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arxiv: 2507.10537 · v2 · submitted 2025-07-14 · ✦ hep-th · gr-qc

Resolving Degeneracies in Complex mathbb{R}times S³ and θ-KSW

Manishankar Ailiga , Shubhashis Mallik , Gaurav Narain This is my paper

Pith reviewed 2026-05-19 04:29 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Gauss-Bonnet gravityLorentzian path integralmini-superspacePicard-Lefschetz methodsNo-boundary geometriesKSW criterioncomplex deformationanti-linear symmetry
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The pith

Complex deformation of G ħ resolves type-1 degeneracies from anti-linear symmetry in the lapse action and modifies the KSW criterion.

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

The paper examines the Lorentzian gravitational path integral for Gauss-Bonnet gravity in a mini-superspace ansatz. The gauge-fixed integral is computed exactly with Airy functions, with dominant contributions from No-boundary geometries. Degeneracies of type-1 arise because anti-linear symmetry in the lapse action causes overlapping flow lines between saddles, blocking unambiguous Picard-Lefschetz analysis. Complex deformation of G ħ breaks this symmetry, lifts the remaining degeneracies after partial quantum corrections, and allows a deformed contour for the WKB approximation. The same deformation changes the KSW criterion, which then imposes a constraint on the deformation parameter if No-boundary geometries are to remain allowed.

Core claim

The authors show that every type-1 degeneracy traces to anti-linear symmetry present in the lapse action. Any breaking of that symmetry resolves the overlap of flow lines and removes ambiguity in identifying relevant saddles. After quantum fluctuations of the scale factor lift type-2 degeneracies completely and type-1 only partially, the complex deformation of G ħ completes the resolution. The resulting contour deformation permits consistent application of Picard-Lefschetz and WKB methods, while the modified KSW criterion requires a strong bound on the deformation to keep No-boundary geometries allowed.

What carries the argument

Complex deformation of (G ħ) that breaks anti-linear symmetry in the lapse action, thereby removing type-1 degeneracies and altering the KSW criterion.

If this is right

  • The lapse integral can be evaluated along a unique deformed contour without ambiguity from overlapping flow lines.
  • Quantum fluctuations of the scale factor plus the deformation together resolve all degeneracies, allowing reliable WKB results.
  • No-boundary geometries satisfy the KSW criterion only when the deformation parameter obeys the derived bound.
  • Artificial defects become unnecessary once anti-linear symmetry is broken by the complex deformation.

Where Pith is reading between the lines

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

  • The same symmetry-breaking deformation may be tested in other gravitational models that exhibit similar anti-linear symmetries in their lapse actions.
  • The approach could inform contour choices for Lorentzian path integrals beyond the mini-superspace truncation.
  • Higher-dimensional or non-minisuperspace calculations would check whether the required constraint on the deformation persists.

Load-bearing premise

The mini-superspace ansatz together with the chosen Robin boundary condition captures the dominant physics of the full four-dimensional Lorentzian path integral.

What would settle it

An explicit saddle-point calculation for a fixed boundary parameter in which the deformed G ħ violates the new KSW bound yet No-boundary geometries still dominate the integral would falsify the claimed constraint.

Figures

Figures reproduced from arXiv: 2507.10537 by Gaurav Narain, Manishankar Ailiga, Shubhashis Mallik.

Figure 1
Figure 1. Figure 1: FIG. 1. Plot depicting type-1 degeneracy where the flow-lines overlap and get resolved by the [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plot depicting type-2 degeneracy of second kind, where not only there is overlap of flow [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plot depicting type-2 degeneracy of first kind, where not only there is overlap of flow [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A plot similar to the Fig. (3) is shown for [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plot illustrating the breaking of the residual [PITH_FULL_IMAGE:figures/full_fig_p036_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Illustration of the Ridge criterion implemented in the complexified [PITH_FULL_IMAGE:figures/full_fig_p045_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. This plots shows a path that connects the end points [PITH_FULL_IMAGE:figures/full_fig_p045_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. This figure provides an explicit demonstration of the Extremal Curve Test applied to the [PITH_FULL_IMAGE:figures/full_fig_p047_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Illustration of KSW numerical analysis results. The area covered with pink dots repre [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Similar to Fig. 9, this figure corresponds to the degenerate case with [PITH_FULL_IMAGE:figures/full_fig_p050_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. This plot illustrates the allowability of background no-boundary saddles under the Ex [PITH_FULL_IMAGE:figures/full_fig_p051_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. This figure shows the [PITH_FULL_IMAGE:figures/full_fig_p053_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. A similar plot to Fig. 12, with [PITH_FULL_IMAGE:figures/full_fig_p054_13.png] view at source ↗
read the original abstract

