Finite-cutoff geometry in double-scaled SYK: SYK deformations, branes and CMC-Krylov complexity

Sergio E. Aguilar-Gutierrez

REVIEW

Finite-cutoff geometry in double-scaled SYK: SYK deformations, branes and CMC-Krylov complexity

Sergio E. Aguilar-Gutierrez

High Energy Physics - Theory · Submitted 14 Oct 2026

Pith's verdict: An ambitious construction of finite-cutoff holography in DSSYK; the central mapping from SYK deformations to chord Hamiltonians is plausible but relies on an unproven assumption about non-polynomial functions. Deserves referee time.

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CitationSergio E. Aguilar-Gutierrez. "Finite-cutoff geometry in double-scaled SYK: SYK deformations, branes and CMC-Krylov complexity." arXiv:2610.20703 (2026). https://pith.science/paper/U2UGRJJF

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The paper does one thing well that is genuinely new: it constructs explicit SYK deformations (KM and TFD-based) whose double-scaling limit reproduces the T² and T²+Λ1 chord Hamiltonians in the ensemble average. That is a concrete step toward a microscopic foundation for finite-cutoff holography within a solvable model. The connection to sine-dilaton gravity at finite cutoff is worked out in detail, including a dictionary with CMC-Krylov complexity and ETW branes. The scope is impressive and the technical execution is careful throughout.

The soft spot is exactly where the stress-test lands. The central moment-matching (3.14) relies on extending the polynomial SYK/chord dictionary to non-polynomial functions like bPT and P^(η)_PT. The paper states this as an assumption (sentence b142) and gives a plausibility argument based on moment convergence and bounded spectra. But no rigorous proof is provided, and the η=-1 case involves a square root that requires spectral control the moment argument alone does not guarantee. If this extension fails, the microscopic derivation of the dual chord Hamiltonian collapses. The authors are upfront about it, which is good, but it is a load-bearing gap.

A secondary concern is that the CMC-Krylov complexity measure (4.13) is chosen to match the bulk CMC slice length, so the 'prediction' that it reproduces that length is built in. This is not a flaw if presented as a dictionary entry, but it limits the claim of explanation. The improved operator ordering (3.26) is also acknowledged as ad hoc (p.22), and the lack of numerical checks or code weakens reproducibility.

This paper is for specialists in DSSYK, JT/sine-dilaton gravity, and holographic complexity. It deserves a serious referee because the central idea—that finite-cutoff holography can be realized microscopically in SYK—is important and the construction is otherwise coherent. The referee should press on the non-polynomial extension and ask for either a rigorous justification or a clear statement of scope where the result holds without it.

Verdict: Conditional accept with major revision required on the foundational assumption.

Referee report

Major objections (5)

