Pith. sign in

REVIEW 3 major objections 4 minor 40 references

Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $AdS_2/CFT_1$ setup

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For a free scalar plus Majorana fermion in global AdS2, the paper establishes that the entanglement entropy between two conformal quantum mechanics copies on the two boundaries equals the logarithm of a regularized bulk geodesic length…

desk verdict The boundary-side calculation is clean and the chain 4ΔS = log(L/2πa) = SvN is genuinely new, but the bulk horizon entropy rests on an unproven flat-space transfer and the 'precisely matches' claim oversells the result. read the letter →

arxiv 2502.01144 v2 pith:SF2Q5MUK submitted 2025-02-03 hep-th

Gabriel Lopes Cardoso , Bernardo Moniz Martins , Suresh Nampuri This is my paper · ORCID
classification hep-th
keywords AdS2/CFT1correspondenceRyu-TakayanagiconjectureentanglemententropyconformalquantummechanicsDFFmodelthermofielddoubleone-loopeffectiveactionblackholehorizon
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that in a free conformal field theory on global AdS2—one massless scalar plus one massless Majorana fermion, central charge $c=3/2$—the one-loop effective action is governed by the same data as two copies of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics model, one on each boundary. In the thermofield double state at global time $\tau \to -\infty$, the entanglement entropy between the two copies evaluates to $S_{\text{vN}} = \log(L/(2\pi a))$, where $L$ is the regularized length of a circular geodesic near the Euclidean AdS2 boundary. The paper shows that this boundary entropy equals four times $\Delta S = \frac14 \log(a/\epsilon)$, the logarithmic term in the regularized one-loop effective action, and that the same value is obtained by cutting Lorentzian AdS2 along its null horizons and applying the flat-space interval entanglement formula to the four resulting triangular wedges. If the calculation is right, this is a concrete AdS2/CFT1 realization of the Ryu-Takayanagi conjecture in which the bulk geodesic length and the bulk horizon entanglement are checked against a direct boundary entropy computation.

What carries the argument

The load-bearing structure is the thermofield double state $|\delta\rangle$ of the DFF model—a one-dimensional conformal quantum mechanics with Hamiltonian $H = (p^2 + g/q^2)/2$—together with the character-integral expression for the bulk one-loop partition function. The bulk heat-kernel spectral densities $\rho_B(\nu)=\nu\tanh(\pi\nu)$ and $\rho_F(\nu)=2\nu\coth(\pi\nu)$ are Fourier-transformed into $W_B(u)$ and $W_F(u)$, whose pole sums reproduce the DFF partition function $Z_{\text{CQM}}(u)=e^{-u}/(1-e^{-u})$; the regularized one-loop action then yields $\Delta S = \frac14 \log(a/\epsilon)$. The boundary state $|\delta\rangle$ is constructed as the $\tau\to-\infty$ limit of states on the two global boundaries, and its reduced density matrix is diagonal with Boltzmann weights $e^{-\beta E_n}$ and $\beta=2\epsilon/a$, giving $S_{\text{vN}}=\log(a/(2\epsilon))$. The bulk-to-boundary link is completed by the identification $\epsilon=a\sinh x_0$, which turns the bulk cutoff into the geodesic length $L=2\pi a^2/\epsilon$.

What would settle it

Perform a direct replica calculation of the entanglement entropy for the free massless scalar and Majorana fermion across a single null horizon segment in Lorentzian AdS2, evaluating the appropriate two-point function on the null surface with a position-dependent cutoff, without invoking the flat-space interval formula. If the result per segment differs from $c/6\log(a/\epsilon)$ with $c=3/2$, or if the total is not $\log(a/\epsilon)$, then Eq. (5.78) fails on the bulk side.

