arxiv: 2605.05305 · v1 · submitted 2026-05-06 · ✦ hep-th

Recognition: unknown

Positivity of the gravitational path integral implies the axionic weak gravity conjecture

Gabriele Di Ubaldo , Luca V. Iliesiu , Henry W. Lin , Cynthia Yan

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:18 UTC · model grok-4.3

classification ✦ hep-th

keywords gravitational path integralaxion weak gravity conjecturewormholespositivityshift symmetryswamplandinstantonsquantum gravity

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The pith

Positivity of the gravitational path integral requires that axions obey a sharp weak gravity bound with precise numerical constants.

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 inner products between states of open and closed universes, as computed by the gravitational path integral, must be positive semi-definite for a consistent Hilbert space. When an axion has an exact shift symmetry, certain wormhole contributions violate this positivity under mild assumptions. If those wormholes remain perturbatively stable in the low-energy theory, positivity instead demands a non-perturbative instability that breaks the symmetry. This yields a concrete version of the axionic weak gravity conjecture. A reader would care because the result derives a swampland constraint directly from the consistency of the gravitational path integral rather than from string theory examples.

Core claim

If the axion has an exact shift symmetry, a combined positivity constraint on closed and open universes is violated when certain wormholes are included. In low-energy effective theories where these wormholes are perturbatively stable, positivity requires that the wormholes have a non-perturbative instability that breaks the shift symmetry. This leads to a sharp version of the axion weak gravity conjecture, including precise numerical constants. The bound is related to possible extensions of other swampland conjectures, such as an imaginary continuation of the distance conjecture.

What carries the argument

Positivity of inner products between open- and closed-universe states computed via the gravitational path integral, which forces non-perturbative breaking of axion shift symmetry through wormhole instabilities.

If this is right

Wormholes in axion theories must develop non-perturbative instabilities to preserve positivity.
The resulting bound on axion instanton actions includes specific numerical constants.
The argument extends to related swampland statements such as an imaginary continuation of the distance conjecture.
The constraint applies to axions arising in string theory compactifications.

Where Pith is reading between the lines

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

The same positivity logic may constrain other scalars with approximate shift symmetries in quantum gravity.
Phenomenological models with very light axions would need to incorporate the minimal instanton effects required by the bound.
Extensions could test whether similar wormhole effects appear in higher-dimensional or supersymmetric settings.

Load-bearing premise

That certain wormholes are included in the path integral and remain perturbatively stable in the low-energy effective theory while the axion shift symmetry is exact.

What would settle it

An explicit computation of the inner product that remains positive when stable wormholes with exact axion shift symmetry are included, or the discovery of a consistent theory containing such stable wormholes without symmetry-breaking instabilities.

Figures

Figures reproduced from arXiv: 2605.05305 by Cynthia Yan, Gabriele Di Ubaldo, Henry W. Lin, Luca V. Iliesiu.

**Figure 1.** Figure 1: FIG. 1. The Lorentzian Poincare patch with metric d view at source ↗

read the original abstract

The gravitational path integral can compute inner products between different states of open and closed universes. To have a well-defined Hilbert space, these inner products should be positive semi-definite, which is not manifest in the low-energy effective theory. In this letter, we analyze the constraints that the positivity of inner products imposes on gravitational theories coupled to axions. If the axion has an exact shift symmetry, we show that, under mild assumptions, a combined positivity constraint on closed and open universes is violated when one includes certain wormholes. In low-energy effective theories where these wormholes are perturbatively stable, positivity requires that the wormholes have a non-perturbative instability that breaks the shift symmetry. This leads to a sharp version of the axion weak gravity conjecture, including precise numerical constants. We relate the bound to possible extensions of other swampland conjectures, arguing for an imaginary continuation of the distance conjecture. We comment on how the bound applies to axions in string theory and discuss phenomenological implications.

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

The paper derives a sharp numerical axion WGC from path-integral positivity via wormhole contributions, but the sign flip relies on mild assumptions about saddles that still need explicit checking.

read the letter

The central claim is that an exact axion shift symmetry leads to a violation of positivity in the combined inner products from closed and open universes once certain wormholes are included. Under the stated assumptions this forces a non-perturbative instability that breaks the symmetry and produces a concrete bound with numerical constants rather than just an order-of-magnitude statement. That is the new piece: a direct extraction of the axionic weak gravity conjecture from the requirement that the gravitational path integral define a positive semi-definite inner product on states. They also sketch a possible link to an imaginary continuation of the distance conjecture and note implications for string theory axions and phenomenology. Those connections are useful even if they remain at the level of comments. The logic from positivity constraint to required instability is presented cleanly. The soft spot is precisely the load-bearing step flagged in the stress-test note. The violation appears only when particular wormholes dominate and the contour or relative weighting of open versus closed topologies is chosen in a specific way. If other saddles or a different prescription flips the sign of the contribution, the positivity constraint is no longer violated and the bound does not follow. The abstract does not display the explicit inner-product matrix or determinant calculation, so it is hard to judge how sensitive the negative eigenvalue is to those choices. The assumptions are labeled mild, but they carry the result. This is aimed at swampland and axion model builders who already know the wormhole literature. A reader working on concrete bounds for cosmology or string compactifications will get the most from it. The claim is sharp enough to justify sending the paper to a serious referee who can check the wormhole saddles and the contour details. I would bring it to a reading group to discuss whether the assumptions can be relaxed or made more explicit.

