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Smoothing the Bouncing-Geodesic Singularity in AdS/CFT

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Mingxuan Liu, Shan-Ming Ruan

High Energy Physics - Theory · Submitted 2 Nov 2026 · 78 references

Pith's verdict: A careful, explicit mechanism paper whose main generic claim rests on an unproven meromorphic-dominance assumption; the Liouville example is real but does not test that assumption.

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CitationMingxuan Liu, Shan-Ming Ruan. "Smoothing the Bouncing-Geodesic Singularity in AdS/CFT." arXiv:2611.03347 (2026). https://pith.science/paper/HIRCW7GT
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The paper

The paper claims

At finite central charge, deformed quasinormal-mode trajectories replace the sharp complex-time singularities of bouncing geodesics with smooth profiles of width $\beta/\sqrt{c}$, while the OPE- and KMS-protected singularities survive.

The reading

In holographic CFTs the classical black hole singularity leaves a sharp imprint on the analytically continued thermal correlator: an infinite lattice of power-law singularities whose complex-time positions are fixed by the bouncing-geodesic time $T_{\rm crt}$ and by the asymptotically linear quasinormal-mode (QNM) spectrum. This paper argues that those additional singularities are a strict semiclassical artifact and are smoothed out at finite central charge $c$ (finite $G_N$): a quadratic bending of the QNM trajectories, $\omega_n = a_0 + a_1 n + (a_2/c)n^2$, produces a Gaussian suppression $\exp(-\kappa\omega^2)$ in frequency space that replaces each pole $1/(T-T_*)^{\nu+1}$ with a smooth bump of width $O(\beta/\sqrt{c})$. The smoothing is selective: the seed singularities fixed by the boundary OPE and, for thermal correlators, by the exact KMS condition are preserved. The paper makes the mechanism explicit in Liouville CFT, where the exact DOZZ three-point coefficient places the poles on a parabola whose curvature vanishes as $c\to\infty$, recovering the linear BTZ QNM spectrum, and it identifies a second, independent Gaussian smoothing arising from energy fluctuations in the canonical spectral decomposition. If correct, the finite-$c$ correlator cleanly separates boundary-fixed analytic structure from interior-geometry data, turning singularity resolution into a question about $1/c$ corrections to QNM data.

What carries the argument

The load-bearing object is the thermal product formula of ref. [28]: an order-one Hadamard product over the QNM pole quartets that reconstructs the meromorphic two-sided correlator $G_{LR}(\omega,k)$ from its pole data. Semiclassically, an asymptotically linear family $\omega_n = a_0 + a_1 n$ turns that product into Gamma functions whose Stirling and reflection asymptotics factor into exponentials that translate the OPE seed singularity through the complex-time lattice $T_{n_+,n_-} = i\beta/2 + n_+ v_+ + n_- v_-$, with $v_\pm = \mp|T_{\rm crt}| + i\beta/2$; this translation is what encodes the bouncing geodesic. The finite-$c$ deformation is introduced as a polynomial QNM trajectory with a leading quadratic correction, $\omega_n = a_0 + a_1 n + (a_2/c)n^2$; inverting to $z_{\rm sc}(\omega)$ and feeding the Stokes factor $\exp(2\pi i\, m\, z_{\rm sc}(\omega))$ converts the quadratic term into the Gaussian damping $\exp(-\kappa_m\omega^2)$, $\kappa_m = 2\pi i m a_2/(a_1^3 c)$, and the Fourier integral into a parabolic-cylinder (erfc) profile of width $O(\beta/\sqrt{c})$. Two further components carry the argument: the exact Liouville DOZZ example, whose Upsilon-function zeros put the poles on a parabola and provide the pole-zero cancellations that remove the $\omega\log\omega$ and $\sqrt{\omega}$ terms the OPE forbids; and the ETH-based spectral decomposition, whose microcanonical saddle with $S''(E_\beta) = -\beta^2/C_V$ produces the universal Gaussian kernel $\exp(-\omega^2/4(\Delta E)^2)$.