Lorentzian gravitational path integral for the Gauss-Bonnet gravity in $4D$ is studied in the mini-superspace ansatz for metric. The gauge-fixed path-integral for Robin boundary choice is computed exactly using {\it Airy}-functions, where the dominant contribution comes from No-boundary geometries. The lapse integral is further analysed using saddle-point methods to compare with exact results. Picard-Lefschetz methods are utilized to find the {\it relevant} complex saddles and deformed contour of integration, thereby using WKB methods to compute the integral along the deformed contour in the saddle-point approximation. However, their successful application is possible only when system is devoid of degeneracies, which in present case appear in two types: {\bf type-1} where the flow-lines starting from neighbouring saddles overlap leading to ambiguities in deciding the {\it relevance} of saddles, {\bf type-2} where saddles merge for specific choices of boundary parameters leading to failure of WKB. Overcoming degeneracies using artificial {\it defects} introduces ambiguities due to the choice of {\it defects} involved. Corrections from quantum fluctuations of scale-factor overcome degeneracies only partially (lifts {\bf type-2} completely with partial resolution of {\bf type-1}), with the residual lifted fluently by complex deformation of $(G\hbar)$. {\it Anti-linear} symmetry present in various forms in the lapse action is the reason behind all the {\bf type-1} degeneracies. Any form of {\it defect} or {\it deformation} breaking anti-linearity resolves {\bf type-1} degeneracies, indicating complex deformation of $(G\hbar)$ as an ideal choice. Compatibility with the KSW criterion is analyzed after symmetry breaking. Complex deformation of $(G\hbar)$ modifies the KSW criterion, imposing a strong constraint on the deformation if No-boundary geometries are required to be always KSW-allowed.

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 studies the Lorentzian gravitational path integral for Gauss-Bonnet gravity in 4D under a mini-superspace ansatz for the metric. It computes the gauge-fixed path integral exactly using Airy functions for a Robin boundary condition, with the dominant contribution arising from No-boundary geometries. The lapse integral is analyzed via saddle-point methods and Picard-Lefschetz theory to identify relevant complex saddles and deformed contours. Two types of degeneracies are identified: type-1 (overlapping flow lines from neighboring saddles due to anti-linear symmetry in the lapse action) and type-2 (merging saddles for specific boundary parameters). Quantum fluctuations of the scale factor resolve type-2 completely and type-1 partially; the residual type-1 degeneracies are lifted by a complex deformation of (G ħ) that breaks anti-linearity. The paper then examines compatibility with the KSW criterion after this symmetry breaking, finding that the deformation modifies the criterion and imposes a constraint to keep No-boundary geometries KSW-allowed.

Significance. If the results hold, the exact Airy-function evaluation of the gauge-fixed integral supplies a concrete benchmark for mini-superspace path integrals in quantum gravity. The identification of anti-linear symmetry as the origin of type-1 degeneracies and the explicit demonstration that any anti-linearity-breaking deformation resolves them clarifies the application of Picard-Lefschetz methods to gravitational integrals. The subsequent KSW analysis after deformation could constrain parameters in no-boundary proposals. The manuscript ships an exact, non-perturbative result, which strengthens the technical contribution.

major comments (2)
  1. Abstract and the paragraph on gauge-fixed path-integral and lapse integral analysis: the central claim that complex deformation of (G ħ) resolves type-1 degeneracies and modifies the KSW criterion rests on the mini-superspace ansatz with Robin boundary conditions faithfully reproducing the dominant saddle structure of the unreduced 4D theory. The manuscript does not provide an argument or test showing that additional graviton and matter fluctuation modes in the full theory would not lift the same degeneracies without deformation or reintroduce them afterward; if the degeneracies are reduction artifacts, the derived constraint on the deformation parameter for KSW compatibility does not necessarily carry over.
  2. Abstract: the statement that complex deformation of (G ħ) is an 'ideal choice' because it breaks anti-linearity is presented as resolving the ambiguity in saddle relevance, yet the deformation is introduced precisely to address the degeneracy problem identified in the lapse action rather than being derived from an external physical principle or from the structure of the full Lorentzian path integral.
minor comments (2)
  1. The abstract mentions that 'explicit numerical checks' are absent for the deformed contour; adding a brief statement of the range of deformation parameters for which No-boundary saddles remain dominant would improve clarity.
  2. Notation for the complex shift in (G ħ) and the Robin boundary parameter should be introduced with a single consistent symbol when first appearing in the lapse-action analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We respond to the major comments below and have revised the manuscript to clarify the scope of our results within the mini-superspace approximation.

read point-by-point responses

  1. Referee: Abstract and the paragraph on gauge-fixed path-integral and lapse integral analysis: the central claim that complex deformation of (G ħ) resolves type-1 degeneracies and modifies the KSW criterion rests on the mini-superspace ansatz with Robin boundary conditions faithfully reproducing the dominant saddle structure of the unreduced 4D theory. The manuscript does not provide an argument or test showing that additional graviton and matter fluctuation modes in the full theory would not lift the same degeneracies without deformation or reintroduce them afterward; if the degeneracies are reduction artifacts, the derived constraint on the deformation parameter for KSW compatibility does not necessarily carry over.