[§3.1, Eq. (3.8b) and (3.14b)] The extension of the polynomial dictionaries (2.13) and (2.21) to the non-polynomial functions bPT (3.8b) and P^{(η)}_{PT} (3.14b) is not rigorously justified. The authors state in sentence [b142] that it follows from moment convergence and bounded spectra, but for functions with branch points — especially the square root in the η=-1 (T²+Λ₁) case — weak convergence of spectral measures does not guarantee convergence of the functional calculus on the relevant spectral subspace where the radicand is non-negative. The paper itself acknowledges this as an assumption. Since the central matching of moments (3.14) and hence the entire microscopic duality depend on this extension, the paper must either provide a rigorous argument (e.g., strong resolvent convergence on the relevant subspace) or clearly state that the microscopic mapping is conjectural pending such a proof. The authors acknowledge this: “Hereandbelowweassumethatthedictionaries(2.13)and(2.21), derivedforpolynomials, extend to continuous functions ofˆHRX (respectively ˆHMQ) such as bPT; this follows from the convergence of the joint moments of(ˆHSYK, ˆHRX)together with the fact that the spectrum of ˆHRX is bounded uniformly inN, by the same argument as forP(η)PT below (3.14).”
[§3.4, Eq. (3.24)–(3.26)] The operator ordering choice (3.26) that incorporates the improvement term K₀−K∞ is described by the authors as 'too ad hoc' (p.22) and is not derived from a microscopic SYK deformation. While it reproduces the low-energy ordering of Ref. [54], the paper provides no symmetry or representation-theoretic argument for this choice in the chord Hamiltonian context (unlike the JT case where sl(2,R) motivates it). The chord Hamiltonian HPT′ is therefore a candidate rather than a derived dual. This weakens the claim that the finite-cutoff dictionary is uniquely determined. The authors acknowledge this as future work, but the current presentation should more clearly separate what is derived from what is postulated. The authors acknowledge this: “It would be interesting to investigate whether there is a symmetry argument in favor of the specific operator ordering choice in (3.26) in finite-cutoff sine dilaton gravity, in analogy with the JT case.”
[§4.2, Eq. (4.13)] The CMC-Krylov complexity C_r(t) is defined by explicitly choosing the cost function (4.14) so that its semiclassical evaluation reproduces the bulk CMC slice length relation (4.9). This is a construction rather than a derivation: the observable was selected to match the bulk relation. The paper should discuss more clearly whether this constitutes a 'boundary interpretation' or simply a transcription of bulk geometry into Krylov language. The claim that it is 'one member in the family of cost functions in [122]' does not by itself provide a microscopic motivation. A sharper separation between definition and prediction is needed.
[§4.1 and App. D] The paper derives analytic formulas for CMC slice lengths in the non-perturbative (ε=+1) branches (Table 2, Eqs. (4.4c), (4.4d), (4.10)), but later argues in §4.1 and App. D that these branches are 'ill-behaved at the quantum mechanical level' with a Hamiltonian unbounded from below and no sensible Hilbert space. The paper should reconcile these statements: are the ε=+1 length formulas only valid as classical approximations, or do they have a quantum interpretation? The current presentation treats them as valid geometric results without sufficiently flagging their status given the later quantum objections.
[§5.6] The existence of bound states at finite cutoff is assumed based on numerical results in App. J, but the paper states that 'there are no analytic formulas for this' for the Heun equation. The analytic condition (5.45) is given, but whether it has solutions for finite Φ_b is not rigorously demonstrated. The paper should clarify the reliability of the numerical evidence and whether the qualitative picture (eternal traversable wormhole) depends on non-perturbative effects that might be sensitive to the finite-cutoff corrections discussed in §5.5. The authors acknowledge this: “Below, we will assume that there exist discrete energy states even at finiteΦb based on the numerical results at finite cutoff in App.”

Minor objections (8)

[General] The notation is overloaded: ε labels both the perturbative/non-perturbative branch in the flow equation and a sign in the CMC slice functional (2.35). These should be distinguished (e.g., by using a different symbol for the CMC sign). See Table 2 and Fig. 3.
[§3.1, Eq. (3.8b)] The explicit functional form of bPT(x) involves a square root and a pole. The text should state more plainly that the operator is defined via the spectral theorem on the spectral subspace where the radicand is non-negative, which is a restriction of the Hilbert space.
[§3.4, Eq. (3.26)] The definition of K₀ and K∞ uses sin p̂_n, but the chord algebra (2.8) expresses e^{±i p̂_n} in terms of ladder operators. The expression is clear but the relationship to the standard chord representation could be given for completeness.
[§4.2, Eq. (4.14)] The cost function coefficients c_n involve a power series expansion in λ. The paper should note that the series converges for n such that the argument of the inverse sinh is less than π/2 (or the relevant domain), especially when r < -1 is considered.
[§5.3, Eq. (5.21)] The identification λC₀ = 2L is stated but the factor of 2 could be motivated by the one-sided geometry. A brief note on the canonical identification {2L, P/2} = {L,P} would help readability.
[App. C] The logarithmic approach to the JT partition function is interesting but the details are dense. A summary sentence connecting (C.7) to (3.28) would improve clarity.
[Fig. 4 and Fig. 5] The panels are small and the curves for different ν are hard to distinguish. Consider using a consistent color scheme and increasing font size for axis labels.
[Table 1] The entry 'Fake temperature sinθ/(2π)' should reference Sec. 2.2 or 3.1 where θ is defined as the energy parameter (3.10).

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The paper

The paper claims

This paper constructs microscopic deformations of the double-scaled SYK model whose ensemble-averaged theory describes wormhole states in finite-cutoff sine dilaton gravity, covering both AdS and dS regions.

The reading

The paper aims to establish a holographic duality between deformations of the double-scaled SYK model and finite-cutoff sine dilaton gravity. By constructing explicit SYK deformations of Kourkoulou-Maldacena and thermofield-double states, the author shows that their double-scaling limits reproduce lower-dimensional analogues of $T^{2}$ and $T^{2}$+Λ1 deformations. The ensemble-averaged theory describes wormhole states in a finite-cutoff sine dilaton gravity, with the low-energy regime corresponding to AdS geometry and the high-energy regime to dS geometry. The construction provides a microscopic origin for the finite-cutoff holographic dictionary and connects Krylov complexity to constant mean curvature slice lengths.