Watch

Extended reading notes

Core claim

The central claim is the chain of equalities in Eq. (5.78): $S_{\text{total}} = 4\,\Delta S = \log(L/(2\pi a)) = S_{\text{vN}}$. Here $\Delta S$ is the one-loop logarithmic contribution to the regularized effective action of the scalar-plus-Majorana theory in Euclidean AdS2; $S_{\text{total}}$ is the total entanglement entropy associated with the four null horizon segments obtained by cutting global Lorentzian AdS2 into isosceles right triangles; $L$ is the length of a closed circular geodesic at a small cutoff distance from the boundary; and $S_{\text{vN}}$ is the von Neumann entropy of the reduced density matrix obtained from the thermofield double state $|\delta\rangle$ of two DFF copies at $\tau \to -\infty$. The paper constructs the DFF model at coupling $g=3/4$ from the bulk spectral data, identifies the regularized boundary state on both boundaries, and computes $S_{\text{vN}}=\log(a/(2\epsilon))$, which matches the geometric quantity $\log(L/(2\pi a))$ up to $\epsilon$-independent constants.

Load-bearing premise

The whole bulk side of the equality rests on transferring the flat-space interval entanglement formula of a 1+1 CFT to the four null horizon wedges cut out of curved AdS2; the paper supplies no derivation of that transfer, and if the curved-null-surface entanglement differs from the flat-space result, $S_{\text{total}}$ and hence the equality with $S_{\text{vN}}$ lose their bulk support. A secondary hinge is the choice $r_0=1$, which fixes the DFF coupling to $g=3/4$ and is what makes the boundary model match the bulk one-loop spectrum.

Editorial extensions

If this is right

  • The logarithmic one-loop correction to the AdS2 effective action is reinterpreted as a genuine entanglement entropy of the dual CFT1, connecting thermodynamic entropy counting to a quantum information quantity.
  • The equality $S_{\text{total}}=S_{\text{vN}}$ gives a two-sided check: the bulk horizon entanglement and the boundary state entanglement are the same number, computed independently by replica tricks on each side.
  • The correspondence extends to $n$ multiplets of the scalar-plus-Majorana theory: the central charge becomes $c=3n/2$ and the boundary model becomes $n$ copies of DFF, so the equality survives the addition of matter content.
  • The cutoff identification $\epsilon = a\sinh x_0$ makes the boundary UV cutoff a bulk geometric quantity, so the entropy is literally a function of a geodesic length; this is the concrete content of the Ryu-Takayanagi conjecture in this one-dimensional setting.

Reading between the lines

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

  • A direct curved-space replica computation across the null horizon would either confirm the triangle decomposition or reveal a correction; this is the most accessible independent test of the paper's bulk claim.
  • The same construction suggests that other one-dimensional conformal models with the same spectral data might reproduce the same entropy, turning the relation into a statement about the universal part of the one-loop partition function rather than a special feature of DFF.
  • The Fisher-information metric sketched in the concluding section could be explored for other conformal quantum mechanics models; if the entropy-gradient relation $dS_{\text{vN}} = -e^{S_{\text{vN}}}\,d\epsilon/a$ is generic, it would give a geometric meaning to the UV cutoff flow.
  • The equality is established at one loop for free fields; a natural next step is to check whether including the gravitational one-loop determinant or interactions shifts both sides by the same amount, which would be a stronger test of the bulk-boundary correspondence.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper computes the one-loop effective action for a free massless scalar plus a massless Majorana fermion in Euclidean AdS2 using the character-integral method, extracts the logarithmic divergence DeltaS = (1/4) log(a/epsilon), and rewrites it in terms of the regularized length L of a closed geodesic near the boundary as DeltaS = (1/4) log(L/(2 pi a)). It then constructs a thermofield-double state for two copies of the DFF conformal quantum mechanics at coupling g = 3/4, computes the von Neumann entropy SvN = log(a/(2 epsilon)), and claims Stotal = 4 DeltaS = log(L/(2 pi a)) = SvN, where Stotal is a bulk entanglement entropy obtained by applying the flat-space Calabrese-Cardy interval formula to four triangular wedges bounded by null horizons in global AdS2. The paper views the relation between SvN and log L as an AdS2/CFT1 version of the Ryu-Takayanagi conjecture.

