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arxiv: 2605.28737 · v1 · pith:KOSPUEFBnew · submitted 2026-05-27 · ✦ hep-th

Complex BPS Black Holes in AdS₃times S³

Finn Larsen , Kartik Sharma This is my paper

Pith reviewed 2026-06-29 11:06 UTC · model grok-4.3

classification ✦ hep-th
keywords supersymmetric indexBPS black holesAdS3 x S3complex saddlesBTZ geometrySTU modelsupersymmetry preservationblack hole thermodynamics
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The pith

Smooth complex black hole solutions represent the supersymmetric index in AdS₃×S³ rather than Euclidean continuations.

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

The paper establishes that the gravitational saddles for the supersymmetric index in AdS₃×S³ are smooth complex solutions obtained as finite-temperature BTZ × S³ geometries. These are constructed in two ways from four-dimensional two-center BPS solutions with complex dipole charges and from six-dimensional black strings with the BPS charge relation. The resulting smoothness and periodicity conditions are exactly those needed for supersymmetry to be globally well-defined and for matching the thermodynamic potentials of the index. A reader would care because this clarifies the proper way to compute supersymmetric state counts using gravity in this geometry.

Core claim

The correct gravitational representation of the supersymmetric index is a smooth complex solution, rather than the naïve Euclidean continuation of the Lorentzian BPS black hole. We construct such saddles for black holes with AdS₃×S³ boundary conditions in the STU model. We arrive at the same finite-temperature BTZ × S³ geometries in two independent ways: from two-center four-dimensional BPS solutions with complex dipole charges, and from six-dimensional black strings after imposing the BPS relation among conserved charges. We analyze the resulting smoothness and periodicity conditions and show that they are precisely those required for globally well-defined supersymmetry and for the thermody

What carries the argument

Finite-temperature BTZ × S³ geometries from complex dipole charges in two-center 4D BPS solutions or from BPS charge relations in 6D black strings, enforcing the periodicity for supersymmetry.

If this is right

  • The thermodynamic potentials of the supersymmetric index are reproduced by these geometries.
  • Supersymmetry is globally well-defined on the resulting periodic geometries.
  • The same geometries emerge from both the four-dimensional two-center and six-dimensional constructions.
  • These saddles provide the gravitational representation for the index under AdS₃×S³ boundary conditions in the STU model.

Where Pith is reading between the lines

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

  • If these complex saddles are the correct ones, similar non-real geometries may be needed for supersymmetric indices in other AdS setups.
  • The matching of the two constructions could be used to test consistency in models with different matter content.
  • The requirement of complex charges might extend to computations of indices for black holes with different horizon topologies.

Load-bearing premise

The two independent constructions produce identical geometries whose periodicity conditions exactly match those demanded by supersymmetry preservation without additional assumptions about the model or boundary conditions.

What would settle it

Showing that the periodicity conditions obtained from the two-center four-dimensional solutions differ from those required for supersymmetry preservation in the six-dimensional black string construction would falsify the claim.

read the original abstract

The correct gravitational representation of the supersymmetric index is a smooth complex solution, rather than the na\"{\i}ve Euclidean continuation of the Lorentizan BPS black hole. We construct such saddles for black holes with \(\mathrm{AdS}_3\times S^3\) boundary conditions in the STU model. We arrive at the same finite-temperature BTZ \(\times S^3\) geometries in two independent ways: from two-center four-dimensional BPS solutions with complex dipole charges, and from six-dimensional black strings after imposing the BPS relation among conserved charges. We analyze the resulting smoothness and periodicity conditions and show that they are precisely those required for globally well-defined supersymmetry and for the thermodynamic potentials of the supersymmetric index.

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 paper claims that the correct gravitational representation of the supersymmetric index for AdS₃×S³ black holes in the STU model consists of smooth complex solutions, rather than the naïve Euclidean continuation of the Lorentzian BPS black hole. These finite-temperature BTZ×S³ geometries are obtained in two independent ways—from 4D two-center BPS solutions with complex dipole charges and from 6D black strings after imposing the BPS relation on conserved charges—and are shown to satisfy precisely the smoothness and periodicity conditions required for globally well-defined supersymmetry and the thermodynamic potentials of the index.