    Authors: We agree that the analysis is performed strictly within the mini-superspace ansatz and that we do not demonstrate how additional fluctuation modes in the unreduced 4D theory would affect the degeneracies. The exact Airy-function evaluation and the resolution via complex (G ħ) deformation are results specific to this reduced model. We have revised the abstract, introduction, and conclusion to explicitly state that the degeneracy resolution and the resulting constraint on the deformation parameter for KSW compatibility are derived within the mini-superspace framework, and that their status in the full theory remains an open question requiring further investigation. revision: yes

  2. Referee: Abstract: the statement that complex deformation of (G ħ) is an 'ideal choice' because it breaks anti-linearity is presented as resolving the ambiguity in saddle relevance, yet the deformation is introduced precisely to address the degeneracy problem identified in the lapse action rather than being derived from an external physical principle or from the structure of the full Lorentzian path integral.

    Authors: We accept the point that the deformation is introduced to resolve the type-1 degeneracies arising from anti-linear symmetry in the lapse action. The description as an 'ideal choice' was meant to contrast it with arbitrary artificial defects, since any anti-linearity-breaking modification resolves the overlap of flow lines. We have revised the abstract to clarify the motivation as arising directly from the degeneracy analysis in the mini-superspace model rather than from an independent external principle. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper first obtains an exact Airy-function expression for the gauge-fixed path integral under the mini-superspace ansatz and Robin boundary conditions. It then performs an independent saddle-point and Picard-Lefschetz analysis of the lapse integral, identifies an anti-linear symmetry in the lapse action as the source of type-1 degeneracies, and notes that any anti-linearity-breaking deformation (including a complex shift in Għ) lifts those degeneracies. The subsequent KSW-compatibility check is performed on the deformed theory. Because the exact Airy result, the symmetry identification, and the degeneracy-lifting property are all derived directly from the model's equations without reducing to a fitted parameter renamed as a prediction or to a self-citation chain, the central claims do not collapse to their inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The calculation rests on the mini-superspace reduction, the validity of the gauge-fixed Robin boundary path integral, and the applicability of Picard-Lefschetz theory to the lapse integral; no new particles or forces are postulated.

free parameters (1)
  • Robin boundary parameter
    Boundary condition parameter appearing in the exact Airy-function result; its specific value affects the location of saddles and degeneracies.
axioms (2)
  • domain assumption Mini-superspace ansatz reduces the full 4D metric degrees of freedom to a single scale factor
    Invoked at the start of the abstract to define the model.
  • domain assumption Picard-Lefschetz theory correctly identifies relevant complex saddles for the Lorentzian gravitational path integral
    Used to select the deformed contour after degeneracies are resolved.

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Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · cited by 1 Pith paper · 18 internal anchors

  1. [1]

    Feynman Diagrams for the Yang-Mills Field,

    L. D. Faddeev and V. N. Popov, “Feynman Diagrams for the Yang-Mills Field,” Phys. Lett. B 25 (1967), 29-30 doi:10.1016/0370-2693(67)90067-6

    work page doi:10.1016/0370-2693(67)90067-6 1967

  2. [2]

    One loop divergencies in the theory of gravitation,

    G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in the theory of gravitation,” Ann. Inst. H. Poincare A Phys. Theor. 20 (1974), 69-94

    work page 1974

  3. [3]

    Laporta and E

    S. Deser, H. S. Tsao and P. van Nieuwenhuizen, “Nonrenormalizability of Einstein Yang-Mills Interactions at the One Loop Level,” Phys. Lett. B 50 (1974), 491-493 doi:10.1016/0370- 2693(74)90268-8

    work page doi:10.1016/0370- 1974

  4. [4]

    One Loop Divergences of Quantized Einstein-Maxwell Fields,

    S. Deser and P. van Nieuwenhuizen, “One Loop Divergences of Quantized Einstein-Maxwell Fields,” Phys. Rev. D 10 (1974), 401 doi:10.1103/PhysRevD.10.401

    work page doi:10.1103/physrevd.10.401 1974

  5. [5]

    QUANTUM GRAVITY AT TWO LOOPS,

    M. H. Goroff and A. Sagnotti, “QUANTUM GRAVITY AT TWO LOOPS,” Phys. Lett. B 160 (1985), 81-86 doi:10.1016/0370-2693(85)91470-4

    work page doi:10.1016/0370-2693(85)91470-4 1985

  6. [6]

