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arxiv: 2606.22810 · v1 · pith:LLCCTV4Bnew · submitted 2026-06-22 · ✦ hep-th · gr-qc· math-ph· math.MP

A Linearized Obstruction to the Supersymmetric Extension of Conformal Boundary Conditions in Euclidean Gravity

Xingyang Yu This is my paper

Pith reviewed 2026-06-26 08:04 UTC · model grok-4.3

classification ✦ hep-th gr-qcmath-phmath.MP
keywords conformal boundary conditionssupersymmetryEuclidean gravitygravitino boundary conditionslinearized supergravityWitten boundary conditionobstructionelliptic boundary value problem
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The pith

Witten's conformal boundary conditions for Euclidean gravity have no half-supersymmetric extension in linearized minimal supergravity.

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

The paper establishes that for fixed conformal bosonic boundary data, no choice of half-dimensional gravitino boundary condition can preserve the full chiral supersymmetry. Supersymmetry acting on the natural local chiral gravitino datum generates the trace-free extrinsic curvature, which the conformal prescription deliberately leaves free as a response. This mismatch creates a pointwise obstruction in tangential momentum space that survives gauge transformations. The result is a highest-derivative linearized no-go, independent of any particular elliptic ansatz or global topology. A sympathetic reader would care because it limits the perturbative Euclidean path integral constructions that can be made supersymmetric at the linear level.

Core claim

For fixed conformal bosonic data, no half-dimensional gravitino boundary condition, whether local or pseudodifferential and including APS-type conditions with compatible ghost conditions, closes the preserved chiral supersymmetry. Supersymmetry selects the chiral gravitino datum whose image under the supersymmetry transformation is the trace-free extrinsic curvature left unfixed by the conformal boundary condition, producing an obstruction at the highest derivative order that persists even after compensating gauge transformations.

What carries the argument

The linearized supersymmetry transformation that maps the chiral gravitino boundary datum to the trace-free extrinsic curvature, acting pointwise in tangential momentum.

If this is right

  • Any attempt to extend the conformal boundary condition to supersymmetry must fail at linear order without additional structures.
  • The obstruction is independent of the choice of elliptic boundary condition for the gravitino.
  • Nonlinear supercovariant boundary terms may evade the obstruction by relating the trace-free extrinsic curvature to gravitino bilinears.
  • The result applies specifically to the perturbative Euclidean gravitational path integral with fixed conformal class and mean curvature.

Where Pith is reading between the lines

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

  • Similar obstructions might appear in other supersymmetric extensions of gravitational boundary conditions beyond minimal supergravity.
  • Investigating whether the nonlinear theory allows tying the bosonic response to fermionic terms could resolve the issue for finite boundaries.
  • Extensions to higher-derivative or non-minimal supergravities might bypass the linearized obstruction through different field content.

Load-bearing premise

Any viable supersymmetric extension must preserve the chiral supersymmetry exactly through a boundary condition on the gravitino at linear order without nonlinear supercovariant corrections or structures linking the trace-free extrinsic curvature to fermionic bilinears.

What would settle it

An explicit construction of a half-dimensional gravitino boundary condition that preserves the full chiral supersymmetry while leaving the trace-free extrinsic curvature as an unfixed response to the conformal data.

read the original abstract

Witten's conformal boundary condition \cite{Witten:2018lgb} provides an elliptic boundary-value problem for the finite-boundary perturbative Euclidean gravitational path integral: one fixes the boundary conformal class and the mean curvature, while the trace-free extrinsic curvature is left free as the conjugate response. We show that this perturbative construction admits no half-supersymmetric extension in linearized minimal supergravity. For fixed conformal bosonic data, no half-dimensional gravitino boundary condition (local or pseudodifferential, APS-type included, with any compatible ghost condition at highest-derivative order) closes the full preserved chiral supersymmetry. Supersymmetry first selects the natural local chiral gravitino datum. Acting back on this datum then produces the trace-free extrinsic curvature, precisely the response that the conformal prescription leaves unfixed. The obstruction is therefore not the failure of a particular elliptic ansatz: even the chiral/Robin completion that is LS-elliptic and BRST-compatible at highest-derivative order would impose Dirichlet control on a Neumann response. The obstruction is pointwise in tangential momentum and survives compensating gauge transformations. It is a linearized, highest-derivative obstruction, not a global or nonlinear no-go; nonlinear supercovariant boundary terms may evade it by tying the trace-free extrinsic curvature to gravitino bilinears.

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

0 major / 1 minor

Summary. The paper claims that Witten's conformal boundary condition for the perturbative Euclidean gravitational path integral—which fixes the boundary conformal class and mean curvature while leaving the trace-free extrinsic curvature as a free response—admits no half-supersymmetric extension in linearized minimal supergravity. For any fixed conformal bosonic data, no half-dimensional gravitino boundary condition (local or pseudodifferential, including APS-type, with compatible ghost conditions) closes the preserved chiral supersymmetry. Supersymmetry maps the natural chiral gravitino datum onto the trace-free extrinsic curvature; the obstruction is pointwise in tangential momentum, survives gauge compensations, and is a highest-derivative linearized effect (with nonlinear supercovariant corrections noted as a possible evasion).

Significance. If the central mapping holds, the result is significant for Euclidean supergravity and boundary-value problems in path integrals. It supplies a parameter-free, direct derivation from the action of supersymmetry transformations on boundary data, explicitly restricted to linear order, that rules out a broad class of gravitino conditions without hidden circularity. This constrains attempts to supersymmetrize conformal boundaries and identifies a concrete highest-derivative obstruction that future nonlinear constructions must address.

minor comments (1)
  1. The abstract states the obstruction clearly but could briefly indicate the precise dimension of the gravitino space relative to the bosonic data for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately captures the central claim and its implications for Euclidean supergravity boundary-value problems.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central derivation is a direct computation of the action of linearized supersymmetry transformations on the gravitino boundary datum, which maps it onto the trace-free extrinsic curvature left unfixed by the conformal bosonic boundary condition. This holds pointwise in tangential momentum and is independent of any fitted parameters, self-referential definitions, or load-bearing self-citations. The argument explicitly restricts itself to linear order and notes that nonlinear corrections could evade the obstruction, confirming the result is self-contained against the stated assumptions without reducing to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; no free parameters are mentioned, and the axioms are standard domain assumptions of linearized supergravity.

axioms (2)
  • domain assumption Analysis restricted to linearized minimal supergravity
    The obstruction is explicitly stated to be linearized.
  • domain assumption Boundary conditions considered are local or pseudodifferential with compatible ghost conditions at highest derivative order
    The paper rules out all such classes.

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discussion (0)

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