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arxiv: 2604.26662 · v2 · pith:HGUT3IVEnew · submitted 2026-04-29 · ✦ hep-th

Complex Geodesics in the Nariai Geometry

Lars Aalsma , Mir Mehedi Faruk This is my paper

Pith reviewed 2026-05-19 17:26 UTC · model grok-4.3

classification ✦ hep-th
keywords Nariai geometrycomplex geodesicstwo-point correlation functionsanalytic continuationde Sitter spaceheat kernelheavy scalar fields
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The pith

The two-point correlation function in Nariai geometry equals a sum over complex geodesics obtained by analytic continuation from a sphere product, with phases retained to eliminate spurious singularities.

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

The paper derives the two-point functions of heavy scalar fields in the Nariai geometry by starting from the geodesic approximation on a product of spheres. Analytic continuation of one sphere then produces the desired Nariai result. The outcome appears as a sum over complex rather than real geodesics. Retaining the phase of each contribution proves essential, otherwise the correlator develops unphysical singularities. This construction directly extends earlier geodesic-sum results obtained in pure de Sitter space.

Core claim

Utilizing the heat kernel formalism, the two-point correlation function is obtained from a geodesic approximation on a product of spheres. By analytically continuing one of the spheres, the correlation function in the Nariai geometry is derived as a sum over complex geodesics. The phase of each geodesic contribution must be accounted for to prevent spurious singularities.

What carries the argument

Analytic continuation of the geodesic sum (via heat kernel) from the sphere product to the Nariai geometry, keeping complex phases.

If this is right

  • The correlator is expressed as an explicit sum over complex geodesics rather than real ones.
  • Each term must carry its full complex phase; dropping the phase creates artificial singularities.
  • The result reduces to known de Sitter expressions when the appropriate limit is taken.
  • The heat-kernel route captures the heavy-mass limit without further corrections in this geometry.

Where Pith is reading between the lines

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

  • The same continuation procedure could be applied to other near-horizon or product geometries that admit sphere-factor descriptions.
  • Phase retention may be required in holographic calculations that rely on complex saddle points to obtain physically sensible correlators.
  • Numerical checks in the large-mass limit could verify whether the continued sum matches direct field-theoretic computations.

Load-bearing premise

The geodesic approximation to the two-point function remains valid under analytic continuation from the product of spheres to the Nariai geometry.

What would settle it

An independent calculation of the same two-point function by direct mode expansion in Nariai coordinates that produces a different functional form or different singularity locations.

read the original abstract

We study two-point correlation functions of heavy scalar fields in the Nariai geometry. Utilizing the heat kernel formalism, we obtain this result from a geodesic approximation to the two-point function on a product of spheres. By analytically continuing one of the spheres, we obtain the correlation function in the Nariai geometry. This result involves a sum over complex geodesics, extending previous results in pure de Sitter space. We emphasize the important role of the phase of each geodesic contribution, which needs to be taken into account to avoid spurious singularities in the correlator.

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 paper computes two-point correlation functions of heavy scalar fields in the Nariai geometry by applying the heat kernel formalism to obtain a geodesic approximation on the product of two spheres, then analytically continuing one sphere factor to reach the Nariai metric. The resulting correlator is expressed as a sum over complex geodesics, extending prior de Sitter results, with the phase of each geodesic contribution retained to eliminate spurious singularities.

Significance. If the analytic continuation is shown to preserve the geodesic saddles and phases without introducing new contributions from the altered global structure or curvature, the result would usefully generalize complex-geodesic methods from pure de Sitter to the Nariai background. This could aid calculations of heavy-field correlators in spacetimes with horizons and different topology, provided the heavy-mass limit and phase continuity are rigorously controlled.

