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Perturbing a quantum black hole

Cartwright, Casey, Gürsoy, Umut, Pedraza, Juan F., Planas, Guim Planella·2025·Journal of High Energy Physics
DOI 10.1007/jhep03(2025)039
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Abstract

A bstract: We analyze the analytic structure of correlators in the field theory dual to the quantum Bañados-Teitelboim-Zanelli (qBTZ) black hole, a braneworld model incorporating exact backreaction from quantum conformal matter. We first compute the quasi-normal mode (QNM) spectrum of operators with dimension ∆ and spin s = 0, ±1/2. The leading QNMs and their overtones display qualitatively different behavior depending on the branch of qBTZ solution, which corresponds to distinct CFT states: branch 1 is a conical singularity dressed with a horizon while branch 2 is a quantum-corrected BTZ black hole. Consequently, the relaxation of probe matter effectively differentiates the CFT states and identifies the corresponding bulk descriptions. We then turn to pole-skipping locations where Green’s functions are not unique. At these points, frequency is proportional to temperature, but momentum exhibits complex temperature dependence due to quantum effects. Under the assumption that the pole-skipping point closest to the origin reflects quantum chaos, we infer the likely behavior of the quantum Lyapunov exponent and butterfly velocity in the dual theory. Finally, we examine pole collisions in complex momentum space, showing that quantum corrections imprint a unique signature on the analytic structure of the poles in retarded Green’s functions, resulting in level-crossing phenomena that differ notably from the level-touching phenomena in the uncorrected BTZ geometry.

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Figure 1 Penrose diagrams for the geometries we consider in this work. Of note, after backreaction both the qCone and qBTZ solutions have the same conformal structure. The conical singularity is displayed as a line while the genuine curvature singularity of the qCone/qBTZ solution is displayed as a wavy line. Their CFT interpretation, however, differs: before backreaction, the conical singularity is dual to a one-particle state O ( z ) | 0 ⟩ in the DCFT 2 , which is pure, while the BTZ solution corresponds to a thermal state. When backreaction is turned on, the DCFT 2 couples with the BCFT 3 and the state of the DCFT 2 is thus described by a density matrix. In this case, thermality arises due to entanglement between the DCFT 2 and the BCFT 3 .
Figure 1 Penrose diagrams for the geometries we consider in this work. Of note, after backreaction both the qCone and qBTZ solutions have the same conformal structure. The conical singularity is displayed as a line while the genuine curvature singularity of the qCone/qBTZ solution is displayed as a wavy line. Their CFT interpretation, however, differs: before backreaction, the conical singularity is dual to a one-particle state O ( z ) | 0 ⟩ in the DCFT 2 , which is pure, while the BTZ solution corresponds to a thermal state. When backreaction is turned on, the DCFT 2 couples with the BCFT 3 and the state of the DCFT 2 is thus described by a density matrix. In this case, thermality arises due to entanglement between the DCFT 2 and the BCFT 3 .CC-BY · From the PDF
Figure 2 Temperature: The temperature of the quantum corrected black hole is displayed as a function of the 8 G 3 M with ℓ 3 = 1. The colors display different values of the quantum backreaction, from ℓ = 1 / 100 in red, to ℓ = 1 / 10 in blue in steps of δℓ = 1 / 100. The thin dashed black line displays the temperature of the BTZ black hole. Notice that the BTZ black hole on the brane is limited to a value of 8 G 3 M = 1 / 3, even at vanishing quantum backreaction. Hence we continue this line outward with a different dashing to emphasize that the temperature of the BTZ black hole continues for larger black hole mass, but the uncorrected BTZ black hole on the brane is limited to 8 G 3 M = 1 / 3. The thin gray lines continue to display the temperature in steps of δℓ = 1 / 100 but are grayed out to emphasize that we will not consider solutions with that particular value of ℓ in this work. The horizontal line in the figure corresponds to the choice of temperature used throughout this work and corresponds to 4 πT = 1.
