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Three dimensional Janus and time-dependent black holes

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Dongsu Bak, Michael Gutperle, Shinji Hirano

High Energy Physics - Theory · Submitted 11 Jan 2007 · 24 references

Pith's verdict: A clean, modest extension of the Janus program to AdS3 plus a genuinely useful family of exact time-dependent black holes; the soft spots are presentational, not structural.

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arXiv IDhep-th/0701108 (version 2)
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CitationDongsu Bak, Michael Gutperle, Shinji Hirano. "Three dimensional Janus and time-dependent black holes." arXiv:hep-th/0701108 (2007). https://pith.science/paper/ZBTV322R
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The paper

The paper claims

The Janus deformation of AdS$_3$ is shown to fit inside ten-dimensional type IIB supergravity, with a two-dimensional interface CFT as its proposed dual and a family of time-dependent black holes as a by-product.

The reading

The Janus deformation takes a maximally symmetric anti-de Sitter space and splits it along a domain wall, letting a scalar field (here the dilaton) approach one constant value on one side and a different constant value on the other. It was known in five dimensions; this paper asks whether the same construction exists for $AdS_3$, the geometry dual to a two-dimensional conformal field theory built from D1- and D5-branes. The authors write an ansatz in which $AdS_3$ is sliced by $AdS_2$ leaves, the three-sphere and the internal four-manifold are warped by powers of the dilaton, and a three-form flux threads the whole thing, then show that the ten-dimensional type IIB equations of motion reduce to a single first-order equation for one scale factor. Solving it in closed form (Jacobi elliptic functions, or elementary functions in a better-chosen coordinate) gives an exact dilaton profile that runs between two boundary values, so the dual field theory is an interface CFT: a two-dimensional theory with a marginal coupling switched on in one half-plane only. The same machinery is then repurposed, by replacing the $AdS_3$ slices with BTZ or with a cosmological slicing, to produce black holes whose horizon area grows with time and whose two asymptotic boundaries carry different couplings. If the construction holds, it gives one of the few nonsupersymmetric deformations of holography where both sides are computable in closed form, in the lowest dimension where the dual theory is a tractable orbifold CFT.

What carries the argument

The load-bearing device is the reduction of the ten-dimensional Janus ansatz to three-dimensional Einstein gravity coupled to a scalar with a negative cosmological constant, $I = (1/16pi G_3)int d^3x sqrt{g}(R_3 - g^{ab}partial_a varphi partial_b varphi + 2)$, followed by the observation that the resulting equation for the scale factor is the energy-conservation law of a particle in the potential $V(f) = -4f(f^2 - f + gamma^2/2)$. Integrability of that first-order equation is what makes the whole construction exact: the solution is written as $f = alpha_+^2 / mathrm{sn}^2(alpha_+(mu+mu_0),k)$ with $k = alpha_-/alpha_+$, and the dilaton follows from $varphi' = gamma/f^{1/2}$. The second device is a slicing substitution: because only $R_{ab} = -2g_{ab}$ is used, the $AdS_3$ factor can be swapped for any three-metric with the same Ricci tensor, which is how the BTZ black hole and the higher-dimensional cosmological black holes are generated almost for free. The third device is the identification of the operator dual to the dilaton in the D1-D5 orbifold CFT, taken to be the marginal operator $O_varphi = -(1/4pi)sum_{i,a} :partial X^{i,a} barpartial X^{i,a}:$.

Extended reading notes

Core claim

The paper's central claim is that the three-dimensional Janus geometry embeds into type IIB supergravity on $AdS_3$ times $S^3$ times $M_4$ (with $M_4$ either $T^4$ or $K3$). The ansatz is $ds^2 = e^{2varphi}(dmu^2 + ds^2_{AdS_2}) + e^{2varphi}ds^2_{S^3} + e^{-2varphi}ds^2_4$ with a dilaton depending only on the slicing coordinate and a three-form flux proportional to the $AdS_2$ and $S^3$ volume forms. The ten-dimensional equations collapse to $f'^2 = 4f^3 - 4f^2 + 2gamma^2 f$, the zero-energy condition for a particle moving in an effective cubic potential, which is integrable. For $gamma^2 < 1/2$ the coordinate runs over a finite interval and the dilaton takes two distinct constant values $varphi_pm$ at the two ends, so the boundary is two half-planes glued at a line and the dual is an interface CFT with a marginal deformation active in one half. The authors further show that the same slicing trick, applied with a BTZ metric or with $-dtau^2 + cos^2tau, ds^2_Sigma$ in place of $AdS_3$, yields exact black-hole solutions whose horizon area grows monotonically in time and whose two disconnected boundaries carry couplings $e^{varphi_+}$ and $e^{varphi_-}$.

Load-bearing premise

The identification of the operator dual to the six-dimensional dilaton as the normal-ordered marginal operator $-(1/4pi)sum :partial X barpartial X:$ is, in the authors' own words, a natural guess, and the whole interface-CFT reading rests on it.