    The Ultraviolet Behavior of Einstein Gravity,

    M. H. Goroff and A. Sagnotti, “The Ultraviolet Behavior of Einstein Gravity,” Nucl. Phys. B 266 (1986), 709-736 doi:10.1016/0550-3213(86)90193-8

    work page doi:10.1016/0550-3213(86)90193-8 1986

  7. [7]

    Two loop quantum gravity,

    A. E. M. van de Ven, “Two loop quantum gravity,” Nucl. Phys. B 378 (1992), 309-366 doi:10.1016/0550-3213(92)90011-Y

    work page doi:10.1016/0550-3213(92)90011-y 1992

  8. [8]

    Renormalization of Higher Derivative Quantum Gravity

    K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D 16 (1977), 953-969 doi:10.1103/PhysRevD.16.953

    work page doi:10.1103/physrevd.16.953 1977

  9. [9]

    Remarks on High-energy Stability and Renormalizability of Gravity Theory,

    A. Salam and J. A. Strathdee, “Remarks on High-energy Stability and Renormalizability of Gravity Theory,” Phys. Rev. D 18 (1978), 4480 doi:10.1103/PhysRevD.18.4480 61

    work page doi:10.1103/physrevd.18.4480 1978

  10. [10]

    Short Distance Freedom of Quantum Gravity

    G. Narain and R. Anishetty, “Short Distance Freedom of Quantum Gravity,” Phys. Lett. B 711 (2012), 128-131 doi:10.1016/j.physletb.2012.03.070 [arXiv:1109.3981 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2012.03.070 2012

  11. [11]

    Exorcising Ghosts in Induced Gravity

    G. Narain, “Exorcising Ghosts in Induced Gravity,” Eur. Phys. J. C 77 (2017) no.10, 683 doi:10.1140/epjc/s10052-017-5249-z [arXiv:1612.04930 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-017-5249-z 2017

  12. [12]

    Physical Running of Couplings in Quadratic Gravity,

    D. Buccio, J. F. Donoghue, G. Menezes and R. Percacci, “Physical Running of Couplings in Quadratic Gravity,” Phys. Rev. Lett. 133 (2024) no.2, 021604 doi:10.1103/PhysRevLett.133.021604 [arXiv:2403.02397 [hep-th]]

    work page doi:10.1103/physrevlett.133.021604 2024

  13. [13]

    On Gauss-bonnet gravity and boundary conditions in Lorentzian path integral quantization,

    G. Narain, “On Gauss-bonnet gravity and boundary conditions in Lorentzian path integral quantization,” JHEP 05, 273 (2021) doi:10.1007/JHEP05(2021)273 [arXiv:2101.04644 [gr-qc]]

    work page doi:10.1007/jhep05(2021)273 2021

  14. [14]

    Surprises in Lorentzian path-integral of Gauss-Bonnet gravity,

    G. Narain, “Surprises in Lorentzian path-integral of Gauss-Bonnet gravity,” JHEP 04 (2022), 153 doi:10.1007/JHEP04(2022)153 [arXiv:2203.05475 [gr-qc]]

    work page doi:10.1007/jhep04(2022)153 2022

  15. [15]

    Lorentzian Robin Universe,

    M. Ailiga, S. Mallik and G. Narain, “Lorentzian Robin Universe,” JHEP 01 (2024), 124 doi:10.1007/JHEP01(2024)124 [arXiv:2308.01310 [gr-qc]]

    work page doi:10.1007/jhep01(2024)124 2024

  16. [16]

    Lorentzian path-integral of Robin Universe,

    M. Ailiga, S. Mallik and G. Narain, “Lorentzian path-integral of Robin Universe,” [arXiv:2407.16692 [gr-qc]]

    work page arXiv

  17. [17]

    Path Integrals and the Indefiniteness of the Gravitational Action,

    G. W. Gibbons, S. W. Hawking and M. J. Perry, “Path Integrals and the Indefiniteness of the Gravitational Action,” Nucl. Phys. B 138 (1978), 141-150 doi:10.1016/0550-3213(78)90161-X

    work page doi:10.1016/0550-3213(78)90161-x 1978

  18. [18]

    The Einstein tensor and its generalizations,

    D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. 12, 498-501 (1971) doi:10.1063/1.1665613

    work page doi:10.1063/1.1665613 1971

  19. [19]

    The four-dimensionality of space and the einstein tensor,

    D. Lovelock, “The four-dimensionality of space and the einstein tensor,” J. Math. Phys. 13, 874-876 (1972) doi:10.1063/1.1666069

    work page doi:10.1063/1.1666069 1972

  20. [20]