Significance. If the equivalence of the two constructions is verified and the periodicity conditions are shown to match those demanded by supersymmetry without additional assumptions, the result would clarify the saddle-point contributions to the supersymmetric index, distinguishing complex saddles from standard Euclidean continuations and providing a firmer gravitational interpretation of the index's thermodynamics in AdS₃.

major comments (2)
  1. [Abstract; §3 (two-center solutions) and §4 (black-string reduction)] The central claim of equivalence between the 4D two-center construction (with complex dipole charges) and the 6D black-string construction (with BPS charge relation) is load-bearing, yet the manuscript provides no explicit parameter mapping in the STU model that confirms the resulting metrics, horizons, and Killing spinor periodicities are identical. This mapping is required to establish that both routes produce the same finite-temperature BTZ×S³ geometry without hidden identifications or boundary-condition adjustments.
  2. [§5] §5 (periodicity analysis): the assertion that the smoothness and periodicity conditions are 'precisely those required for globally well-defined supersymmetry' is not supported by a direct side-by-side comparison of the Killing spinors obtained from each construction; without this, it remains unclear whether the thermodynamic potentials of the index are reproduced identically.
minor comments (1)
  1. [Notation throughout] The notation distinguishing the complex dipole charges in the 4D construction from the conserved charges in the 6D construction should be made more uniform across sections to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where greater explicitness would strengthen the manuscript. We address each major comment below and will incorporate the requested details in a revised version.

read point-by-point responses

  1. Referee: [Abstract; §3 (two-center solutions) and §4 (black-string reduction)] The central claim of equivalence between the 4D two-center construction (with complex dipole charges) and the 6D black-string construction (with BPS charge relation) is load-bearing, yet the manuscript provides no explicit parameter mapping in the STU model that confirms the resulting metrics, horizons, and Killing spinor periodicities are identical. This mapping is required to establish that both routes produce the same finite-temperature BTZ×S³ geometry without hidden identifications or boundary-condition adjustments.

    Authors: We agree that an explicit parameter mapping between the two constructions was not supplied. In the revised manuscript we will add a dedicated subsection (or appendix) that gives the direct dictionary between the 4D two-center parameters (including the complex dipole charges) and the 6D black-string charges (subject to the BPS relation). This mapping will be used to verify that the resulting metrics, horizon locations, and periodicity conditions coincide exactly, with no additional identifications required. revision: yes

  2. Referee: [§5] §5 (periodicity analysis): the assertion that the smoothness and periodicity conditions are 'precisely those required for globally well-defined supersymmetry' is not supported by a direct side-by-side comparison of the Killing spinors obtained from each construction; without this, it remains unclear whether the thermodynamic potentials of the index are reproduced identically.

    Authors: We accept that a direct, side-by-side comparison of the Killing spinors from the two constructions is absent. The revised version will include an explicit comparison of the Killing spinors (or their periodicity properties) obtained from each route, confirming that both yield the same supersymmetry-preserving boundary conditions and therefore the same thermodynamic potentials for the index. revision: yes

Circularity Check

0 steps flagged

Two independent constructions of the same geometries with no reduction to inputs by construction

full rationale

The abstract states that the finite-temperature BTZ × S³ geometries are obtained via two separate routes (4D two-center BPS solutions with complex dipole charges; 6D black strings after imposing the BPS charge relation) and that the resulting smoothness/periodicity conditions match those required by supersymmetry. No equations, self-citations, or parameter fittings are presented in the provided text that would make any central claim equivalent to its own inputs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; central claim rests on the assumption that complex dipole charges and BPS charge relations produce globally supersymmetric geometries whose periodicity matches the index requirements. No explicit free parameters, ad-hoc axioms, or new entities are named in the abstract.

axioms (2)
  • domain assumption Supersymmetry preservation imposes specific periodicity conditions on the complex geometry.
    Invoked when stating that the smoothness conditions are precisely those required for globally well-defined supersymmetry.
  • domain assumption The STU model admits consistent complex extensions of BPS solutions.
    Underlying the construction of two-center solutions with complex dipole charges.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Index saddle for the D1-D5-P black string and its decoupling limit

    hep-th 2026-06 unverdicted novelty 5.0

    Constructs index saddle for D1-D5-P black string via BPS limit and uplift, with decoupling to BTZ x S^3 computing CFT index.

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