Significance. The technical core of the paper is a coherent and explicit character-integral evaluation of the AdS2 one-loop action, and the cancellation of the 1/epsilon^2 divergences between the scalar and Majorana contributions is a clean, verifiable result. If the bulk horizon-entropy step could be derived rather than imported, the paper would provide a simple, concrete AdS2/CFT1 analogue of the Ryu-Takayanagi correspondence, with the bonus that the boundary entropy is computed directly from a thermofield-double state rather than postulated. The paper is also transparent about the universal log term and about which parts of the calculation are regularization-dependent. The principal weakness, and the reason this is not yet acceptable, is the unsupported transfer of the flat-space Calabrese-Cardy formula to curved AdS2 null surfaces in Section 4.2; this step is load-bearing for the claimed bulk-boundary equality.

major comments (3)
  1. [Section 4.2, Eqs. (4.16)-(4.18)] The bulk entanglement entropy Stotal is the sole bulk quantity in the final equality (5.78), but it is not derived from the free scalar-plus-Majorana system on AdS2. Instead, the flat-space Calabrese-Cardy interval formula is applied to four right triangles in global AdS2 with ell = pi a / sqrt(2) and Lhat = sqrt(2) pi a. Global AdS2 is only conformal to a flat strip (Eq. (2.12)), and the entanglement entropy of a 2D CFT is not invariant under Weyl rescalings when c is nonzero, so the vacuum entanglement across the null diagonals need not equal the flat-space expression. Please provide a derivation, for example a replica computation on the fixed AdS2 background, or explicitly demote the equality Stotal = 4 Delta S to a conjecture. As written, Eq. (5.78) has no bulk support if this transfer fails.
  2. [Eqs. (4.22), (5.75), (5.78)] The central relation is stated as an equality, but the intermediate expressions give 4 Delta S = log(a/epsilon) while SvN = log(a/(2 epsilon)) = log(a/epsilon) - log 2. The paper acknowledges this by adding up to epsilon-independent constants in Eq. (5.78), yet the abstract and Section 6 state that the bulk horizon entropy precisely matches the boundary entanglement entropy. Since epsilon-independent constants are non-universal in entanglement entropy, the log 2 discrepancy is not fatal, but the claims of exact agreement and the final sentence of the abstract should be calibrated to what is actually established.
  3. [Section 5.3 and Eq. (5.66)] The thermofield-double state |delta> is constructed with coefficients e^{-beta n/2}, i.e. with En = n, while the DFF spectrum in Eq. (5.4) has R|n> = (n+1)|n>. The shift is absorbed by the normalization 1/sqrt(Z(beta)) and cancels in the reduced density matrix, so it does not affect the final entropy, but the text should state explicitly that this is an energy-zero shift and explain why it is harmless, especially because Eq. (5.76) uses the unshifted spectrum to obtain the same entropy.
minor comments (4)
  1. [Eq. (4.17)] With ell = pi a / sqrt(2) and Lhat = sqrt(2) pi a, one has sin(pi ell / Lhat) = 1, so the Calabrese-Cardy formula gives S = (c/6) log(sqrt(2) a / epsilon) = (1/4) log(sqrt(2) a / epsilon); the text simplifies this to (c/6) log(a/epsilon) = (1/4) log(a/epsilon) by dropping an O(1) constant. This is consistent with the stated up to additive constants convention, but it should be flagged in the displayed equation.
  2. [Section 5.3, Eqs. (5.64)-(5.66)] The construction of |delta> uses an epsilon-expansion and retains only the leading order in beta = 2 epsilon / a. Please clarify that the identification with a thermofield-double state of the DFF model is a leading-order statement in beta and that subleading corrections are not controlled.
  3. [Section 4.2, Eq. (4.20)] The identification epsilon = a sinh x0 is an ad hoc IR/UV regulator identification; its status as a choice, rather than a derived relation, should be stated explicitly so that the geometric rewriting Delta S = (1/4) log(L/(2 pi a)) is not mistaken for a parameter-free prediction.
  4. [Abstract and Section 6] The phrase precisely matches in the abstract overstates the relation, which holds only up to epsilon-independent constants. Rephrasing to matches up to non-universal constants would align the abstract with Eqs. (5.75)-(5.78).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the bulk and boundary entropy computations are independent, and the DFF/g=3/4 identification is a dictionary construction rather than a fitted prediction.