Figure 2 Temperature: The temperature of the quantum corrected black hole is displayed as a function of the 8 G 3 M with ℓ 3 = 1. The colors display different values of the quantum backreaction, from ℓ = 1 / 100 in red, to ℓ = 1 / 10 in blue in steps of δℓ = 1 / 100. The thin dashed black line displays the temperature of the BTZ black hole. Notice that the BTZ black hole on the brane is limited to a value of 8 G 3 M = 1 / 3, even at vanishing quantum backreaction. Hence we continue this line outward with a different dashing to emphasize that the temperature of the BTZ black hole continues for larger black hole mass, but the uncorrected BTZ black hole on the brane is limited to 8 G 3 M = 1 / 3. The thin gray lines continue to display the temperature in steps of δℓ = 1 / 100 but are grayed out to emphasize that we will not consider solutions with that particular value of ℓ in this work. The horizontal line in the figure corresponds to the choice of temperature used throughout this work and corresponds to 4 πT = 1.CC-BY · From the PDF
Figure 3 Mode transition from BTZ to qBTZ: The QNM frequencies of a BTZ black hole are displayed as one slowly turns on quantum backreaction with each different curve representing a different overtone. The modes are displayed for fixed n = 0 and 4 πT = 1 while ℓ varies from ℓ = 0 to ℓ = 0 . 1. The figure appears to display that there are no modes for ℓ = 0 of the fourth overtone, however, this is just a reflection of the numerics breaking down. The curves are shown at a fixed value of temperature.
Figure 3 Mode transition from BTZ to qBTZ: The QNM frequencies of a BTZ black hole are displayed as one slowly turns on quantum backreaction with each different curve representing a different overtone. The modes are displayed for fixed n = 0 and 4 πT = 1 while ℓ varies from ℓ = 0 to ℓ = 0 . 1. The figure appears to display that there are no modes for ℓ = 0 of the fourth overtone, however, this is just a reflection of the numerics breaking down. The curves are shown at a fixed value of temperature.CC-BY · From the PDF
Figure 4 Mode transition from BTZ to qBTZ: Thermalization time The imaginary part of the lowest QNM frequency of a BTZ black hole is displayed as one slowly turns on quantum backreaction. The modes are displayed for fixed n = 0 and 4 πT = 1 as ℓ varies from ℓ = 0 to ℓ = 0 . 1. The curve is displayed at fixed temperature.
Figure 4 Mode transition from BTZ to qBTZ: Thermalization time The imaginary part of the lowest QNM frequency of a BTZ black hole is displayed as one slowly turns on quantum backreaction. The modes are displayed for fixed n = 0 and 4 πT = 1 as ℓ varies from ℓ = 0 to ℓ = 0 . 1. The curve is displayed at fixed temperature.CC-BY · From the PDF
Figure 5 Mode transition from quantum dressed conical singularity to quantum corrected BTZ: The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction. The modes are displayed for fixed n = 0 and 4 πT = 1 as ℓ varies from ℓ = 1 / 100 displayed in green (triangles for qCone, diamonds for qBTZ) to ℓ = 1 / 10 displayed in red (triangles for qCone, diamonds for qBTZ). Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1).
Figure 5 Mode transition from quantum dressed conical singularity to quantum corrected BTZ: The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction. The modes are displayed for fixed n = 0 and 4 πT = 1 as ℓ varies from ℓ = 1 / 100 displayed in green (triangles for qCone, diamonds for qBTZ) to ℓ = 1 / 10 displayed in red (triangles for qCone, diamonds for qBTZ). Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1).CC-BY · From the PDF
Figure 6 Origin of the quantum correction: Thermalization time The imaginary part of the lowest QNM frequency of a quantum corrected geometry is displayed as one slowly increases the quantum backreaction. The modes are displayed for fixed n = 0 and 4 πT = 1 as ℓ varies from ℓ = 0 to ℓ = 0 . 1.
Figure 6 Origin of the quantum correction: Thermalization time The imaginary part of the lowest QNM frequency of a quantum corrected geometry is displayed as one slowly increases the quantum backreaction. The modes are displayed for fixed n = 0 and 4 πT = 1 as ℓ varies from ℓ = 0 to ℓ = 0 . 1.CC-BY · From the PDF
Figure 7 Mode transition from quantum dressed conical singularity to quantum corrected BTZ: The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction for operators obeying h R -h L = ± 1 / 2. The modes are displayed for fixed n = 1 and 4 πT = 1 as ℓ varies from ℓ = 1 / 100 (green triangles) to ℓ = 1 / 10 (red triangles). Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1). Note that the Green dots on the black curves are at Re( w ) = 2 due to our choice of units.
