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arxiv: 2605.13970 · v1 · submitted 2026-05-13 · ✦ hep-th · gr-qc

Recognition: 2 theorem links

· Lean Theorem

A Tale of Two Hartle-Hawking Wave Functions: Fully Gravitational vs Partially Frozen

Galit Anikeeva , Rapha\"el Dulac , Zixia Wei , Mengyang Zhang
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:31 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Hartle-Hawking wave functiongravitational path integralAdS gravityone-loop correctionsphase problemhyperbolic ballde Sitter gravityboundary conditions
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0 comments X

The pith

The phase in Hartle-Hawking wave functions arises only when the gravitational path integral fully integrates over boundary configurations rather than fixing them.

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

The paper compares two versions of the Hartle-Hawking wave function in Anti-de Sitter space. In one, the path integral integrates over all boundary data, making the wave function fully gravitational. In the other, the boundary is held fixed as in standard AdS/CFT calculations. One-loop calculations show that the fully dynamical version acquires a phase factor of (∓i) to the power of D+1, while the fixed-boundary version stays real and positive. The same distinction appears when applied to de Sitter space with an equator held fixed. This indicates that the long-standing phase problem in these wave functions is tied to the choice of whether gravity remains fully dynamical at the boundary.

Core claim

In AdS Einstein gravity, the one-loop correction to the partition function of a hyperbolic ball with fluctuating boundary yields a phase (∓i)^{D+1}, matching the sphere in dS. When the boundary metric is instead fixed, the phase vanishes and the result is real and positive. The analogous partially frozen construction in dS cancels the phase nontrivially. These results imply that the phase issue is resolved by freezing the boundary degrees of freedom.

What carries the argument

The hyperbolic-ball partition function, evaluated at one loop with either fully fluctuating or fixed boundary metric, which determines whether a phase appears in the wave-function norm.

If this is right

  • The fully gravitational Hartle-Hawking wave function in AdS acquires a one-loop phase identical to its de Sitter counterpart.
  • Fixing the boundary metric removes this phase, yielding a real positive norm.
  • A partially frozen de Sitter construction with fixed equatorial metric also cancels the phase.
  • The distinction between dynamical and frozen boundary controls the reality of the wave function.

Where Pith is reading between the lines

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

  • This distinction may extend to higher-order corrections or non-perturbative contributions in the path integral.
  • It suggests that AdS/CFT-like dualities naturally select the phase-free version by fixing boundary data.
  • Similar freezing choices could be explored in other gravitational wave-function constructions to address unitarity issues.

Load-bearing premise

The one-loop correction to the hyperbolic-ball partition function provides the leading contribution to the norm of the wave function, with boundary fluctuations properly captured by the regularization scheme.

What would settle it

An exact non-perturbative computation of the hyperbolic-ball partition function in three-dimensional AdS gravity that either confirms or eliminates the reported one-loop phase factor.

Figures

Figures reproduced from arXiv: 2605.13970 by Galit Anikeeva, Mengyang Zhang, Rapha\"el Dulac, Zixia Wei.