Extended reading notes

Core claim

The central claim is that the complex-time singularities associated with bouncing geodesics exist only in the strict $c\to\infty$ limit: they are generated by the asymptotically linear QNM spectrum through the thermal (Hadamard) product formula, and any nonlinear finite-$c$ deformation of the QNM trajectories modifies the Stokes sectors of that product, dressing the large-frequency exponential $e^{i\omega T_*}$ with a damping factor. For the leading quadratic deformation the dressing is a Gaussian, $\exp(-\kappa_m\omega^2)$ with $\kappa_m\sim\beta^2/c$, and the would-be singularity becomes a smooth, entire profile of width $O(\beta/\sqrt{c})$ (eqs. (3.36)-(3.43)); for spectra whose inverse trajectory is sublinear or sufficiently superlinear in frequency, the displaced singularity is likewise absent. The mechanism is selective rather than destructive: the OPE singularity at the fundamental KMS seed and its thermal images are protected by the boundary OPE plus exact KMS, and only the additional bouncing lattice is smoothed (section 3.4). Liouville CFT supplies an explicit finite-$c$ realization: the analytically continued DOZZ coefficient has poles on the parabola $\mathrm{Re}\,\omega = -(\mathrm{Im}\,\omega)^2/(8\kappa^2)$, a quadratic deformation of the BTZ QNM line, with zeros on the negative real axis whose combined Hadamard data satisfy the OPE's ultraviolet constraints. A complementary mechanism follows from the thermal spectral decomposition: the concavity of the black hole entropy, $S''(E_\beta) = -\beta^2/C_V$, yields a Gaussian factor $\exp(-\omega^2/4(\Delta E)^2)$ with $\Delta E\sim 1/\sqrt{G_N}$, which convolves every singularity sector into a smooth profile of time width $O(\sqrt{G_N})$ (eqs. (5.20)-(5.21)).

Load-bearing premise

The central claim collapses if, in the intermediate frequency window $1 \ll \beta|\omega| \ll c$, the exact finite-$c$ correlator is not dominated by a meromorphic component whose QNM poles follow a polynomial trajectory with a leading quadratic deformation, because then the Gaussian suppression kernel and the predicted $\beta/\sqrt{c}$ smoothing profile would not be the generic mechanism operating in real holographic CFTs.

Editorial extensions

If this is right

  • The semiclassical singularity lattice $T = \pm m|T_{\rm crt}| + i\beta(2p+m+1)/2$ is resolved at finite $c$ into smooth bumps of width $O(\beta/\sqrt{c})$, so the black hole interior leaves a finite, calculable signature in the correlator instead of a divergence.
  • The genuine OPE singularity and its exact KMS images survive the smoothing, so finite-$c$ effects erase only the bouncing-geodesic lattice, separating boundary-fixed analytic structure from interior-geometry data.
  • The Liouville DOZZ example shows the mechanism concretely: exact poles lie on a parabola at finite $c$ and reproduce the linear BTZ QNM spectrum only as $c\to\infty$, demonstrating that quadratic QNM bending is realized, not merely assumed, in a solvable CFT.
  • At finite $c$ zero families (absent in the semiclassical product formula) are required to cancel $\omega\log\omega$ and $\sqrt{\omega}$ contributions forbidden by the OPE, opening the possibility of bootstrapping the finite-$c$ pole-zero data from UV constraints plus thermal analyticity.
  • The spectral-decomposition mechanism adds a sector-independent Gaussian damping $\exp(-\omega^2/4(\Delta E)^2)$ with $\Delta E\sim 1/\sqrt{G_N}$, so together the two mechanisms supply complementary quantum resolutions of the singularity's holographic imprint.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The specific Gaussian shape is an artifact of the quadratic truncation: any nonlinearity in the inverse QNM trajectory eliminates the displaced singularity, so the paper's message is best read as 'the bouncing-geodesic singularity is unstable to finite-$c$ effects', with the Gaussian profile being the leading, not the universal, form of the smoothing.
  • The protected-seed principle suggests a general diagnostic: any feature of the analytically continued correlator not pinned by the OPE plus (for thermal states) KMS is semiclassically unstable, so the bouncing lattice is best understood as Stokes data of a QNM product rather than exact spectral data of the CFT.
  • The same bending mechanism should shift other QNM-built observables, notably the late-time linear growth of the black hole interior and complexity-like quantities, since the lowest QNM controls that growth rate; $1/c$ corrections to the QNM trajectory would then produce a subleading shift, a testable effect in exactly solvable cases like JT gravity.
  • A natural extension the paper leaves implicit is to include the $\omega$-dependence of the ETH spectral function $F_O(E,\omega)$ instead of freezing it at the saddle, which would couple the two smoothing mechanisms and could produce a regime where the total suppression is the product of two Gaussian kernels with widths $\beta/\sqrt{c}$ and $\sqrt{G_N}$.
Original abstract