major comments (2)
  1. [analytic continuation step (post-§2)] The central step of analytic continuation from S²×S² to the Nariai geometry (described in the abstract and presumably detailed after the heat-kernel setup) assumes that the complex geodesics and their phases remain valid without additional saddles or jumps in the imaginary part when the metric signature changes. No explicit path specification, demonstration of path-independence, or cross-check against the pure dS₂ limit is indicated, which is load-bearing for the claim that retaining phases cancels spurious singularities.
  2. [heat kernel formalism section] The heat-kernel derivation on the sphere product is used to justify the heavy-field limit, but the manuscript provides no error estimates or verification that the geodesic approximation survives the continuation without corrections arising from the changed curvature or global structure (as required for the Nariai case).
minor comments (2)
  1. [results section] Clarify the notation for the complex geodesic lengths and phases in the final sum; ensure they are defined consistently with the continuation parameter.
  2. [discussion] Add a brief comparison table or plot showing the Nariai correlator against the pure de Sitter limit in a controlled regime to illustrate the extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their detailed and insightful comments, which have helped us identify areas where the manuscript can be improved. We address each major comment below and commit to making the necessary revisions to clarify the analytic continuation and strengthen the justification of the approximations used.

read point-by-point responses

  1. Referee: [analytic continuation step (post-§2)] The central step of analytic continuation from S²×S² to the Nariai geometry (described in the abstract and presumably detailed after the heat-kernel setup) assumes that the complex geodesics and their phases remain valid without additional saddles or jumps in the imaginary part when the metric signature changes. No explicit path specification, demonstration of path-independence, or cross-check against the pure dS₂ limit is indicated, which is load-bearing for the claim that retaining phases cancels spurious singularities.

    Authors: We thank the referee for highlighting this important point. Upon review, we recognize that the details of the analytic continuation path were not sufficiently explicit in the original manuscript. In the revised version, we will specify the analytic continuation path in the complex plane for the sphere radius parameter, demonstrate that the complex geodesics and their phases continue smoothly without introducing new saddles or jumps in the imaginary part, and provide a cross-check by recovering the known pure de Sitter results in the appropriate limit. We will also argue for path-independence by showing that the relevant geodesics lie in the same homotopy class and that the phase is determined by the continuous variation of the geodesic length. This will reinforce that retaining the phases indeed cancels the spurious singularities as claimed. revision: yes

  2. Referee: [heat kernel formalism section] The heat-kernel derivation on the sphere product is used to justify the heavy-field limit, but the manuscript provides no error estimates or verification that the geodesic approximation survives the continuation without corrections arising from the changed curvature or global structure (as required for the Nariai case).

    Authors: We agree that error estimates and verification of the approximation's robustness under continuation would enhance the manuscript. The heat kernel expansion on the sphere product yields the geodesic approximation with corrections that are exponentially suppressed in the heavy mass limit. In the revision, we will include explicit error bounds for this approximation and provide a discussion verifying that these bounds persist under the analytic continuation to the Nariai geometry. Since the continuation affects the metric continuously and the Nariai space has constant curvature in each factor, no new corrections from curvature changes or global topology are expected at leading order. We will add this analysis to the heat kernel section. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard analytic continuation from sphere product via heat kernel

full rationale

The paper obtains the Nariai two-point function by first applying the geodesic approximation and heat kernel formalism on the product of spheres, then analytically continuing one sphere factor. This is a conventional extension of prior de Sitter results and does not reduce any claimed prediction or phase retention to a self-definition, fitted input, or self-citation chain within the paper's own equations. The derivation remains self-contained against external benchmarks such as the known sphere-product heat kernel and the analytic continuation procedure itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the geodesic approximation in the heavy-field limit and the legitimacy of analytic continuation between the sphere product and Nariai geometry. No free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption Geodesic approximation accurately captures the two-point function of heavy scalars via the heat kernel on the sphere product.
    Invoked to obtain the correlator before analytic continuation.
  • domain assumption Analytic continuation from the sphere product yields the correct Nariai correlator without introducing uncontrolled errors.
    Central step that maps the simpler geometry to Nariai space.

pith-pipeline@v0.9.0 · 5611 in / 1432 out tokens · 42783 ms · 2026-05-19T17:26:49.344626+00:00 · methodology

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