Figure 7 Mode transition from quantum dressed conical singularity to quantum corrected BTZ: The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction for operators obeying h R -h L = ± 1 / 2. The modes are displayed for fixed n = 1 and 4 πT = 1 as ℓ varies from ℓ = 1 / 100 (green triangles) to ℓ = 1 / 10 (red triangles). Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1). Note that the Green dots on the black curves are at Re( w ) = 2 due to our choice of units.CC-BY · From the PDF
Figure 8 Ratio of the frequency to momentum of the nearest pole-skipping point to the origin of the complex plane v b = | w ∗ / q ∗ | . The black line refers to the value of the ratio for ℓ = 0.
Figure 8 Ratio of the frequency to momentum of the nearest pole-skipping point to the origin of the complex plane v b = | w ∗ / q ∗ | . The black line refers to the value of the ratio for ℓ = 0.CC-BY · From the PDF
Figure 9 Pole-skipping points of Green's functions of single trace scalar operators dual to massive scalar fields in BTZ geometry For this image ∆ = 2 . 5, ℓ = 1 and ¯ r h = 1.
Figure 9 Pole-skipping points of Green's functions of single trace scalar operators dual to massive scalar fields in BTZ geometry For this image ∆ = 2 . 5, ℓ = 1 and ¯ r h = 1.CC-BY · From the PDF
Figure 10 Scalar pole-skipping: Lowest pole-skipping momentum of Green's functions of single trace scalar operators dual to massive scalar fields in qBTZ geometry. Here we have taken ∆ = 2 . 5 and ℓ = 1 / 3 and hence the strength of the quantum backreaction ν = 1 / 3.
Figure 10 Scalar pole-skipping: Lowest pole-skipping momentum of Green's functions of single trace scalar operators dual to massive scalar fields in qBTZ geometry. Here we have taken ∆ = 2 . 5 and ℓ = 1 / 3 and hence the strength of the quantum backreaction ν = 1 / 3.CC-BY · From the PDF
Figure 11 Lowest pole-skipping momentum of Green's functions of single trace scalar operators dual to massive scalar fields in qBTZ geometry. Here we have taken ∆ = 2 . 5 while allowing the strength of the quantum backreaction ℓ to vary.
Figure 11 Lowest pole-skipping momentum of Green's functions of single trace scalar operators dual to massive scalar fields in qBTZ geometry. Here we have taken ∆ = 2 . 5 while allowing the strength of the quantum backreaction ℓ to vary.CC-BY · From the PDF
Figure 12 Pole-skipping momentum of Green's functions of single trace scalar operators dual to massive scalar fields in qBTZ geometry displayed as a function of ℓ . Here we have taken ∆ = 2 and 4 πT = 1 while allowing the strength of the quantum backreaction ℓ to vary. Left: The lowest pole-skipping momentum. Right: The first higher pole-skipping momentum.
Figure 12 Pole-skipping momentum of Green's functions of single trace scalar operators dual to massive scalar fields in qBTZ geometry displayed as a function of ℓ . Here we have taken ∆ = 2 and 4 πT = 1 while allowing the strength of the quantum backreaction ℓ to vary. Left: The lowest pole-skipping momentum. Right: The first higher pole-skipping momentum.CC-BY · From the PDF
Figure 13 Lowest pole-skipping momentum of Green's functions of single trace fermion operators dual to massive scalar fields in qBTZ geometry. We take m = √ 1 . 25 and ℓ = 1 / 3, hence the strength of the quantum backreaction ν = 1 / 3. Each image displays the pole-skipping momentum associated to different Matsubara frequencies for n = 0 , 1 , 2 , 3, with the top displaying n = 0 , 1 from left to right, and the bottom displaying n = 2 , 3 from left to right.
Figure 13 Lowest pole-skipping momentum of Green's functions of single trace fermion operators dual to massive scalar fields in qBTZ geometry. We take m = √ 1 . 25 and ℓ = 1 / 3, hence the strength of the quantum backreaction ν = 1 / 3. Each image displays the pole-skipping momentum associated to different Matsubara frequencies for n = 0 , 1 , 2 , 3, with the top displaying n = 0 , 1 from left to right, and the bottom displaying n = 2 , 3 from left to right.CC-BY · From the PDF
Figure 14 Lowest pole-skipping momentum of Green's functions of single trace fermion operators dual to massive spinor fields in qBTZ geometry. Here we have taken m = √ 1 . 25, fixing ℓ 3 = 3 allowing the strength of the quantum backreaction ν = ℓ/ℓ 3 to vary.
