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arxiv: 2606.27843 · v1 · pith:VCZVKWY6new · submitted 2026-06-26 · ✦ hep-th · gr-qc

Classification of Killing Horizons in D=11 Supergravity

Jan Gutowski , Chettha Saelim , Martin Wolf This is my paper

Pith reviewed 2026-06-29 04:12 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Killing horizonsD=11 supergravitysupersymmetrylightcone chiralitynear-horizon geometriespp-wavesKilling spinorsdegenerate horizons
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The pith

Supersymmetric degenerate Killing horizons in D=11 supergravity divide into two classes by lightcone chirality of the Killing spinor.

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

The paper classifies supersymmetric degenerate Killing horizons with closed spatial cross sections in eleven-dimensional supergravity without assuming the near-horizon limit. Solutions separate into two classes according to whether the negative lightcone chirality component of the Killing spinor is non-zero or vanishes on the horizon. In the first class every solution is isometric to a supersymmetric near-horizon geometry. In the second class the spinorial Lie derivative along the horizon-generating vector vanishes and any solution with more than thirteen supersymmetries must be a pp-wave. The result constrains possible geometries for supersymmetric black-hole horizons by partitioning them into known families or tightly restricted cases.

Core claim

We prove that all such solutions fall into two distinct classes, depending on lightcone chirality with respect to a Gaussian null coordinate system. For the first class of solutions, the negative lightcone chirality part of the Killing spinor is non-zero on the Killing horizon, and we prove that all such solutions are isometric to supersymmetric near-horizon geometries. In the second class, the negative lightcone chirality part of the Killing spinor vanishes on the Killing horizon. In this case, we prove that the spinorial Lie derivative of the Killing spinor with respect to the Killing vector which generates the Killing horizon vanishes, and that all such solutions with more than 13 supersy

What carries the argument

Lightcone chirality of the Killing spinor defined with respect to a Gaussian null coordinate system, which partitions solutions into the two classes.

If this is right

  • Solutions with non-zero negative lightcone chirality are isometric to supersymmetric near-horizon geometries.
  • In the vanishing-chirality class the spinorial Lie derivative of the Killing spinor vanishes.
  • All solutions in the vanishing-chirality class with more than 13 supersymmetries are pp-waves.
  • The classification applies directly to solutions away from the near-horizon limit.

Where Pith is reading between the lines

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

  • The chirality-partition method may extend to classifications of Killing horizons in other dimensions or supergravity theories.
  • Any non-pp-wave solutions in the vanishing-chirality class must have at most 13 supersymmetries.
  • The result narrows the search for new supersymmetric black-hole solutions to the vanishing-chirality class with lower supersymmetry counts.

Load-bearing premise

A Gaussian null coordinate system exists on the Killing horizon in which lightcone chirality of the Killing spinor can be defined and used to split the solutions.

What would settle it

A supersymmetric degenerate Killing horizon in D=11 supergravity with closed spatial cross section that cannot be placed in either class, or a solution in the vanishing-chirality class possessing more than 13 supersymmetries that is not a pp-wave.

read the original abstract

We initiate the classification of supersymmetric degenerate Killing horizons, with closed spatial cross section, away from the near-horizon limit in D=11 supergravity. We prove that all such solutions fall into two distinct classes, depending on lightcone chirality with respect to a Gaussian null coordinate system. For the first class of solutions, the negative lightcone chirality part of the Killing spinor is non-zero on the Killing horizon, and we prove that all such solutions are isometric to supersymmetric near-horizon geometries. In the second class, the negative lightcone chirality part of the Killing spinor vanishes on the Killing horizon. In this case, we prove that the spinorial Lie derivative of the Killing spinor with respect to the Killing vector which generates the Killing horizon vanishes, and that all such solutions with more than 13 supersymmetries are pp-waves.

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

1 major / 0 minor

Summary. The manuscript initiates the classification of supersymmetric degenerate Killing horizons with closed spatial cross-sections in D=11 supergravity. It proves that all such solutions fall into two distinct classes depending on the lightcone chirality of the Killing spinor with respect to a Gaussian null coordinate system. In the first class (negative lightcone chirality component non-zero on the horizon), all solutions are isometric to supersymmetric near-horizon geometries. In the second class (negative component vanishes), the spinorial Lie derivative of the Killing spinor vanishes, and all solutions with more than 13 supersymmetries are pp-waves.

Significance. If the central claims hold, this work provides a significant advance in the classification of supersymmetric solutions in eleven-dimensional supergravity away from the near-horizon limit. The two-class partition and the pp-wave result for high supersymmetry constitute concrete, falsifiable statements that could guide further searches for black-hole solutions and their near-horizon limits.

major comments (1)
  1. [Abstract] Abstract and opening paragraphs: the claim that 'all such solutions fall into two distinct classes' is load-bearing for the entire classification. It is performed with respect to a Gaussian null coordinate system in which lightcone chirality is defined and constant, yet the manuscript provides no derivation that such a global chart exists for every supersymmetric degenerate Killing horizon with closed spatial cross-section. Topological obstructions on the cross-section or regularity issues at the degenerate surface could prevent a global chart, leaving some solutions outside either class or counted ambiguously.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this foundational point regarding the global existence of Gaussian null coordinates. We address the comment in detail below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the claim that 'all such solutions fall into two distinct classes' is load-bearing for the entire classification. It is performed with respect to a Gaussian null coordinate system in which lightcone chirality is defined and constant, yet the manuscript provides no derivation that such a global chart exists for every supersymmetric degenerate Killing horizon with closed spatial cross-section. Topological obstructions on the cross-section or regularity issues at the degenerate surface could prevent a global chart, leaving some solutions outside either class or counted ambiguously.

    Authors: We agree that the manuscript does not contain an explicit derivation of the global existence of the Gaussian null coordinate system. In the revised manuscript we will add a short preliminary subsection (new Section 2.1) that constructs such coordinates globally in a tubular neighborhood of any degenerate Killing horizon whose spatial cross-section is compact. The construction proceeds by first using the Killing property and degeneracy to obtain a null vector field that is nowhere zero on the horizon, then extending it off the horizon via the exponential map along the orthogonal geodesics; compactness of the cross-section together with the smoothness of the metric guarantees that this yields a globally defined chart without topological obstructions. With this addition the two-class partition is rigorously justified for all solutions under consideration. revision: yes

Circularity Check

0 steps flagged

No circularity; classification follows from spinor equations and coordinate assumptions without reduction to inputs

full rationale

The paper claims to prove a partition of supersymmetric degenerate Killing horizons into two classes based on the lightcone chirality of the Killing spinor in a Gaussian null coordinate system, with further results for each class derived from the Killing spinor equations. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described claims. The derivation is presented as direct from the horizon data and spinor conditions, making it self-contained with no exhibited reduction of outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions from 11D supergravity and differential geometry without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption Killing spinors exist and satisfy the supersymmetry Killing spinor equation in D=11 supergravity
    Invoked throughout the classification of supersymmetric solutions.
  • domain assumption A Gaussian null coordinate system can be introduced on the degenerate Killing horizon to define lightcone chirality
    Used to split solutions into the two classes described in the abstract.

pith-pipeline@v0.9.1-grok · 5675 in / 1462 out tokens · 52308 ms · 2026-06-29T04:12:44.075029+00:00 · methodology

discussion (0)

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Reference graph

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