What carries the argument

The key object is the deformed chord Hamiltonian (3.1), which acts on the chord number basis and encodes the finite-cutoff geometry. The construction relies on a polynomial dictionary between SYK operators and chord operators, extended to non-polynomial functions via convergence of spectral measures. The low-energy limit reproduces the Pöschl–Teller Hamiltonian of finite-cutoff JT gravity, and the full theory matches the ADM Hamiltonian of sine dilaton gravity at finite cutoff.

Extended reading notes

Core claim

The central claim is that microscopic deformations of SYK states, in the double-scaling limit, yield chord Hamiltonians that are exactly the $T^{2}$ and $T^{2}$+Λ1 deformations (equations 3.1-3.3). The ensemble-averaged moments of these deformations match those of the chord Hamiltonian, establishing a duality with finite-cutoff sine dilaton gravity. This duality describes both perturbative and non-perturbative branches of the energy spectrum flow equation, with only the perturbative branch admitting physical quantum states. The construction also identifies a boundary interpretation for constant mean curvature slice lengths as a complexity measure in Krylov space.

Load-bearing premise

The paper assumes that the polynomial dictionary between SYK and chord operators extends to non-polynomial functions like the deformation operators, a step not rigorously proven.

Editorial extensions

If this is right

The microscopic construction provides a concrete boundary dual for finite-cutoff sine dilaton gravity, enabling precise tests of the holographic dictionary.
The CMC-Krylov complexity measure (4.13) offers a boundary interpretation of constant mean curvature slice lengths, linking complexity to geometry.
Including end-of-the-world branes leads to a boundary deformation that can describe either black holes or eternal traversable wormholes, depending on bound state existence.
The analysis of non-perturbative branches shows they are ill-defined quantum mechanically, providing constraints on possible bulk duals.
The framework can be extended to include matter, potentially allowing computation of finite-cutoff correlators.

Reading between the lines

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

If the duality is correct, the finite-cutoff holographic dictionary implies that the Lyapunov exponent measured in proper boundary time is given by the Tolman temperature, not the asymptotic temperature, which could be tested in SYK numerics.
The existence of a natural operator ordering (Friedrichs extension) in the low-energy theory suggests that the chord Hamiltonian at finite n automatically imposes a Dirichlet-like boundary condition at L=0, which may have implications for the spectrum of black hole microstates.
The CMC-Krylov complexity framework suggests that higher-dimensional holographic complexity may also be represented by cost functions in Krylov space, potentially extending the complexity=anything conjecture.
The construction of ETW branes at finite cutoff provides a toy model for studying the information paradox in a controlled setting.