    A Remarkable property of the Riemann-Christoffel tensor in four dimensions,

    C. Lanczos, “A Remarkable property of the Riemann-Christoffel tensor in four dimensions,” Annals Math. 39, 842-850 (1938) doi:10.2307/1968467

    work page doi:10.2307/1968467 1938

  21. [21]

    Zwiebach, Phys

    B. Zwiebach, “Curvature Squared Terms and String Theories,” Phys. Lett. B 156, 315-317 (1985) doi:10.1016/0370-2693(85)91616-8

    work page doi:10.1016/0370-2693(85)91616-8 1985

  22. [22]

    The Quartic Effective Action for the Heterotic String,

    D. J. Gross and J. H. Sloan, “The Quartic Effective Action for the Heterotic String,” Nucl. Phys. B 291, 41-89 (1987) doi:10.1016/0550-3213(87)90465-2

    work page doi:10.1016/0550-3213(87)90465-2 1987

  23. [23]

    Three Loop Beta Functions for the Superstring and Heterotic String,

    R. R. Metsaev and A. A. Tseytlin, “Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the Sigma Model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor,” Nucl. Phys. B293, 385-419 (1987) doi:10.1016/0550- 3213(87)90077-0

    work page doi:10.1016/0550- 1987

  24. [24]

    Lorentzian Quantum Cosmology

    J. Feldbrugge, J. L. Lehners and N. Turok, “Lorentzian Quantum Cosmology,” Phys. Rev. D 95 (2017) no.10, 103508 doi:10.1103/PhysRevD.95.103508 [arXiv:1703.02076 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.95.103508 2017

  25. [25]

    Unstable no-boundary fluctuations from sums over regular metrics

    A. Di Tucci and J. L. Lehners, “Unstable no-boundary fluctuations from sums over reg- ular metrics,” Phys. Rev. D 98 (2018) no.10, 103506 doi:10.1103/PhysRevD.98.103506 [arXiv:1806.07134 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.98.103506 2018

  26. [26]

    No smooth beginning for spacetime,

    J. L. Lehners, “No smooth beginning for spacetime,” Int. J. Mod. Phys. D 28 (2018) no.02, 1930005 doi:10.1142/9789811258251 0005

    work page doi:10.1142/9789811258251 2018

  27. [27]

    No smooth beginning for spacetime

    J. Feldbrugge, J. L. Lehners and N. Turok, “No smooth beginning for spacetime,” Phys. Rev. Lett. 119 (2017) no.17, 171301 doi:10.1103/PhysRevLett.119.171301 [arXiv:1705.00192 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.119.171301 2017

  28. [28]

    No Rescue for the No Boundary Proposal

    J. Feldbrugge, J. L. Lehners and N. Turok, “No rescue for the no boundary proposal: Pointers to the future of quantum cosmology,” Phys. Rev. D 97 (2018) no.2, 023509 doi:10.1103/PhysRevD.97.023509 [arXiv:1708.05104 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.97.023509 2018

  29. [29]

    No-boundary prescriptions in Lorentzian quan- tum cosmology,

    A. Di Tucci, J. L. Lehners and L. Sberna, “No-boundary prescriptions in Lorentzian quan- tum cosmology,” Phys. Rev. D 100 (2019) no.12, 123543 doi:10.1103/PhysRevD.100.123543 62 [arXiv:1911.06701 [hep-th]]

    work page doi:10.1103/physrevd.100.123543 2019

  30. [30]

    Wave function of simple universes analytically continued from negative to positive potentials,

    J. L. Lehners, “Wave function of simple universes analytically continued from negative to positive potentials,” Phys. Rev. D 104 (2021) no.6, 063527 doi:10.1103/PhysRevD.104.063527 [arXiv:2105.12075 [hep-th]]

    work page doi:10.1103/physrevd.104.063527 2021

  31. [31]

    Lessons for quantum cosmology from anti–de Sitter black holes,

    A. Di Tucci, M. P. Heller and J. L. Lehners, “Lessons for quantum cosmology from anti–de Sitter black holes,” Phys. Rev. D 102 (2020) no.8, 086011 doi:10.1103/PhysRevD.102.086011 [arXiv:2007.04872 [hep-th]]

    work page doi:10.1103/physrevd.102.086011 2020

  32. [32]

    The No-Boundary Proposal as a Path Integral with Robin Boundary Conditions

    A. Di Tucci and J. L. Lehners, “No-Boundary Proposal as a Path Integral with Robin Boundary Conditions,” Phys. Rev. Lett. 122 (2019) no.20, 201302 doi:10.1103/PhysRevLett.122.201302 [arXiv:1903.06757 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.122.201302 2019

  33. [33]