Figure 14 Lowest pole-skipping momentum of Green's functions of single trace fermion operators dual to massive spinor fields in qBTZ geometry. Here we have taken m = √ 1 . 25, fixing ℓ 3 = 3 allowing the strength of the quantum backreaction ν = ℓ/ℓ 3 to vary.CC-BY · From the PDF
Figure 15 QNM spectrum of Green's functions of single trace scalar operators dual to massless scalar fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In all images, the blue curve corresponds to a value of the momentum for which q = q 1 < q c while the gray curves denote momentum above the critical momentum q 2 > q c and the temperature is held fixed to 4 πT = 1. Top: The modes computed on a BTZ background geometry ( ℓ = 0). Middle: The modes computed on a qBTZ background geometry with ℓ = 1 / 100 for which the strength of the quantum backreaction. Bottom: The modes computed on a qBTZ background geometry with ℓ = 1 / 10.
Figure 15 QNM spectrum of Green's functions of single trace scalar operators dual to massless scalar fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In all images, the blue curve corresponds to a value of the momentum for which q = q 1 < q c while the gray curves denote momentum above the critical momentum q 2 > q c and the temperature is held fixed to 4 πT = 1. Top: The modes computed on a BTZ background geometry ( ℓ = 0). Middle: The modes computed on a qBTZ background geometry with ℓ = 1 / 100 for which the strength of the quantum backreaction. Bottom: The modes computed on a qBTZ background geometry with ℓ = 1 / 10.CC-BY · From the PDF
Figure 16 The lowest critical point in the dispersion relation of Green's functions of single trace scalar operators dual to massless scalar fields is tracked as the quantum backreaction ℓ is turned on. Here κ = -1 ensures a smooth connection to the BTZ geometry.
Figure 16 The lowest critical point in the dispersion relation of Green's functions of single trace scalar operators dual to massless scalar fields is tracked as the quantum backreaction ℓ is turned on. Here κ = -1 ensures a smooth connection to the BTZ geometry.CC-BY · From the PDF
Figure 17 QNM spectrum of Green's functions of single trace scalar operators dual to massless scalar fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In all three images the blue curve corresponds Branch 1b while the red curve corresponds to Branch 2. Top: | q | = | q 1 | < | q (1 b ) c | . Middle: | q (1 b ) c | < | q | = | q 2 | < q (2) c | . Bottom: | q (2) c | < | q | = | q 3 | . In all three images ℓ = 1 / 10 and x 1 is chosen as in table 1 to fix 4 πT = 1.
Figure 17 QNM spectrum of Green's functions of single trace scalar operators dual to massless scalar fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In all three images the blue curve corresponds Branch 1b while the red curve corresponds to Branch 2. Top: | q | = | q 1 | < | q (1 b ) c | . Middle: | q (1 b ) c | < | q | = | q 2 | < q (2) c | . Bottom: | q (2) c | < | q | = | q 3 | . In all three images ℓ = 1 / 10 and x 1 is chosen as in table 1 to fix 4 πT = 1.CC-BY · From the PDF
Figure 18 QNM spectrum of Green's functions of single trace scalar operators dual to massless scalar fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In the image, the blue curves corresponds to a value of the momentum for which q = q 1 < q c while the gray curves denote momentum above the critical momentum q 2 > q c . Top: The modes computed on a qBTZ background geometry. Bottom: The modes computed on a qCone background geometry. In both images, ℓ = 1 / 10 and the value of x 1 is chosen as in table 1 to fix 4 πT = 1.
Figure 18 QNM spectrum of Green's functions of single trace scalar operators dual to massless scalar fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In the image, the blue curves corresponds to a value of the momentum for which q = q 1 < q c while the gray curves denote momentum above the critical momentum q 2 > q c . Top: The modes computed on a qBTZ background geometry. Bottom: The modes computed on a qCone background geometry. In both images, ℓ = 1 / 10 and the value of x 1 is chosen as in table 1 to fix 4 πT = 1.CC-BY · From the PDF
Figure 19 QNM spectrum of Green's functions of single trace scalar operators dual to massless spinor field at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , 2 π ], so that q 2 goes through 4 π , the curves display circuits traced out beginning at the QNM located at φ = 0. In all images, the blue curve corresponds to a value of the momentum for which q = q 1 < n c while the gray curves denote momentum above the critical momentum q 2 > q c and the temperature is held fixed to 4 πT = 1. Top: The modes computed on a BTZ background geometry. Middle: The modes computed on a qBTZ background geometry with ℓ = 1 / 100. Bottom: The modes computed on a qBTZ background geometry with ℓ = 1 / 10.