Referee Report

1 major / 0 minor

Summary. The paper claims that the positivity of inner products computed via the gravitational path integral between states of open and closed universes imposes strong constraints on axion theories. For an exact axion shift symmetry, under mild assumptions, certain wormhole saddles violate a combined positivity constraint; in low-energy EFTs where these wormholes are perturbatively stable, positivity then requires a non-perturbative instability that breaks the shift symmetry. This yields a sharp version of the axionic weak gravity conjecture with precise numerical constants. The work also relates the bound to possible extensions of other swampland conjectures (including an imaginary continuation of the distance conjecture) and discusses applications to string theory axions and phenomenology.

Significance. If the central derivation holds, the result is significant: it derives a quantitative axionic WGC directly from the requirement of positive semi-definite inner products in the gravitational path integral, without additional string-theoretic input or fitted parameters. The approach supplies sharp numerical bounds and a concrete mechanism (wormhole instability) that could generalize to other swampland statements. Credit is given for the attempt to obtain a parameter-free prediction from path-integral consistency and for linking the result to an extension of the distance conjecture.

major comments (1)

The load-bearing step is the demonstration that, under the stated mild assumptions, the wormhole contributions produce a negative eigenvalue (or determinant) in the combined closed+open inner-product matrix when the axion shift symmetry is exact. The manuscript must explicitly define these assumptions—including the choice of integration contour, the relative weighting of open versus closed topologies, and the absence or sub-dominance of other saddles—because, as the stress-test notes, a different contour or weighting can flip the sign and eliminate the violation, leaving the WGC bound un-derived. This clarification is required before the central claim can be accepted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. Their comment has led us to improve the clarity of the assumptions in the revised version, which we believe strengthens the presentation of the central result without altering its content.

read point-by-point responses

Referee: The load-bearing step is the demonstration that, under the stated mild assumptions, the wormhole contributions produce a negative eigenvalue (or determinant) in the combined closed+open inner-product matrix when the axion shift symmetry is exact. The manuscript must explicitly define these assumptions—including the choice of integration contour, the relative weighting of open versus closed topologies, and the absence or sub-dominance of other saddles—because, as the stress-test notes, a different contour or weighting can flip the sign and eliminate the violation, leaving the WGC bound un-derived. This clarification is required before the central claim can be accepted.

Authors: We agree that greater explicitness regarding the assumptions is warranted to make the derivation fully robust and to address potential sensitivities to contour or weighting choices. In the revised manuscript we have inserted a new subsection (now labeled 2.2) immediately after the statement of the main positivity constraint. There we enumerate the assumptions as follows: (i) the integration contour is the steepest-descent contour in the complexified metric-axion field space that passes through the dominant wormhole saddle while preserving the reality conditions on the boundary data; (ii) the relative weighting between open and closed topologies is fixed by the standard gravitational path-integral measure, with closed-universe contributions scaled by the spatial volume factor and open-universe contributions normalized to the boundary area; (iii) all other candidate saddles (higher-genus surfaces, multi-wormhole configurations, or instantons with additional axion windings) are either absent by boundary conditions or exponentially suppressed relative to the leading wormhole in the regime where the axion decay constant is large compared with the Planck scale. We also include a short paragraph discussing the stress-test scenarios, explaining that contours or weightings that remove the negativity would correspond to a different theory whose low-energy Hilbert space is already positive-definite without any instability, which lies outside the class of EFTs we consider. This explicit listing does not change the numerical bound but removes any ambiguity about the domain of validity of the argument. revision: yes

Circularity Check

0 steps flagged

Derivation from positivity of gravitational inner products is self-contained with no reduction to inputs by construction

full rationale

The paper begins from the independent requirement that the gravitational path integral must produce positive semi-definite inner products between states of open and closed universes. It then shows that exact axion shift symmetry, combined with specific wormhole saddles, violates this positivity under stated mild assumptions on contour and topology weighting. This forces a non-perturbative instability breaking the symmetry, yielding the axion WGC bound with explicit numerical constants. No equation or step equates the final bound to a fitted parameter, a self-defined quantity, or a load-bearing self-citation whose validity depends on the present result. The argument remains falsifiable via the sign of the inner-product matrix and does not rename or smuggle prior results as new derivations. The mild assumptions are explicit physical inputs rather than hidden tautologies.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on standard quantum-gravity assumptions about the path integral and wormholes rather than new free parameters or invented entities; the only inputs are the positivity requirement and the perturbative stability of wormholes in the low-energy theory.

axioms (3)

domain assumption The gravitational path integral computes inner products between states of open and closed universes.
Explicitly stated as the starting point for defining a Hilbert space.
domain assumption These inner products must be positive semi-definite.
Required for a well-defined quantum theory; invoked to obtain the constraint.
domain assumption Certain wormholes are perturbatively stable in the low-energy effective theory.
Used to show that positivity is violated unless a non-perturbative instability is present.

pith-pipeline@v0.9.0 · 5481 in / 1606 out tokens · 63694 ms · 2026-05-08T17:18:25.916472+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

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as we approach the divergence. While we expect the three-boundary wormhole to have a similar divergence to the two-boundary wormhole once two of the boundaries come close to one another (givingκ 3 ∼ e−#/GN κ2 as we approach the divergence) we do not expect a faster divergence to be possible. 5 that all axion wormholes should be excluded from the gravitati...
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