Black hole singularities remain a central open problem in quantum gravity. In the AdS/CFT correspondence, bulk spacelike extremal surfaces can probe the black hole interior, and information about the near-singularity geometry is encoded in analytically continued boundary correlators. As the bouncing geodesic approaches the black hole singularity, it becomes nearly null and gives rise to characteristic divergences at complex times. Importantly, these complex-time singularities emerge only in the strict semiclassical limit ($G_N\to 0$ or $c\to\infty$) and are governed by the asymptotically linear quasinormal mode (QNM) spectrum. At finite $c$, nonlinear deformations of the QNM trajectories produce suppression kernel that smooths out the complex-time singularities. Using Liouville CFT as an analytically solvable example, we explicitly demonstrate the finite-$c$ deformation of the QNM spectrum and its associated pole-zero structure. We also identify a complementary Gaussian smoothing mechanism arising from finite-energy fluctuations in the thermal spectral decomposition. These results show how finite $G_N$ quantum corrections modify the holographic imprints of the classical black hole singularity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-04: 1 request, 6,316 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.23 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Black hole singularities remain a central open problem in quantum gravity. In the AdS/CFT correspondence, bulk spacelike extremal surfaces can probe the black hole interior, and information about the near-singularity geometry is encoded in analytically continued boundary correlators. As the bouncing geodesic approaches the black hole singularity, it becomes nearly null and gives rise to characteristic divergences at complex times. Importantly, these complex-time singularities emerge only in the strict semiclassical limit ($G_N\to 0$ or $c\to\infty$) and are governed by the asymptotically linear quasinormal mode (QNM) spectrum. At finite $c$, nonlinear deformations of the QNM trajectories produce suppression kernel that smooths out the complex-time singularities. Using Liouville CFT as an analytically solvable example, we explicitly demonstrate the finite-$c$ deformation of the QNM spectrum and its associated pole-zero structure. We also identify a complementary Gaussian smoothing mechanism arising from finite-energy fluctuations in the thermal spectral decomposition. These results show how finite $G_N$ quantum corrections modify the holographic imprints of the classical black hole singularity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Black hole singularities remain a central open problem in quantum gravity. background 1.00 0.03 0.02 asserted 1.00
s1 In the AdS/CFT correspondence, bulk spacelike extremal surfaces can probe the black hole interior, and information about the near-singularity geometry is encoded in analytically continued boundary correlators. background 0.97 0.09 0.04 asserted 1.00
s2 As the bouncing geodesic approaches the black hole singularity, it becomes nearly null and gives rise to characteristic divergences at complex times. result 0.52 0.27 0.04 asserted 1.00
s3 Importantly, these complex-time singularities emerge only in the strict semiclassical limit ($G_N\to 0$ or $c\to\infty$) and are governed by the asymptotically linear quasinormal mode (QNM) spectrum. result 0.64 0.35 0.09 asserted 1.00
s4 At finite $c$, nonlinear deformations of the QNM trajectories produce suppression kernel that smooths out the complex-time singularities. result 0.96 0.69 0.51 asserted 1.00
s5 Using Liouville CFT as an analytically solvable example, we explicitly demonstrate the finite-$c$ deformation of the QNM spectrum and its associated pole-zero structure. contribution 0.71 0.96 0.10 asserted 1.00
s6 We also identify a complementary Gaussian smoothing mechanism arising from finite-energy fluctuations in the thermal spectral decomposition. result 0.80 0.90 0.08 asserted 1.00
s7 These results show how finite $G_N$ quantum corrections modify the holographic imprints of the classical black hole singularity. interpretation 0.98 0.88 0.82 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Bouncing geodesics in the two-sided AdS black hole. As the conserved geodesic energy is increased, the turning point of a real radial spacelike geodesic moves toward the curvature singularity. In the limiting configuration the geodesic becomes almost null and approaches a trajectory that bounces off the singularity. In terms of the two-sided boundary time, the corresponding critical Lorentzian time is denoted by Tcrt. corresponding bulk propagator. In the large-mass, or equivalently large-∆, limit, the bulk propagator admits a Wentzel–Kramers–Brillouin (WKB) expansion whose leading…