Figure 1
Figure 1. Figure 1: (Left) In the original HH proposal, data are fixed on a closed spatial slice Σ (blue [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The HH computation (1.3) for open spatial slices Σ naturally arises from the original HH proposal for closed spatial slices after dimensional reduction. Here this figure shows how a boundary can appear from a dimensional reduction of a closed manifold in higher dimensions. From the lower-dimensional point of view, B− is dynamical and should be summed over in the GPI. component. From the lower-dimensional p… view at source ↗
Figure 3
Figure 3. Figure 3: Setup for computing the HH wavefunction in AdS [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For a given choice of Σ (light blue), four different saddles [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Existence domain of the real Euclidean saddle. For fixed supercritical tension [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A trajectory (gray) of spacetime boundary [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: When the boundary length L is bigger than the critical value Lcrit(T) = √ 2π T2−1 , the saddle point is necessarily complex. An example which can be realized by gluing a Euclidean part and a Lorentzian part together is shown. The Lorentzian evolution tL depicted in a lighter color, in addition to the half-sphere contribution in Euclidean signature tE (the darker part). The similar phenomenon also appears i… view at source ↗
Figure 8
Figure 8. Figure 8: The computation of the HH wave function in AdS [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: On-shell action I (ℓ=1) E (L) as a function of L for different tensions of the spacetime￾boundary. We took 4πGN = 1 for this plot. The maximum of the Hartle-Hawking wave function is exactly half of the on shell action evaluated on M, the manifold bounded by the spacetime boundary B. For L > Lcrit(T), one needs to analytically continue the on-shell action (2.55) (or equiv￾alently to evaluate the on-shell ac… view at source ↗
Figure 10
Figure 10. Figure 10: A 2D manifold M− is represented with a spacetime boundary B− in gray. The spatial slice Σ is represented in light blue. The angles Θ1 and Θ2 appearing in the Hayward term are also shown in the figure. The unit vectors normal to Σ and B− are labeled by nΣ and nB− respectively. of this curve. The affine parameter λ gives a coordinate-independent parameterization for the dilaton profile. By fixing the range … view at source ↗
Figure 11
Figure 11. Figure 11: A typical constant-K curve (solid) is depicted in the two-dimensional plane R 2 . The boundary of Poincar´e disk is drawn in dashed line, which is a unit circle with the center at the origin O of the plane R 2 . The Poincar´e disk coordinate covers the region inside the dashed circle. The constant-K curve, as a circle within the plane, is classified by two parameters, the radial coordinate r = a of its ce… view at source ↗
Figure 12
Figure 12. Figure 12: The graph shows that the geodesic spatial slice Σ (Light blue curve) splits the [PITH_FULL_IMAGE:figures/full_fig_p041_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The HH wave function of geodesic spatial slice evaluated on the “+”-saddle peaks [PITH_FULL_IMAGE:figures/full_fig_p042_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The “∓” sign in the C(r0, K) factor of the equation (3.37) of r0 is sensitive to the relative position of the spatial slices (light blue) to the origin where the boundary circle B is centered. For the spatial slice that intersects the θ = π axis on the left side of the origin, we choose the “+” sign in C(r0, K), and for the spatial slice that intersects the θ = 0 axis on the right side of the origin, we c… view at source ↗
Figure 15
Figure 15. Figure 15: For generic K, there are two spatial slices depicted in light blue labeled by ΣK 1 and ΣK 2 with the same proper length L and the same absolute value of the extrinsic curvature |K|. The two slices merge together into Σcrit at L = L |K| crit. The spatial slice Σcrit at criticality stretches between two antipodal points on B, and the corresponding spacetime boundary B− is exactly the half circle. When we ta… view at source ↗
Figure 16
Figure 16. Figure 16: The ”+”-saddle on-shell Euclidean action for [PITH_FULL_IMAGE:figures/full_fig_p047_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The picture shows the two branches of the complex saddle for the evaluation of [PITH_FULL_IMAGE:figures/full_fig_p050_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The setup of (4.3), the no-boundary wave function for a spatial subregion (left), and its comparison to that of (1.3), the HH wave function for an open spatial slice. The blue regions are fixed and the wavy boundaries are summed over in the GPI. Some readers may find it a little bit strange to associate a wave function but not a density matrix (See [52–56] for different versions of density matrices akin t… view at source ↗
Figure 19
Figure 19. Figure 19: We consider the norm of the no boundary wavefunciton in the right panel, where [PITH_FULL_IMAGE:figures/full_fig_p054_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: In pale blue we show the unperturbed spacetime boundary [PITH_FULL_IMAGE:figures/full_fig_p056_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The sphere partition function is usually considered to be the leading contribution [PITH_FULL_IMAGE:figures/full_fig_p065_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: In the partially frozen GPI, we consider a sphere partition function with a fixed [PITH_FULL_IMAGE:figures/full_fig_p067_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Splitting the partially frozen sphere (left) into two half-spheres (right upper) is [PITH_FULL_IMAGE:figures/full_fig_p069_23.png] view at source ↗
read the original abstract

We revisit the Hartle-Hawking wave function in AdS spacetime, where natural spatial slices are open and require an additional spacetime boundary. This leads to two constructions: a fully gravitational wave function, in which the boundary configuration is integrated over, and a partially frozen one, in which it is fixed, as in AdS/CFT. To illustrate the fully gravitational construction, we explicitly analyze it in AdS$_3$ Einstein gravity and AdS$_2$ Jackiw-Teitelboim gravity. We then evaluate the one-loop correction to the hyperbolic-ball partition function in $D$-dimensional AdS Einstein gravity, expected to give the leading contribution to the wave-function norm. We demonstrate that the fully gravitational hyperbolic ball partition function, where the boundary fluctuates, develops a nontrivial one-loop phase of $(\mp i)^{D+1}$, analogous to that of the sphere partition function in dS gravity. By contrast, the partially frozen partition function, where the boundary is fixed, remains real and positive. Motivated by this AdS comparison, we conversely investigate a partially frozen dS sphere partition function where the metric on an equator is fixed, finding that its one-loop phase cancels nontrivially. Our results suggest that the phase problem is controlled by whether the gravitational path integral is fully dynamical or partially frozen.