    Relativistic S Matrix of Dynamical Systems with Boson and Fermion Constraints,

    I. A. Batalin and G. A. Vilkovisky, “Relativistic S Matrix of Dynamical Systems with Boson and Fermion Constraints,” Phys. Lett. B 69 (1977), 309-312 doi:10.1016/0370-2693(77)90553- 6

    work page doi:10.1016/0370-2693(77)90553- 1977

  34. [34]

    Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points,

    M. Honda, H. Matsui, K. Okabayashi and T. Terada, “Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points,” Phys. Rev. D 110, no.8, 8 (2024) doi:10.1103/PhysRevD.110.083508 [arXiv:2402.09981 [gr-qc]]

    work page doi:10.1103/physrevd.110.083508 2024

  35. [35]

    Jackiw-Teitelboim Gravity and Lorentzian Quantum Cosmology,

    M. Honda, H. Matsui, K. Numajiri and K. Okabayashi, “Jackiw-Teitelboim Gravity and Lorentzian Quantum Cosmology,” [arXiv:2412.20398 [gr-qc]]

    work page arXiv

  36. [36]

    Teitelboim, Quantum mechanics of the gravi- tational field , Phys

    C. Teitelboim, “Quantum Mechanics of the Gravitational Field,” Phys. Rev. D 25 (1982), 3159 doi:10.1103/PhysRevD.25.3159

    work page doi:10.1103/physrevd.25.3159 1982

  37. [37]

    The Proper Time Gauge in Quantum Theory of Gravitation,

    C. Teitelboim, “The Proper Time Gauge in Quantum Theory of Gravitation,” Phys. Rev. D 28 (1983), 297 doi:10.1103/PhysRevD.28.297

    work page doi:10.1103/physrevd.28.297 1983

  38. [38]

    Derivation of the Wheeler-De Witt Equation from a Path Integral for Min- isuperspace Models,

    J. J. Halliwell, “Derivation of the Wheeler-De Witt Equation from a Path Integral for Min- isuperspace Models,” Phys. Rev. D 38 (1988), 2468 doi:10.1103/PhysRevD.38.2468

    work page doi:10.1103/physrevd.38.2468 1988

  39. [39]

    Path integral quantum cosmology: A Class of exactly soluble scalar field minisuperspace models with exponential potentials,

    L. J. Garay, J. J. Halliwell and G. A. Mena Marugan, “Path integral quantum cosmology: A Class of exactly soluble scalar field minisuperspace models with exponential potentials,” Phys. Rev. D 43, 2572-2589 (1991) doi:10.1103/PhysRevD.43.2572

    work page doi:10.1103/physrevd.43.2572 1991

  40. [40]

    Causality Versus Gauge Invariance in Quantum Gravity and Supergravity,

    C. Teitelboim, “Causality Versus Gauge Invariance in Quantum Gravity and Supergravity,” Phys. Rev. Lett. 50 (1983), 705 doi:10.1103/PhysRevLett.50.705

    work page doi:10.1103/physrevlett.50.705 1983

  41. [41]

    Analytic Continuation Of Chern-Simons Theory

    E. Witten, “Analytic Continuation Of Chern-Simons Theory,” AMS/IP Stud. Adv. Math. 50 (2011), 347-446 [arXiv:1001.2933 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  42. [42]

    A New Look At The Path Integral Of Quantum Mechanics

    E. Witten, “A New Look At The Path Integral Of Quantum Mechanics,” [arXiv:1009.6032 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv

  43. [43]

    Resurgence theory, ghost-instantons, and analytic continuation of path integrals

    G. Basar, G. V. Dunne and M. Unsal, “Resurgence theory, ghost-instantons, and ana- lytic continuation of path integrals,” JHEP 10 (2013), 041 doi:10.1007/JHEP10(2013)041 [arXiv:1308.1108 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep10(2013)041 2013

  44. [44]

    An Introduction to Resurgence, Trans-Series and Alien Calculus

    D. Dorigoni, “An Introduction to Resurgence, Trans-Series and Alien Calculus,” Annals Phys. 409, 167914 (2019) doi:10.1016/j.aop.2019.167914 [arXiv:1411.3585 [hep-th]]

    work page doi:10.1016/j.aop.2019.167914 2019

  45. [45]

    Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling

    Y. Tanizaki and T. Koike, “Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling,” Annals Phys. 351 (2014), 250-274 doi:10.1016/j.aop.2014.09.003 [arXiv:1406.2386 [math-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aop.2014.09.003 2014

  46. [46]

    The Lovelock Gravitational Field Equations in Cosmology,

    N. Deruelle and L. Farina-Busto, “The Lovelock Gravitational Field Equations in Cosmology,” Phys. Rev. D 41, 3696 (1990) doi:10.1103/PhysRevD.41.3696

    work page doi:10.1103/physrevd.41.3696 1990

  47. [47]