Figure 2. Analytic continuation as a probe of the black hole interior. Left: In the maximally extended AdS black-hole geometry, the different exterior and interior regions can be represented using complexified Schwarzschild time t, with horizon crossings accompanied by characteristic imaginary-time shifts. Analytic continuation therefore provides a natural way to move inside the black hole interior. Right: The same structure is reflected in the dual boundary theory. The two-sided thermal correlator GLR(T, k) is defined in the physical Kubo–Martin–Schwinger (KMS) strip, while its analytic…
Figure 3. Analytic structure of the two-sided thermal correlator at fixed spatial separation. The central region denotes the principal KMS strip −β/2 < Im T < β/2. Adjacent KMS strips are analytically connected through the spacelike intervals −|x| < Re T < |x| on their shared boundaries, where microcausality makes the two Wightman boundary values coincide. Repeating this gluing across successive thermal boundaries produces a connected principal thermal sheet, while analytic continuation around the lightcone/OPE branch points at T (±,n) OPE leads instead to secondary Riemann sheets. precisely t
Figure 4. Analytic continuation and sheet structure of GLR(T, x). Red dots denote OPE singularities, red wavy lines indicate branch cuts, and red crosses mark possible additional singularities. The upper panels illustrate the first three sheets, with the blue and green curves representing two distinct analytic continuation paths. The cyan regions are continuously connected along the green path, as illustrated in the lower-left panel with a different choice of branch cuts. The lower-right panel shows the limit |x| → 0, where pairs of OPE singularities merge and the corresponding continuation…
Figure 5. Two complementary OPE channels of the HHLL correlator. In the (HH)(LL) channel, the semiclassical heavy-light limit is governed by the vacuum block and exhibits emergent thermal images, including the forbidden singularities. In the crossed (HL)(HL) channel, the exchanged states are organized by the transition energy ω = Ea − EH, whose continuation is governed by the QNM spectrum in the semiclassical black hole regime. where GE(τ, x) denotes the corresponding two-point function in an energy eigenstate. At large entropy, the canonical integral is dominated by the saddle Eβ satisfying…
Figure 6. Left: linear QNM spectrum, i.e., poles of GLR(ω, k) in the complex ω-plane; Right: complex-time singularities of the analytically continued thermal correlator GLR(T, k) in the complex T-plane. transparent form [37] v+ = −Tcrt + i β 2 , v− = Tcrt + i β 2 . (2.90) Their real parts are thus fixed directly by the characteristic bouncing time Tcrt, while their imaginary parts are fixed by the thermal scale β. Using the fundamental seed singularity Tseed = iβ/2, the entire upper-half-plane family can then be written as Tn+,n− = iβ 2 + n+v+ + n−v− = (n+ − n−)|Tcrt| + i β 2 (n+ +
Figure 7. Real part of the Gaussian-regularized integral I0(∆Tm, κm). This is the simplest example illustrating the smoothing of the semiclassical complex-time singularity for Re κm > 0, with |κm| ∼ β 2/c. which remains finite for any nonzero κm satisfying the convergence condition. In contrast, the strict semiclassical limit κm → 0 recovers the original power-law singularity, Iν(∆Tm, 0) = lim ϵ→0+ Γ(ν + 1) (ϵ + i∆Tm) ν+1 ∼ 1 (T − Tm) ν+1 . (3.42) The Gaussian deformation therefore replaces the sharp semiclassical singularity with a regular profile characterized by the width |∆Tm| ∼ p
Figure 8. Comparison of the exact Fourier transform Fc (solid lines) and its quadratic approximation F EFT c (dashed lines) for different values of c, with τ∗ = 1. The left and right panels show the profiles along the imaginary-time and real-time directions, respectively, near the semiclassical displaced singularity Tm = Tseed + iτ∗. The quadratic approximation evaluated along the complex contour accurately reproduces the leading smooth profile in both directions. As c increases, the profile becomes increasingly localized, with a characteristic width O(c −1/2 ) and amplitude O(c 1/2 ), rec
Figure 9. Schematic complex-time singularity structure at finite c and fixed spatial momentum. Left: for the HHLL correlator GH(t, k), the red point at t = 0 denotes the genuine light-light OPE singularity. The additional semiclassical thermal images, indicated by red halos, are smoothed out by finite-c corrections. Right: for the thermal two-sided correlator GLR(T, k), the red points denote the protected OPE/KMS singularities at T = T (±,k) seed , while the gray halos represent the smoothing of additional bouncing-geodesic singularities generated by the deformation of linear QNMs. with a nonv
Figure 10. Pole trajectories ω pole m=0,n of the analytically continued Liouville DOZZ coefficient in the complex ω-plane for different central charges with ∆ = 3 and βH = 2π. The discrete points represent the poles ω pole m=0,n, while the colored curves illustrate their exact parabolic trajectories. As c increases, the parabolas approach the imaginary-frequency axis, recovering the semiclassical BTZ QNM spectrum indicated by the black dotted line. More precisely, we use the effective BTZ temperature TH defined in eq. (2.29), κ = r ∆H 2 − c − 1 24 = s 1 24 + π 2c 6β 2 H , (4.2