Original abstract

We develop a finite-cutoff holographic description of the double-scaled SYK (DSSYK) model. We construct microscopic deformations of Kourkoulou-Maldacena and thermofield-double SYK states whose double-scaling limits reproduce a lower-dimensional analogue of $\text{T}\overline{\text{T}}$ and $\text{T}\overline{\text{T}}+\Lambda_2$ deformations, which we denote $T^2$ and $T^2+\Lambda_1$, respectively. The ensemble-averaged theory describes wormhole states in finite-cutoff sine dilaton gravity, and finite-cutoff anti-de Sitter and de Sitter geometries in the low- and high-energy regimes, respectively. The analysis includes both perturbative and non-perturbative solutions to the energy spectrum flow equation, and the regions behind and beyond the respective horizons. Only the perturbative branch admits physical quantum states. We study how to incorporate end-of-the-world (ETW) branes in the finite cutoff holographic dictionary. The ETW brane leads to an additional boundary theory deformation, which leads to black hole or eternal traversable wormhole geometries depending on the existence of normalizable bound states. We extend the finite cutoff length evaluations for constant mean curvature (CMC) slices, which we interpret as complexity in the boundary theory. We discuss possible extensions with matter. Our results provide a microscopic framework for studying the finite-cutoff boundaries, CMC slices and branes in the holographic dictionary of DSSYK.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.76 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.21 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We develop a finite-cutoff holographic description of the double-scaled SYK (DSSYK) model. We construct microscopic deformations of Kourkoulou-Maldacena and thermofield-double SYK states whose double-scaling limits reproduce a lower-dimensional analogue of $\text{T}\overline{\text{T}}$ and $\text{T}\overline{\text{T}}+\Lambda_2$ deformations, which we denote $T^2$ and $T^2+\Lambda_1$, respectively. The ensemble-averaged theory describes wormhole states in finite-cutoff sine dilaton gravity, and finite-cutoff anti-de Sitter and de Sitter geometries in the low- and high-energy regimes, respectively. The analysis includes both perturbative and non-perturbative solutions to the energy spectrum flow equation, and the regions behind and beyond the respective horizons. Only the perturbative branch admits physical quantum states. We study how to incorporate end-of-the-world (ETW) branes in the finite cutoff holographic dictionary. The ETW brane leads to an additional boundary theory deformation, which leads to black hole or eternal traversable wormhole geometries depending on the existence of normalizable bound states. We extend the finite cutoff length evaluations for constant mean curvature (CMC) slices, and interpret them as complexity in the boundary theory. We discuss possible extensions with matter. Our results provide a microscopic framework for studying the finite-cutoff boundaries, CMC slices and branes in the holographic dictionary of DSSYK.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We develop a finite-cutoff holographic description of the double-scaled SYK (DSSYK) model.	contribution 1.00	0.95	0.14	asserted 1.00
s1	We construct microscopic deformations of Kourkoulou-Maldacena and thermofield-double SYK states whose double-scaling limits reproduce a lower-dimensional analogue of $\text{T}\overline{\text{T}}$ and $\text{T}\overline{\text{T}}+\Lambda_2$ deformations, which we denote $T^2$ and $T^2+\Lambda_1$, respectively.	contribution 0.83	0.94	0.15	asserted 1.00
s2	The ensemble-averaged theory describes wormhole states in finite-cutoff sine dilaton gravity, and finite-cutoff anti-de Sitter and de Sitter geometries in the low- and high-energy regimes, respectively.	result 0.91	0.85	0.22	asserted 1.00
s3	The analysis includes both perturbative and non-perturbative solutions to the energy spectrum flow equation, and the regions behind and beyond the respective horizons.	contribution 0.56	0.90	0.07	asserted 1.00
s4	Only the perturbative branch admits physical quantum states.	result 0.97	0.84	0.12	asserted 1.00
s5	We study how to incorporate end-of-the-world (ETW) branes in the finite cutoff holographic dictionary.	contribution 1.00	0.86	0.05	asserted 0.88
s6	The ETW brane leads to an additional boundary theory deformation, which leads to black hole or eternal traversable wormhole geometries depending on the existence of normalizable bound states.	result 0.93	0.85	0.13	asserted 1.00
s7	We extend the finite cutoff length evaluations for constant mean curvature (CMC) slices, and interpret them as complexity in the boundary theory.	contribution 0.75	0.94	0.09	asserted 1.00
s8	We discuss possible extensions with matter.	future work 0.99	0.53	0.03	hedged possibility 0.83
s9	Our results provide a microscopic framework for studying the finite-cutoff boundaries, CMC slices and branes in the holographic dictionary of DSSYK.	interpretation 0.55	0.89	0.62	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

not examined

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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.** CMC slices (black curves) between Dirichlet boundaries (red curves) in an AdS2 black hole (light orange) dS2 space (light blue). η = +1 denotes the finite regions described by T 2 deformations in the boundary dual, and η = −1 for T 2 + Λ1. Further, ε = −1 corresponds to regions where 0 ≤ Φ ≤ Φb, with Φ = Φb being the dilaton at the boundary; while ε = +1 corresponds to those where Φ ≥ Φb, associated to the different branches in the energy spectrum. Lk denotes the length of CMC slices when ε = −1, L ≡ Lk=0; and L˜ k for ε = +1, while L˜ = L˜ k=0. We also arrived at a boundary interp

**Figure 2.** Structure of this work. We highlight the connection between the different topics, ranging from SYK deformations to ETW branes in sine dilaton gravity, and their location in the main text. The SYK model Originally introduced by [166] and shaped to its more modern form by [60–62], the SYK model is a system with 2N Majorana fermions in (0 + 1)-dimensions, and p-body all-to-all interactions. The theory is given by Hˆ SYK = ip/2X I JIψˆ I , (2.1) where I = i1, . . . , ip is a collective index, 1 ≤ i1 < · · · < ip ≤ 2N; and similarly ψˆ I ≡ ψˆ i1 . . . ψˆ ip is a string of Majorana