    Schwarzschild field in n dimensions and the dimensionality of space prob- lem,

    F. R. Tangherlini, “Schwarzschild field in n dimensions and the dimensionality of space prob- lem,” Nuovo Cim. 27, 636-651 (1963) doi:10.1007/BF02784569 63

    work page doi:10.1007/bf02784569 1963

  48. [48]

    Dimensionality of Space and the Pulsating Universe,

    F. Tangherlini, “Dimensionality of Space and the Pulsating Universe,” Nuovo Cim. 91 (1986), 209-217

    work page 1986

  49. [49]

    Generalized boundary conditions in closed cosmologies,

    D. Brizuela and M. de Cesare, “Generalized boundary conditions in closed cosmologies,” Phys. Rev. D 107 (2023) no.10, 104054 doi:10.1103/PhysRevD.107.104054 [arXiv:2303.04007 [gr-qc]]

    work page doi:10.1103/physrevd.107.104054 2023

  50. [50]

    Boundary choices and one-loop Complex Universe,

    M. Ailiga, S. Mallik and G. Narain, “Boundary choices and one-loop Complex Universe,” [arXiv:2410.19724 [gr-qc]]

    work page arXiv

  51. [51]

    Hamiltonian formulation of f(Riemann) theories of gravity

    N. Deruelle, M. Sasaki, Y. Sendouda and D. Yamauchi, Prog. Theor. Phys. 123 (2010), 169- 185 doi:10.1143/PTP.123.169 [arXiv:0908.0679 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1143/ptp.123.169 2010

  52. [52]

    Robin Gravity

    C. Krishnan, S. Maheshwari and P. N. Bala Subramanian, “Robin Gravity,” J. Phys. Conf. Ser. 883, no.1, 012011 (2017) doi:10.1088/1742-6596/883/1/012011 [arXiv:1702.01429 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1742-6596/883/1/012011 2017

  53. [53]

    Constraining the topological Gauss- Bonnet coupling from GW150914,

    K. Chakravarti, R. Ghosh and S. Sarkar, “Constraining the topological Gauss- Bonnet coupling from GW150914,” Phys. Rev. D 106 (2022) no.4, L041503 doi:10.1103/PhysRevD.106.L041503 [arXiv:2201.08700 [gr-qc]]

    work page doi:10.1103/physrevd.106.l041503 2022

  54. [54]

    Positivity of Curvature-Squared Corrections in Gravity

    C. Cheung and G. N. Remmen, “Positivity of Curvature-Squared Corrections in Grav- ity,” Phys. Rev. Lett. 118 (2017) no.5, 051601 doi:10.1103/PhysRevLett.118.051601 [arXiv:1608.02942 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.118.051601 2017

  55. [55]

    Harlow and T

    D. Harlow and T. Numasawa, “Gauging spacetime inversions in quantum gravity,” [arXiv:2311.09978 [hep-th]]

    work page arXiv

  56. [56]

    Witten,Bras and kets in Euclidean path integrals,Beijing J

    E. Witten, “Bras and Kets in Euclidean Path Integrals,” [arXiv:2503.12771 [hep-th]]

    work page arXiv

  57. [57]

    CPT-Symmetric Universe

    L. Boyle, K. Finn and N. Turok, “CPT-Symmetric Universe,” Phys. Rev. Lett. 121, no.25, 251301 (2018) doi:10.1103/PhysRevLett.121.251301 [arXiv:1803.08928 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.121.251301 2018

  58. [58]

    Structure of Lefschetz thimbles in simple fermionic systems

    T. Kanazawa and Y. Tanizaki, “Structure of Lefschetz thimbles in simple fermionic systems,” JHEP 03, 044 (2015) doi:10.1007/JHEP03(2015)044 [arXiv:1412.2802 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2015)044 2015

  59. [59]

    Some remarks on Lefschetz thimbles and complex Langevin dynamics

    G. Aarts, L. Bongiovanni, E. Seiler and D. Sexty, “Some remarks on Lefschetz thim- bles and complex Langevin dynamics,” JHEP 10, 159 (2014) doi:10.1007/JHEP10(2014)159 [arXiv:1407.2090 [hep-lat]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep10(2014)159 2014

  60. [60]

    Real observers solving imaginary problems,

    J. Maldacena, “Real observers solving imaginary problems,” [arXiv:2412.14014 [hep-th]]

    work page arXiv

  61. [61]

    Physical instabilities and the phase of the Euclidean path integral,

    V. Ivo, J. Maldacena and Z. Sun, “Physical instabilities and the phase of the Euclidean path integral,” [arXiv:2504.00920 [hep-th]]

    work page arXiv

  62. [62]