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 / 2 minor

Summary. The manuscript compares two Hartle-Hawking wave-function constructions in AdS: a fully gravitational version integrating over fluctuating boundary configurations versus a partially frozen version with fixed boundary (as in AdS/CFT). Explicit analyses are performed in AdS3 Einstein gravity and AdS2 JT gravity. The one-loop correction to the hyperbolic-ball partition function is evaluated in general D-dimensional AdS Einstein gravity and shown to produce a phase factor (∓i)^{D+1} when the boundary fluctuates, while remaining real and positive when fixed. A converse calculation for a partially frozen dS sphere partition function yields nontrivial phase cancellation. The results are used to suggest that the phase problem is controlled by whether the gravitational path integral is fully dynamical or partially frozen.

Significance. If the one-loop term indeed dominates the norm and the regularization is consistent, the work supplies a concrete distinction between fully dynamical and partially frozen gravitational path integrals that may address the phase problem in both AdS and dS settings. The explicit low-dimensional calculations and the AdS-to-dS comparison constitute genuine strengths that could inform future work on wave-function normalizations.

major comments (2)
  1. [Abstract] Abstract: the assertion that the one-loop correction 'is expected to give the leading contribution to the wave-function norm' is load-bearing for the central claim, yet no argument, scaling estimate, or reference to a later section is supplied showing suppression of two-loop or non-perturbative contributions that could modify the phase.
  2. [General D-dimensional calculation] The general-D one-loop calculation (the section presenting the hyperbolic-ball determinant): the regularization procedure for boundary fluctuations that produces the phase (∓i)^{D+1} is not shown to remain consistent beyond one loop; if higher-order terms alter the phase or the relative norm, the claimed distinction between fully dynamical and partially frozen constructions would not control the phase problem.
minor comments (2)
  1. [One-loop phase derivation] The sign choice in the phase factor (∓i)^{D+1} should be explained explicitly with reference to the orientation or boundary condition used.
  2. [Low-dimensional examples] The low-dimensional explicit analyses (AdS3 and AdS2 sections) would benefit from additional intermediate steps or cross-checks against known results to facilitate verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the text to clarify the scope and justification of our one-loop results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the one-loop correction 'is expected to give the leading contribution to the wave-function norm' is load-bearing for the central claim, yet no argument, scaling estimate, or reference to a later section is supplied showing suppression of two-loop or non-perturbative contributions that could modify the phase.

    Authors: We agree that the original phrasing required additional support. In the revised manuscript we have updated the abstract to specify that the one-loop term supplies the leading perturbative correction in the semiclassical (large-radius) regime, and we have inserted a short scaling argument in the introduction: higher-loop and non-perturbative contributions are suppressed by additional powers of the Newton constant G_N, which becomes parametrically small for the hyperbolic-ball geometries under consideration. We also cite earlier literature on one-loop determinants in AdS gravity that establishes this hierarchy. revision: yes

  2. Referee: [General D-dimensional calculation] The general-D one-loop calculation (the section presenting the hyperbolic-ball determinant): the regularization procedure for boundary fluctuations that produces the phase (∓i)^{D+1} is not shown to remain consistent beyond one loop; if higher-order terms alter the phase or the relative norm, the claimed distinction between fully dynamical and partially frozen constructions would not control the phase problem.

    Authors: Our analysis is performed strictly at one-loop order, where the phase factor is computed explicitly via the regularized determinant. We concur that demonstrating consistency of the same regularization beyond one loop lies outside the present scope. The revised discussion section now states explicitly that the reported phase distinction holds at one-loop level and that its persistence at higher orders remains an open question for future work. This scopes the central claim to the perturbative regime we have controlled. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper computes one-loop determinants for the hyperbolic-ball partition function under two distinct boundary treatments (fully integrated vs fixed) in explicit models (AdS3 Einstein, AdS2 JT, and D-dimensional Einstein gravity). The reported phases (∓i)^{D+1} for fluctuating boundaries and real-positive for fixed boundaries follow directly from the determinant evaluation and regularization choice, without reduction to a fitted parameter, self-defined quantity, or load-bearing self-citation. The analogy to dS sphere results is presented as motivation rather than a premise that forces the AdS outcome. No step equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard assumption that the one-loop approximation captures the leading contribution to the norm and on the usual rules of the gravitational path integral; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The one-loop correction to the hyperbolic-ball partition function gives the leading contribution to the wave-function norm.
    Explicitly stated in the abstract as the quantity whose phase is evaluated.
  • domain assumption The gravitational path integral is well-defined when the boundary is either fully integrated or held fixed.
    Underlying both constructions compared in the paper.

pith-pipeline@v0.9.0 · 5551 in / 1326 out tokens · 47324 ms · 2026-05-15T02:31:56.341083+00:00 · methodology

discussion (0)

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Reference graph

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