IR behaviour of one-loop complex $\mathbb{R}\times S^3$ saddles

Abstract: Gravitational path-integral over $\mathbb{R}\times S^3$ complex metrics with fluctuations is studied in 4D for Einstein-Hilbert gravity in Lorentzian signature, with the aim to investigate the IR properties of complex saddles for various boundary choices. General covariance doesn't allow arbitrary boundary choices for the background and fluctuations. In the ADM-decomposition, while imposing ``no-boundary'' condition at the initial boundary, two scenarios are considered for the final boundary: Dirichlet and fixed extrinsic curvature. Universe undergoes transition from a Euclidean to Lorentzian phase in either scenario, where the dominant saddle in Euclidean phase correspond to a Euclidean metric (imaginary time), while the Lorentzian phase has two complex metrics as dominant saddles which superimpose. One-loop corrected lapse action is computed using Hurwitz-Zeta regularization. UV-divergences canceled by suitable counter terms lead to a renormalized lapse action. One-loop renormalized Hartle-Hawking wave-function is computed using the Picard-Lefschetz and WKB methods, where the contributions coming from the metric-fluctuations show secularly growing infrared divergences as the Universe expands. This is compared with the situation in pure Lorentzian dS, corresponding to a Universe transitioning from an initial state of vanishing conjugate momenta to final state of fixed extrinsic curvature, thereby giving real saddles. Picard-Lefschetz methods alone are not sufficient to overcome the technical hurdles in the one-loop computation, which needs to be supplemented by an $iε$-prescription, achieved via slight complexification of the cosmological constant $Λ$. The UV renormalized one-loop dS wavefunction has the same leading IR divergence as for the Hartle-Hawking no-boundary Universe. Interestingly for all boundary choices considered, the saddles remain KSW-allowed.

• The paper presents a rigorous analysis of one-loop infrared divergences in ℝ×S³ complex metrics using Picard-Lefschetz theory and Hurwitz-Zeta regularization.
• It demonstrates that exponential parametrization leads to simplified, strictly linear boundary conditions, streamlining the one-loop path-integral computations.
• The study reveals that IR divergences persist universally across both Hartle-Hawking and Lorentzian de Sitter saddles, highlighting robust secular growth.

One-Loop Infrared Behavior of Complex $\mathbb{R}\times S^3$ Saddles in Lorentzian Quantum Gravity

Overview

This essay provides a technical summary of "IR behaviour of one-loop complex $\mathbb{R}\times S^3$ saddles" (2604.21736), which analyzes the gravitational path-integral for $\mathbb{R}\times S^3$ complex metrics in four-dimensional Einstein-Hilbert gravity, with a focus on IR properties of complex semiclassical saddles under various boundary conditions. The study rigorously treats fluctuations, boundary constraints from covariance, and both analytic and numerical aspects of one-loop determinants.

Figure 1: The Hartle-Hawking (H-H) no-boundary wave function $\Psi_{\rm H-H}$, with small metric fluctuations on the final hypersurface.

Boundary Condition Constraints and Parametrization

General covariance imposes stringent restrictions on permissible boundary conditions for both the background metric and spatial fluctuations. The paper introduces a comprehensive analysis using the ADM decomposition, exploring two final boundary scenarios: (1) Dirichlet and (2) fixed extrinsic curvature. Enforcing the no-boundary initial condition, boundary choices for metric fluctuations are dictated by background constraints. A key technical result is that exponential parametrization ($\varepsilon=1$) yields simpler, strictly linear boundary conditions for fluctuations, significantly streamlining one-loop path-integral computations compared to the more generic linear split ($\varepsilon=0$).

Figure 2: Diagram of background no-boundary universe, highlighting the saddle geometry in the complex time plane.

Semiclassical Saddle Analysis

The Universe transitions from Euclidean to Lorentzian phases; dominant Euclidean saddles correspond to purely imaginary time metrics, whereas Lorentzian evolution is mediated by a superposition of two complex saddle metrics. The allowed saddles are classified using Picard-Lefschetz (PL) theory, with explicit checks of their KSW-allowability to ensure path-integral convergence and physical legitimacy.

Figure 3: KSW allowability plots for no-boundary saddles at varying $H_1<H$, demonstrating the available time path below the extremal curve for all $H_1$.

The Picard-Lefschetz method is combined with a supplementary $i$-prescription (complexifying the cosmological constant) to resolve technical obstacles in the one-loop computation, particularly in the Lorentzian regime.