Figure 19 QNM spectrum of Green's functions of single trace scalar operators dual to massless spinor field at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , 2 π ], so that q 2 goes through 4 π , the curves display circuits traced out beginning at the QNM located at φ = 0. In all images, the blue curve corresponds to a value of the momentum for which q = q 1 < n c while the gray curves denote momentum above the critical momentum q 2 > q c and the temperature is held fixed to 4 πT = 1. Top: The modes computed on a BTZ background geometry. Middle: The modes computed on a qBTZ background geometry with ℓ = 1 / 100. Bottom: The modes computed on a qBTZ background geometry with ℓ = 1 / 10.CC-BY · From the PDF
Figure 20 The lowest critical point in the dispersion relation of Green's functions of single trace operators is tracked as the quantum backreaction ℓ is turned on. Here κ = -1 ensures a smooth connection to the BTZ geometry. The black dots correspond to operators with s = 0 (as displayed in figure 16) while the blue diamonds correspond to operators with s = 1 / 2. The dashed lines show the uncorrected value for s = 0 (black) and s = 1 / 2 (blue) to help guide the eye. The two branches that connect smoothly to the qBTZ solution are displayed as empty circles (branch 1b) and empty squares (branch 2).
Figure 20 The lowest critical point in the dispersion relation of Green's functions of single trace operators is tracked as the quantum backreaction ℓ is turned on. Here κ = -1 ensures a smooth connection to the BTZ geometry. The black dots correspond to operators with s = 0 (as displayed in figure 16) while the blue diamonds correspond to operators with s = 1 / 2. The dashed lines show the uncorrected value for s = 0 (black) and s = 1 / 2 (blue) to help guide the eye. The two branches that connect smoothly to the qBTZ solution are displayed as empty circles (branch 1b) and empty squares (branch 2).CC-BY · From the PDF
Figure 21 QNM spectrum of Green's functions of single trace fermion operators dual to massless spinor fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In all three images the blue curve corresponds Branch 1b while the red curve corresponds to Branch 2. Top: | q | = | q 1 | < | q (1 b ) c | Middle: | q (1 b ) c | < | q | = | q 2 | < q (2) c | . Bottom: | q (2) c | < | q | = | q 3 | . In all three images ℓ 3 = 1, ℓ = 1 / 10 and x 1 is chosen as in table 1 to fix 4 πT = 1.
Figure 21 QNM spectrum of Green's functions of single trace fermion operators dual to massless spinor fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In all three images the blue curve corresponds Branch 1b while the red curve corresponds to Branch 2. Top: | q | = | q 1 | < | q (1 b ) c | Middle: | q (1 b ) c | < | q | = | q 2 | < q (2) c | . Bottom: | q (2) c | < | q | = | q 3 | . In all three images ℓ 3 = 1, ℓ = 1 / 10 and x 1 is chosen as in table 1 to fix 4 πT = 1.CC-BY · From the PDF
Figure 22 QNM spectrum of Green's functions of single trace fermion operators dual to massless spinor fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In the image the blue curves corresponds to a value of the momentum for which q = 0 . 16 = q 1 < q (1 b ) c while the gray curves denote momentum above the critical momentum q 2 = 0 . 199 > q (1 b ) c . Top: The modes computed in a branch 1b qBTZ background geometry. Bottom: The modes computed in branch 1a, qCone, background geometry. In both images ℓ = 1 / 10 and the value of x 1 is chosen as in table 1 to fix 4 πT = 1
Figure 22 QNM spectrum of Green's functions of single trace fermion operators dual to massless spinor fields at complex momentum q = q 0 e iφ . As φ scans through the range φ ∈ [0 , π ] the curves denote circuits traced out beginning at the QNM located at φ = 0. In the image the blue curves corresponds to a value of the momentum for which q = 0 . 16 = q 1 < q (1 b ) c while the gray curves denote momentum above the critical momentum q 2 = 0 . 199 > q (1 b ) c . Top: The modes computed in a branch 1b qBTZ background geometry. Bottom: The modes computed in branch 1a, qCone, background geometry. In both images ℓ = 1 / 10 and the value of x 1 is chosen as in table 1 to fix 4 πT = 1CC-BY · From the PDF
Figure 23 Classical BTZ spectrum: A depiction of the classical spectrum of 2+1 dimensional solution given in eq. (A.2a)
Figure 23 Classical BTZ spectrum: A depiction of the classical spectrum of 2+1 dimensional solution given in eq. (A.2a)CC-BY · From the PDF
Figure 24 The QNM frequencies of a BTZ black hole are displayed as one slowly turns on quantum backreaction with each different curve representing a different overtone. The modes are displayed for fixed ℓ 3 = 1, n = 0 and x 1 = 1 / 2 as ℓ varies from ℓ = 0 to ℓ = 1. The figure appears to display that there are no modes for ℓ = 0 of the third overtone, however, this is just a reflection of the numerics breaking down.