**Figure 3.** Asymptotic Dirichlet boundaries (solid red lines) in the AdS2 black hole (left) and in dS2 space (right) can be moved to different locations (red dashed lines), marked I-IV, in the Penrose diagram through a dual T 2 (η = +1) or T 2 + Λ1 (η = −1) deformation in the boundary theory. Each region represents a combination of the η and ε parameters in the energy spectrum (2.23), (2.24), where ε = −1 corresponds to the perturbative solutions and ε = +1 the non-perturbative ones. The regions are denoted: (I) η = +1, ε = −1; (II) η = −1, ε = −1, (III) η = −1, ε = +1; (IV) η = +1, ε = +1. CMC

**Figure 4.** Left: AdS2 black hole Penrose diagrams describing the low-energy bulk dual of the deformed DSSYK for (a) η = +1, ε = −1; (d) η = −1, ε = +1; (g) η = −1 ε = −1, (j) η = +1, ε = +1. Black curves represent geodesic lines. Middle and right: Comparison between the geodesic length L (middle column) and the CMC slice length Lk when k = 100 (right column) as a function of the rescaled time u for different values of ν (displayed in the figure). The convexity of Lk(t) increases with increasing |k|. – 30 –

**Figure 5.** Left: dS2 Penrose diagrams for (a) η = +1, ε = −1; (c) η = −1, ε = +1; (f) η = −1 ε = −1, (h) η = +1, ε = +1. Black curves denote geodesic lines. Middle and right: corresponding plots of the CMC slice length Lk for fixed ν = 0.6 (when η = −1) and ν = 1.1 (η = +1); for ε = −1 the relation (4.11) is the same as in AdS2 and the growth is linear at late times for every k, while for ε = +1 the growth is hyperfast when |k| < 1 and linear when |k| > 1 (4.10). The convexity of Lk(t) increases as |k| increases. – 31 –

**Figure 6.** Left: Exponential late-time growth of L|k|=1 in dS2 space for ε = +1 and (a) η = −1, (c) η = +1. Right: Corresponding Penrose diagram (b) for a fixed CMC slice with curvature k = −1, and (d) k = −1 on the top and k = +1 bottom Milne patches with a Dirichlet boundary. As special case, note that when |k| = 1, for η = ±1: ε = −1 : L|k|=1 = √ 2 sinh−1 √ 2 sinh L 2 , (4.12a) η = −1, ε = +1 : L|k|=1 =2 sin L 2 = 2p 1 − ν 2 cosh u , (4.12b) η = +1, ε = +1 : L|k|=1 =2 sin L 2 = 2p ν 2 − 1 sinh u , (4.12c) where the first line follows from (4.9), (4.11) and the last two from (4.10) (the…

**Figure 7.** Geodesic lengths in AdS2 for the ε = ±1, η = ±1 configurations with a timelike ETW brane, where the boundary anchoring point is translated symmetrically in the (b, c, d) cases. The dashed green line represents the continuation of the brane trajectory past the cutoff boundary at Φ = Φb (red solid curve). Note that the cases with ε = +1 do not interact with the brane. where we use Schwarzschild coordinates (4.1), and KAdS = − tanh σ0, with σ0 the signed geodesic distance from the geodesic with the same charge to the CMC brane, positive when the brane is displaced towards the cutoff…

**Figure 8.** (a) Modified Scarf II potential Um (5.26) for different values of the brane tension (displayed in the figure) at a finite cutoff parameter Φb = 2. When mJT < 0, local minima of Um emerge before the singularity at L = 0. (b) Semiclassical interpretation of the negative energy bound states EADM = −s 2 n in (5.26), where the Penrose diagram uses σ, T global coordinates (5.51). The ETW brane (blue line) is now static in global AdS2 space, and light rays (purple oscillating curves) emitted at the finite boundary (red line) bounce back. Dashed black lines represent the portion of…

