotically AdS6 black holes.

Where Pith is reading between the lines

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

  • Black hole entropy in AdS4 could be extracted directly from field theory localization data without solving the gravitational equations of motion.
  • The prepotential of gauged supergravity might be read off from SCFT indices in the appropriate limit.
  • The same saddle point technique may produce analogous relations in other dimensions or for other classes of black holes.

Load-bearing premise

The saddle point approximation in the Cardy-like limit of small S1 radius and large squashing parameter captures the dominant contribution to the index.

What would settle it

An exact computation of the superconformal index for any specific 3d N=2 SCFT that fails to equal the square of its squashed three-sphere partition function in the small-radius large-squashing limit.

read the original abstract

We use supersymmetric localization for 3d $\mathcal{N}=2$ SCFTs to establish a relation of the form $Z_{S^1\times S^2} \sim |Z_{S^3_b}|^2$ between the superconformal index and the squashed three-sphere partition function. This applies to general 3d $\mathcal{N}=2$ SCFTs and is derived using a saddle point approximation in the Cardy-like limit of small $S^1$ radius and large squashing parameter. We also show a similar relation between the topologically twisted index and the squashed sphere partition function. In the context of holography our results lead to a relation between the partition function of supersymmetric asymptotically AdS$_4$ black holes and the large $N$ limit of $Z_{S^3_b}$ of the dual SCFT. Since the latter encodes the gauged supergravity prepotential, this result is akin to the OSV conjecture for asymptotically flat black holes. We confirm this relation in detail for SCFTs arising from M2-branes using recent large $N$ results from supersymmetric localization. We also briefly discuss a similar relation for 5d SCFTs and its implications for asymptotically AdS$_6$ black holes.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript uses supersymmetric localization on 3d N=2 SCFTs to derive the relation Z_{S¹×S²} ∼ |Z_{S³_b}|^2 (and an analogous relation for the topologically twisted index) via saddle-point evaluation in the simultaneous limit of small S¹ radius and large squashing parameter. Holographically, this is interpreted as relating the partition function of supersymmetric AdS₄ black holes to the large-N limit of the dual SCFT's squashed-sphere partition function (which encodes the gauged supergravity prepotential), yielding an AdS analogue of the OSV conjecture. The relation is checked explicitly for M2-brane SCFTs using existing large-N localization results, with a brief extension to 5d SCFTs and AdS₆ black holes.

Significance. If the saddle-point step can be placed on a rigorous footing with controlled remainders, the result supplies a concrete holographic dictionary between AdS₄ black-hole partition functions and SCFT data already computed by localization, thereby furnishing an OSV-like statement in AdS whose leading large-N behavior is fixed by the prepotential. The explicit verification for M2-brane theories provides a non-trivial consistency check that strengthens the claim.

major comments (2)
  1. [saddle-point evaluation of the superconformal index] The central relation is obtained by saddle-point evaluation of the localized matrix model for the superconformal index in the Cardy-like limit β→0, b→∞. No explicit bound on the remainder of the saddle-point expansion is supplied, nor is it shown that sub-leading terms remain negligible after the large-N limit is taken and the holographic dictionary is applied (see the derivation leading to the relation stated in the abstract and the holographic discussion).
  2. [holographic interpretation] The holographic identification equates the approximated index directly with the black-hole partition function whose leading behavior is captured by the prepotential in Z_{S³_b}. It is not demonstrated that the error incurred by the saddle-point approximation is sub-dominant to the terms retained in the large-N holographic map (abstract and the M2-brane confirmation section).
minor comments (1)
  1. [abstract] The symbol ∼ in the central relation is used without a precise statement of the sense in which the equality holds (leading vs. sub-leading orders, etc.).

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and positive assessment of the significance of our results. Below we respond point by point to the major comments on the saddle-point approximation.

read point-by-point responses

  1. Referee: [saddle-point evaluation of the superconformal index] The central relation is obtained by saddle-point evaluation of the localized matrix model for the superconformal index in the Cardy-like limit β→0, b→∞. No explicit bound on the remainder of the saddle-point expansion is supplied, nor is it shown that sub-leading terms remain negligible after the large-N limit is taken and the holographic dictionary is applied (see the derivation leading to the relation stated in the abstract and the holographic discussion).

    Authors: The saddle-point evaluation extracts the leading exponential behavior in the simultaneous Cardy-like limit, where corrections around the saddle are suppressed by positive powers of the small parameters β and 1/b. These corrections remain sub-dominant once the large-N limit is subsequently taken, as the leading term scales with N to a positive power while corrections do not. The explicit large-N match for M2-brane theories further supports that the retained terms capture the correct leading holographic behavior. We do not supply a fully rigorous bound with controlled remainders, as this would require analytic number theory techniques outside the scope of the localization approach used here. revision: no

  2. Referee: [holographic interpretation] The holographic identification equates the approximated index directly with the black-hole partition function whose leading behavior is captured by the prepotential in Z_{S³_b}. It is not demonstrated that the error incurred by the saddle-point approximation is sub-dominant to the terms retained in the large-N holographic map (abstract and the M2-brane confirmation section).

    Authors: In the large-N regime the holographic dictionary retains only the leading exponential terms fixed by the prepotential; the saddle-point errors are either exponentially smaller or scale as sub-leading powers of N (or logarithms) and are therefore negligible compared with the O(N^{3/2}) or higher leading contributions to the black-hole partition function. This is confirmed by the explicit agreement at leading order in the M2-brane examples. We therefore maintain that the error is sub-dominant for the purposes of the stated relation. revision: no

standing simulated objections not resolved
  • A rigorous mathematical bound on the saddle-point remainder (with controlled errors after the large-N limit) that would place the central relation on fully rigorous footing.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard localization plus saddle-point limit

full rationale

The claimed relation Z_{S^1×S^2} ∼ |Z_{S^3_b}|^2 is obtained by applying supersymmetric localization to the superconformal index followed by an explicit saddle-point evaluation in the simultaneous β→0, b→∞ limit. This step is an approximation whose validity is controlled by the limit itself rather than by redefining the target quantity in terms of itself. The subsequent holographic identification with the AdS4 black-hole partition function is presented as a consequence of the approximated index equaling the large-N prepotential encoded in Z_{S^3_b}, not as an input that forces the result. No self-citation is invoked as a uniqueness theorem, no parameter is fitted to a subset and then relabeled a prediction, and no ansatz is smuggled via prior work. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the applicability of supersymmetric localization to general 3d N=2 SCFTs and the validity of the saddle-point approximation in the stated limit; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption supersymmetric localization computes the superconformal index and squashed-sphere partition function for 3d N=2 SCFTs
    Invoked as the starting point for the derivation in the abstract.
  • domain assumption saddle-point approximation is valid in the Cardy-like limit of small S1 radius and large squashing parameter
    Explicitly used to obtain the relation Z_{S^1×S^2} ∼ |Z_{S^3_b}|^2.

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discussion (0)

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Reference graph

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