One-Loop Determinant, UV Renormalization, and Zeta Regularization

The path-integral is evaluated up to one loop using Hurwitz-Zeta regularization. UV divergences from the metric-fluctuation sector are extracted and canceled by analytically derived counterterms. The resultant one-loop renormalized lapse action is employed to construct the Hartle-Hawking wavefunction.

A salient numerical result is that the one-loop fluctuation determinant demonstrates secularly growing IR divergences as the Universe expands, scaling with spatial volume in the asymptotic regime. This phenomenon is quantitatively analyzed via zeta-regularized infinite sums and Hurwitz-Zeta function techniques, giving the explicit structure of leading and subleading terms in both fixed size and fixed extrinsic curvature boundary cases.

Figure 4: PL plots for saddles in Lorentzian ($H_1<H$) and Euclidean ($\mathbb{R}\times S^3$0) regimes, using complexified Newton's constant to resolve Stokes degeneracy.

Comparative Analysis with Lorentzian de Sitter

The paper benchmarks the Hartle-Hawking no-boundary results against pure Lorentzian de Sitter (dS) wavefunctions, computed with initial vanishing conjugate momenta and fixed extrinsic curvature on the final hypersurface (real saddles). Picard-Lefschetz contours with $\mathbb{R}\times S^3$1-prescription are found necessary to resolve pinch singularities in the functional determinant at real dS saddles. Crucially, the UV-renormalized one-loop dS wavefunction shares the same leading IR divergence as the Hartle-Hawking universe, indicating the IR pathology is not a boundary artifact and is robust to saddle selection.

Figure 5: Plot of $\mathbb{R}\times S^3$2 showing increasing amplitude as $\mathbb{R}\times S^3$3, dominated by one-loop gravity fluctuation growth.

KSW Allowability and Physical Implications

Analysis via the KSW (Kontsevich-Segal-Witten) criterion confirms that all considered saddles—complex no-boundary and complexified dS—remain within the physically allowable domain for all boundary choices, including nonlinear conformal boundary conditions. The allowable region is demonstrated visually and algebraically for a wide range of parameters.

Figure 6: Saddle geometry of expanding real de Sitter ($\mathbb{R}\times S^3$4): from throat at $\mathbb{R}\times S^3$5 to large $\mathbb{R}\times S^3$6, with initial surface fluctuation vanishing.

Figure 7: Picard-Lefschetz analysis for real de Sitter, showing relevant saddles lying on the integration contour.

IR Divergence: Quantitative Results

Strong quantitative findings include:

• The one-loop corrected Hartle-Hawking wave function exhibits exponential growth in the IR limit, with the volume factor in the exponent, irrespective of boundary condition (fixed final size or extrinsic curvature).
• The correction to the phase from the one-loop determinant overtakes the semiclassical phase in the deep IR, especially in fixed curvature cases.
• The wavefunction for both complex and real dS saddles matches in asymptotic behavior, confirming the universality of secular growth.

  Figure 8: Contact divergence arising from the final hypersurface approaching the initial hypersurface; proper distance vanishes as $\mathbb{R}\times S^3$7.

Future Directions

The study opens several technical avenues for further exploration:

• Generalization to other instantons, EAdS asymptotics, and wormhole geometries.
• Deeper investigation of pinch singularities and their resolution in Lorentzian QFT via resummation or Schwinger-Keldysh formalism.
• Analyses of boundary choices and their impact on mixing in fluctuation sectors beyond the Dirichlet/Robin scheme.

  Figure 9: PL plot for complexified $\mathbb{R}\times S^3$8, showing both saddles relevant and highlighting domain deformation due to complexification.

  

Figure 10: KSW allowability plots for complex de Sitter saddles under various parameters: saddles remain below extremal curve and are allowable.

Conclusion

This paper delivers a rigorous treatment of the gravitational path-integral over $\mathbb{R}\times S^3$9 complex metrics and fluctuations in Lorentzian quantum gravity. It establishes, through careful boundary analysis, covariance, and one-loop techniques, that secular IR divergences in the wavefunction are inherent and boundary-condition-independent for both Hartle-Hawking and Lorentzian de Sitter universes. Technical tools including Picard-Lefschetz, Hurwitz-zeta regularization, and KSW analysis are used to provide explicit, numerically precise descriptions of these phenomena. The work has broader implications in cosmological quantum gravity, particularly regarding the robustness of IR divergences and the sensitivity of physical wavefunctions to the choice of fluctuation parametrization and boundary constraints.

Figure 11: $\mathbb{R}\times S^3$0 evaluated as a function of $\mathbb{R}\times S^3$1, illustrating IR divergence and matching one-loop dominance in both Hartle-Hawking and de Sitter cases.