Editorial extensions

If this is right

  • The Janus deformation of the $AdS_3/CFT_2$ correspondence is a concrete, exactly solvable example of an interface CFT, in the lowest dimension where the dual orbifold CFT is under perturbative control.
  • The one-point function of the dual marginal operator is predicted at leading order in the deformation, $langle O_varphi rangle = -gamma Q_1 Q_5, epsilon(x)/(4pi x^2)$, giving a number that conformal perturbation theory on the field-theory side can be checked against.
  • The deformed energy-momentum tensor vanishes, $langle T_{ab}rangle = 0$, so the deformation does not change the central charge even though the asymptotic $AdS_3$ radius appears to differ on the two sides; the dilaton factors cancel against the behaviour of the three-dimensional Newton constant.
  • Time-dependent black holes with a running dilaton exist in three and higher dimensions, with horizon area growing monotonically from a minimum to $2pi r_0$ (three dimensions) or $V_Sigma$ (higher dimensions), and reduce to the static $M=0$, $k=-1$ topological black hole when the deformation is switched off.
  • The three- and five-dimensional members of this family embed into ten-dimensional type IIB supergravity, so their duals are ordinary gauge-theory-like CFTs rather than lower-dimensional toy models.

Reading between the lines

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

  • Because the boundary metric of the higher-dimensional black holes can be chosen either as a cosmological $cos^2tau$ slicing or as flat $R times Sigma$, the same bulk solution can be read as a dual description of either a big-bang/big-crunch cosmology or a thermalization process starting far from equilibrium; the authors mention the second reading but do not compute anything with it.
  • The two-boundary structure of the Janus black hole resembles a traversable-wormhole geometry with unequal couplings on the two ends, which suggests asking whether the dual state is entangled across the two boundaries and whether the growth of horizon area is dual to entanglement growth.
  • Since the SL(2,R) symmetry of the classical supergravity can rotate the dilaton into an axion, the natural next case has both the gauge coupling and the theta angle jumping at the interface; the authors note this but do not write the rotated solution.
  • The construction only uses $R_{ab} = -2g_{ab}$ for the sliced factor, so any three-dimensional Einstein metric with negative cosmological constant should work as a seed; this suggests a broader family of Janus black holes with rotation or with different horizon topologies than the ones written down here.
Original abstract

We show that the three dimensional Janus geometry can be embedded into the type IIB supergravity and discuss its dual CFT description. We also find exact solutions of time dependent black holes with a nontrivial dilaton field in three and higher dimensions as an application of the Janus construction.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,262 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We show that the three dimensional Janus geometry can be embedded into the type IIB supergravity and discuss its dual CFT description. We also find exact solutions of time dependent black holes with a nontrivial dilaton field in three and higher dimensions as an application of the Janus construction.

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 We show that the three dimensional Janus geometry can be embedded into the type IIB supergravity and discuss its dual CFT description. contribution 0.94 0.93 0.54 asserted 1.00
s1 We also find exact solutions of time dependent black holes with a nontrivial dilaton field in three and higher dimensions as an application of the Janus construction. result 0.80 0.96 0.26 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 v2
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. Penrose diagram for the time dependent black hole with two couplings. The τ (∈ [−π/2, π/2]) coordinate runs vertically upward and µ (∈ [−µ0, µ0]) to the right horizontally 5. Time dependent topological black holes in higher dimensions In this section, we would like to generalize the black hole solution of previous section to higher dimensions. The construction is again fairly straightforward. We begin with a dilaton Einstein gravity described by I = 1 16πGd Z √ gd  Rd − g ab d ∂aφ∂bφ + (d − 1)(d − 2) . (5.1) with d ≥ 3. As we just described in the previous sections, any sol

Figures reproduced from arXiv:hep-th/0701108 by the authors.

References
How later papers used it

67 later papers cite it

Evidence: substantial. 51 of the citing papers share no author with it, and 9 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 128 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 of 4 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

64 arXiv papers cite this in 121 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 64 sentences without a confident label

Builds on it 4 papers
Uses its method or tool 2 papers
  • First by choosing the conformal factor $h^2= cos^2τ/f(μ)$, the boundary metric for the CFT is given by [ref] ds^2_B = -dt^2 + ds^2_Σ .

    Time dependent black holes and thermal equilibration · Correspondence · 0.82

  • In the presence of the exactly marginal perturbation, we can employ the Janus solution [ref] as the gravity dual to $ρ^(2)1|2_A$.

    SVD Entanglement Entropy · Holographic SVD entropy from Janus Solution · 0.83

Uses its data or values 3 papers
  • In addition, the area of the horizon is given by [ref] 𝒜 (τ) = 2 πL r_h α_+ sin≤π/2 - τ/sn ≤α_+ ≤π/2 - τ, k t_B →∞⟶ 2 πL r_h .

    Volume complexity for Janus $\mathrm{AdS}_3$ geometries · Time dependence of the volume · 0.93

  • In the explicit example of 3D Janus solution given in [ref] , the metric reads e^2h(ρ)=1/2+1-2γ^2/2cosh(2ρ), where $γ$ parameterizes the Janus deformation.

    Aspects of Pseudo Entropy in Field Theories · Holographic Pseudo Entropy in Gapless Phases via Janus Solutions · 0.87

  • In an explicit example of 3D Janus solution of Einstein-dilaton theory [ref] , we have e^2h(ρ)=1/2+1-2γ^2/2cosh(2ρ).

    Pseudo Entropy in Free Quantum Field Theories · Holographic Analysis · 0.83

Compares with it 2 papers
Cites it as support 1 paper
Cites it as background 34 papers

Discussion

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