Figure 24 The QNM frequencies of a BTZ black hole are displayed as one slowly turns on quantum backreaction with each different curve representing a different overtone. The modes are displayed for fixed ℓ 3 = 1, n = 0 and x 1 = 1 / 2 as ℓ varies from ℓ = 0 to ℓ = 1. The figure appears to display that there are no modes for ℓ = 0 of the third overtone, however, this is just a reflection of the numerics breaking down.CC-BY · From the PDF
Figure 25 The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction. The modes are displayed for fixed ℓ 3 = 1, n = 0 and x 1 = 1 / 2 as ℓ varies from ℓ = 1 / 10 (green triangles) to ℓ = 4 (red triangles) Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1).
Figure 25 The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction. The modes are displayed for fixed ℓ 3 = 1, n = 0 and x 1 = 1 / 2 as ℓ varies from ℓ = 1 / 10 (green triangles) to ℓ = 4 (red triangles) Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1).CC-BY · From the PDF
Figure 26 Mode transition from quantum dressed conical singularity to quantum corrected BTZ: The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction for operators obeying h R -h L = ± 1 / 2. The modes are displayed for fixed ℓ 3 = 1, n = 1 and x 1 = 1 / 2 as ℓ varies from ℓ = 1 / 10 (green triangles) to ℓ = 4 (red triangles). Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1).
Figure 26 Mode transition from quantum dressed conical singularity to quantum corrected BTZ: The lowest QNM frequencies are displayed as one slowly turns on quantum backreaction for operators obeying h R -h L = ± 1 / 2. The modes are displayed for fixed ℓ 3 = 1, n = 1 and x 1 = 1 / 2 as ℓ varies from ℓ = 1 / 10 (green triangles) to ℓ = 4 (red triangles). Blue dots represent the quantum dressed conical singularity ( κ = 1) while the black dots represent the quantum corrected BTZ geometry ( κ = -1).CC-BY · From the PDF
Figure 27 Convergence: The QNMs were calculated with n = 0 and 4 πT = 1. The various shaped markers indicate the different branches and the colors indicate different values of ℓ ranging from ℓ = 0 . 1 in red to ℓ = 0 in black. In the image, the rainbow-colored markers correspond to the code used throughout the draft, while the dots correspond to modes obtained from QNMspectral.
Figure 27 Convergence: The QNMs were calculated with n = 0 and 4 πT = 1. The various shaped markers indicate the different branches and the colors indicate different values of ℓ ranging from ℓ = 0 . 1 in red to ℓ = 0 in black. In the image, the rainbow-colored markers correspond to the code used throughout the draft, while the dots correspond to modes obtained from QNMspectral.CC-BY · From the PDF
Figure 28 Comparison of results from QNMspectral: The QNMs were calculated with n = 0, ℓ 3 = 3 and ℓ = 1. In the image, the rainbow-colored markers correspond to the code used throughout the draft, while the dots correspond to modes obtained from QNMspectral.
Figure 28 Comparison of results from QNMspectral: The QNMs were calculated with n = 0, ℓ 3 = 3 and ℓ = 1. In the image, the rainbow-colored markers correspond to the code used throughout the draft, while the dots correspond to modes obtained from QNMspectral.CC-BY · From the PDF

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Authors
Cartwright, C., Gürsoy, U., Pedraza, J. F., & Planas, G. P.
Affiliations
  • Utrecht University
  • Instituto de Física Teórica UAM/CSIC
ORCID
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2025-03-06
Volume
2025
Issue
3
ISSN
1029-8479
DOI
10.1007/jhep03(2025)039
Cited by
7
References
6
Language
en
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