    Testing the Black-Hole Area Law with GW150914,

    M. Isi, W. M. Farr, M. Giesler, M. A. Scheel and S. A. Teukolsky, “Testing the Black-Hole Area Law with GW150914,” Phys. Rev. Lett. 127, no.1, 011103 (2021) doi:10.1103/PhysRevLett.127.011103 [arXiv:2012.04486 [gr-qc]]

    work page doi:10.1103/physrevlett.127.011103 2021

  63. [63]

    Review of the no-boundary wave function,

    J. L. Lehners, “Review of the no-boundary wave function,” Phys. Rept. 1022 (2023), 1-82 doi:10.1016/j.physrep.2023.06.002 [arXiv:2303.08802 [hep-th]]

    work page doi:10.1016/j.physrep.2023.06.002 2023

  64. [64]

    Real Tunneling Geometries and the Large Scale Topology of the Universe,

    G. W. Gibbons and J. B. Hartle, “Real Tunneling Geometries and the Large Scale Topology of the Universe,” Phys. Rev. D 42 (1990), 2458-2468 doi:10.1103/PhysRevD.42.2458

    work page doi:10.1103/physrevd.42.2458 1990

  65. [65]

    The no boundary density matrix,

    V. Ivo, Y. Z. Li and J. Maldacena, “The no boundary density matrix,” JHEP 02, 124 (2025) doi:10.1007/JHEP02(2025)124 [arXiv:2409.14218 [hep-th]]

    work page doi:10.1007/jhep02(2025)124 2025

  66. [66]

    Maldacena,Comments on the no boundary wavefunction and slow roll inflation, arXiv:2403.10510

    J. Maldacena, “Comments on the no boundary wavefunction and slow roll inflation,” [arXiv:2403.10510 [hep-th]]

    work page arXiv

  67. [67]

    Witten,A Note On Complex Spacetime Metrics, arXiv:2111.06514

    E. Witten, “A Note On Complex Spacetime Metrics,” [arXiv:2111.06514 [hep-th]]

    work page arXiv

  68. [68]

    doi: 10.1093/qmath/haab027 arXiv: 2105.10161 [hep-th] 13, 18 References156

    M. Kontsevich and G. Segal, “Wick Rotation and the Positivity of Energy in Quantum Field Theory,” Quart. J. Math. Oxford Ser. 72 (2021) no.1-2, 673-699 doi:10.1093/qmath/haab027 [arXiv:2105.10161 [hep-th]]. 64

    work page doi:10.1093/qmath/haab027 2021

  69. [69]

    Complex actions in two-dimensional topology change

    J. Louko and R. D. Sorkin, “Complex actions in two-dimensional topology change,” Class. Quant. Grav. 14 (1997), 179-204 doi:10.1088/0264-9381/14/1/018 [arXiv:gr-qc/9511023 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/14/1/018 1997

  70. [70]

    Uses of complex metrics in cosmology,

    C. Jonas, J. L. Lehners and J. Quintin, “Uses of complex metrics in cosmology,” JHEP 08 (2022), 284 doi:10.1007/JHEP08(2022)284 [arXiv:2205.15332 [hep-th]]

    work page doi:10.1007/jhep08(2022)284 2022

  71. [71]

    Sur une mani` ere d’´ etendre le th´ eor` eme de la moyenne aux ´ equations diff´ erentielles du premier ordre.Mathematische Annalen

    Michel Petrovitch, M. Sur une mani` ere d’´ etendre le th´ eor` eme de la moyenne aux ´ equations diff´ erentielles du premier ordre.Mathematische Annalen. 54 pp. 417-436 (1901)

    work page 1901

  72. [72]

    Kontsevich-Segal Criterion in the No- Boundary State Constrains Inflation,

    T. Hertog, O. Janssen and J. Karlsson, “Kontsevich-Segal Criterion in the No- Boundary State Constrains Inflation,” Phys. Rev. Lett. 131 (2023) no.19, 191501 doi:10.1103/PhysRevLett.131.191501 [arXiv:2305.15440 [hep-th]]

    work page doi:10.1103/physrevlett.131.191501 2023

  73. [73]

    Kontsevich-Segal criterion in the no- boundary state constrains anisotropy,

    T. Hertog, O. Janssen and J. Karlsson, “Kontsevich-Segal criterion in the no- boundary state constrains anisotropy,” Phys. Rev. D 111 (2025) no.4, 046008 doi:10.1103/PhysRevD.111.046008 [arXiv:2408.02652 [hep-th]]

    work page doi:10.1103/physrevd.111.046008 2025

  74. [74]

    The gravitational index and allowable complex metrics,

    P. Benetti Genolini and S. Murthy, “The gravitational index and allowable complex metrics,” J. Phys. A 58, no.21, 215401 (2025) doi:10.1088/1751-8121/add7a7 [arXiv:2503.20866 [hep- th]]. 65

    work page doi:10.1088/1751-8121/add7a7 2025