**Figure 9.** Potential U(L) of the Schrödinger equation associated with the non-perturbative branches in the T 2 and T 2 + Λ1 deformed energy spectrum with the operator ordering prescription by [54] in (D.2) (orange solid curve), and the minimal case in (D.8). where A1/2 are constant coefficients, u ≡ 1 2+ q 1 4 − 4Φ2 b . The same argument as in the previous part applies. The behavior of the wavefunctions near the edges is f1 ∼ √ π Γ( 1 2 − u) Γ( 1−u 2 − s)Γ( 1−u 2 + s) ϵ+ 2 u + √ π Γ(u − 1 2 ) Γ( u 2 − s)Γ( u 2 + s) ϵ+ 2 1−u , (D.10a) f2 ∼ − √ π Γ( 1 2 − u) 2Γ(1 − u 2 − s)Γ(1 − u 2 + s)…

**Figure 10.** Accumulation surfaces in finite-cutoff holography. (a) Penrose diagram of the AdS2 black holes, with a CMC slice evolving in boundary time reaching the accumulation surface Σ in the late time regime. (b) Illustration in terms of the effective potential (E.17) for a fixed conserved constant c, where Φh = 3, Φb = 8, c = 10. The accumulation surface corresponds to the radial location where the potential reaches its maximum, such that a particle traveling from the cutoff at r = Φb can at most lose all its kinetic energy (i.e. it is a turning point (E.15)) when it reaches the Σ;…

**Figure 11.** Representation of the shockwave geometries in the AdS2 black hole and dS2 space, including the one-sided lengths (F.5) and Schwarzschild coordinates (4.1). where H(x) is the Heaviside step function, reflecting that the shockwave is sent from x − = 0, and α is called the shockwave parameter. F.1 Geodesic lengths For concreteness, let us consider embedding coordinates in AdS to describe the length of spacelike geodesics between its boundaries; an analogous treatment can be done in dS space. The geodesic length between spacelike points in AdS (and similarly dS) can be computed in…

**Figure 12.** Evolution of the semiclassical G4 (F.20) when ∆ is fixed, and (a) α is fixed, and (b) ν is fixed. Parameters in the plots: ∆ = 1, α = 1 (left) ν = 1.01 (right). The magnitude of the generating function of the length increases as we approach the horizon limit (left), and the time to reach its asymptotic value increases with α (right). tL −tR Φh k < 0 k > 0 M (a) tL −tR (b)

**Figure 13.** Different CMC slices Σ± anchored to the boundaries of an AdS2 black hole with black hole radius Φh (dashed black line). (a) Spacetime region M bounded by the CMC slices Σ± in (2.35), where the sign of k of the CMC slice is relative to the future-oriented normal vector to the slice. (b) We display the CMC slices for different boundary times when we include a shockwave along one of the horizons, which is represented in the discontinuous coordinates (F.24). Relative to the asymptotic boundary, a CMC slice with a given k moves further away as we increase the backreaction parameter α. d…

**Figure 14.** (a) Penrose diagram of the ETW brane (green solid curve) in the AdS black hole with a finite Dirichlet boundary (red curve). (b) Growth of the geodesic length (5.3) as a function of time for different values of ν (shown in the plot), where we set mJT = 1. The main difference with respect to the shockwave case in (F.47) is that here we describe a strongly backreacting particle at late times, instead of a conformal operator with dimension ∆ inserted in the HH preparation of state at t = 0. While the shockwave can be described as a limiting case of the operator insertion, which is…

**Figure 15.** Potential energy term U(L) (5.6) for different choices of parameters. In (a) we set Φb = 1.2, while in (b) Φb = 5 (the maximum of U is Φ 2 b at mJT = 0), and different values for the brane tension are displayed in the figure. As expected from (I.4), the maximum and minimum are more pronounced as we increase Φb or |mJT|, respectively. Asymptotic boundary limit At fixed x = 8Φbe −L, we can expand (I.1) in e −2L = x 2/(64Φ2 b ), which gives U = U∞(x) + U2(x) Φ2 b + O

**Figure 16.** (a) Discrete states sn of (5.26), determined numerically from (5.45), as a function of mJT for different Φb; the dashed lines are (5.47). (b) Difference between the phase of A+ in (5.34) and its asymptotic boundary limit, arg AABL + = arg Γ(4is)−arg Γ( 1 2 + 2is+ 2mJT)− 2s log(8Φb), given by the Whittaker function (5.41) (mod π), at mJT = −1. The difference is small for s ≪ Φb and becomes O(1) for s ∼ Φb, where the region near L = 0 is no longer classically forbidden. [4] L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with T T, JHEP 04 (2018) 010 [1611.03470]. [5

Figures reproduced from arXiv:2610.20703 by the authors.

References

Discussion

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