Figure 11. Pole–zero structure of the analytically continued Liouville DOZZ coefficient in the complex-frequency plane at c = 100. The four pole branches, labeled by two integers (m, n), follow the parabolic trajectory (4.34), while the zeros lie on the negative real-frequency axis, beginning at ω = −2κ 2 . The leading finite-c correction contains a quadratic real contribution, which bends the semiclassical BTZ QNM line into the complex-frequency plane. In fact, the quadratic deformation of the QNM spectrum can be seen directly from the exact pole trajectory. Since κ, b, and α1 are all real in
Figure 12. Penrose diagrams and geodesic configurations. Left: BTZ. The red limiting null configuration is not approached by spacelike geodesics at finite boundary time. The blue curve is the spacelike geodesic at T = 0. Right: a higher-dimensional black hole. A family of blue spacelike geodesics approaches the bouncing geodesic presented by the red curve. not approach a bouncing null configuration at finite boundary time. Formally taking d = 2 in the critical bouncing time (2.43) gives Tcrt = − β 2 cot π d d=2 = 0 . (6.1) Naively, as shown in figure 12, this critical value can be understoo
Figure 13. Locations of complex-time singularities of the analytically continued thermal correlator GLR(T, k) in the complex T-plane. which are the lightcone/OPE singularities and their thermal images. Analytic continuation around these singularities changes the monodromy of the chiral factors without introducing additional singularity locations on secondary sheets. By contrast, in higher-dimensional black holes, analytic continuation can access distinct geodesic branches, including bouncing geodesics that approach the spacelike singularity. These trajectories give rise to additional…
Figure 14. Pole distributions of GLR(ω, k). Poles in the lower half of the complex ω-plane correspond to QNMs. with the local behavior GBTZ LR (T, k) ∼ 1 (T −TN ) 2∆−1 . As explained in Appendix A, these singularities arise from the thermal images and collisions of the chiral singularities. By contrast, higher-dimensional black holes exhibit semi-classical linear QNM trajectories with a complex slope. As derived in eq. (2.90), the two shifts are v± = ±|Tcrt| + iβ/2. This gives rise to additional complex-time singularities displaced from the imaginary axis by the characteristic bouncing time
Figure 15. Toy-model coupling of QNMs |n⟩ to perturbative graviton states |q⟩ and virtual black-hole states |p⟩. The central black disk represents the fixed background |H⟩. state sum. Because the QNM frequencies ωn are complex, the effective Hamiltonian is generally non-Hermitian: Heff =
Figure 16. Two contour-pinching mechanisms in the complex x plane. Blue curves denote integration cycles, crosses denote poles, and black dots denote branch points; dashed lines indicate branch cuts. (a,b) The cycle γ encloses z but excludes −z. Their collision at z = 0 pinches the cycle and produces a 1/z singularity. (c,d) The cycle γ01 surrounds the cut [0, 1] but excludes the third branch point z. As z → 1, the external branch point collides with the enclosed endpoint and pinches the cycle, producing a logarithmic singularity. Red circles mark the pinch locations. In more spatial…
Figure 17. Illustration of singularity collision in x plane for the BTZ case. Here red dots denote singularities of GLR(T, x) in x plane, according to A.7, and red curly lines denote cuts between them. Light blue semicircle is the Fourier contour for the x to k Fourier integral A.1 in the case of k < 0. It decomposes into blue cut cycles. When T varies, these singularities will move in the x plane, and collide with each other at T = TN = i βN 2 . Here we show the configuration of N = 0, 1, 2. In the case N = 0, the contour is actually not pinched and one can deform the contour (e.g., back to…
Figure 18. Picard–Lefschetz analysis of the quartic UV model (B.2) and its quadratic EFT approximation, with taking c = 64 and ξ = i(T − Tm) √ c = 0.3 + 0.8i. Panels (a) and (b) correspond to α = 1 + 2i and α = 1, respectively. The black horizontal lines represent the original physical integration contours Cphys, while the blue and orange solid curves denote the UV Lefschetz thimbles CUV and the quadratic EFT contours CEFT, respectively. The dashed arcs near the outer boundaries indicate their corresponding asymptotic convergence sectors. The blue dots and orange circles mark the local UV and…
Figure 19. Contours for the exact square-root transform and its quadratic, cubic, and quartic approximations, shown in the three columns. Here τ∗ = 1, c = 64, and s = −1 + 0.3/ √ c on the upper continuation. The upper row shows frequencies of order c, while the lower row resolves the smoothing region at frequencies of order √ c. Blue solid and orange dashed curves join the endpoint to the local saddle and then follow a descending arm of the exact and truncated exponents, respectively. Blue circles and orange stars mark the local saddles, while purple stars mark additional saddles of the…

Figures reproduced from arXiv:2611.03347 by the authors.

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