full rationale

The paper's central chain is not circular. The bulk quantity Delta S is obtained from the regularised one-loop effective action (4.11)-(4.15), which is built from spectral densities and character integrals (3.14)-(3.16), (4.1)-(4.3); the boundary entanglement entropy SvN is computed independently from an explicit thermofield-double state and reduced density matrix (5.61)-(5.75). The equality Stotal = 4Delta S = log(L/2pi a) = SvN in Eq. (5.78) therefore compares two independent calculations, up to additive constants. The identification of the DFF model at g = 3/4 is a dictionary construction: the paper states 'We set r0 = 1 in view of (4.7)' in Section 5, but the leading logarithmic divergence of SvN is independent of r0, so this choice does not force the target entropy. Likewise, the relation epsilon = a sinh x0 in (4.20) is a standard IR/UV cutoff identification, not a fit to the final entropy. The self-citations [36] and [38] appear only in concluding remarks and are not load-bearing. The main caveat is not circularity but an unproven transfer: Eqs. (4.16)-(4.18) apply the flat-space Calabrese-Cardy interval formula 'Following eq. (23) in [31]' to null triangles in curved global AdS2 without deriving the Weyl/anomaly transformation; if that transfer fails, Stotal lacks independent bulk support. That is a correctness risk, not a circular reduction, because the formula is an external benchmark and the paper does not derive it from the target equality.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two chosen inputs: the DFF parameter r0 = 1 / g = 3/4 fixed by matching the bulk one-loop trace, and the regulator identification sinh x0 = epsilon/a that converts the UV cutoff into the geodesic length. It also assumes two domain facts: that the flat-space interval entropy applies to null horizons in AdS2, and that the boundary state is a thermal TFD at the stated temperature. No new physical entities are introduced.

free parameters (2)
  • DFF vacuum weight r0 and derived coupling g = r0 = 1, g = 3/4
    Set by choosing r0 = 1 to match the bulk-derived trace ZCQM(u) = sum_n e^{-(n+1)u} in eqs. (4.5)-(4.7); the coupling is not independently measured or predicted.
  • IR cutoff identification sinh x0 = epsilon/a = x0 = epsilon/a (epsilon to 0)
    Eq. (4.20) sets the bulk IR regulator equal to the UV cutoff; this converts Delta S into log(L/(2 pi a)) and is the point where the geometric length enters. Different identifications would change the relation.
assumptions (4)
  • standard math The Fourier integral W(u) of the spectral density is evaluated by deforming the contour to the positive imaginary axis and summing simple poles; this assumes no other singularities contribute.
    Used in Appendix A and Section 3 to obtain (3.18)-(3.19).
  • domain assumption The regularized one-loop effective action can be split as beta Delta E - Delta S, with the x0-independent part identified as Delta S, following [5,6].
    Section 4.2, eqs. (4.14)-(4.15); the thermodynamic interpretation of the x0-independent term is assumed.
  • domain assumption The flat-space Calabrese-Cardy interval entropy applies to each of the four triangular regions of global AdS2 bounded by null horizons.
    Section 4.2, eqs. (4.16)-(4.18); no curved-space or null-horizon derivation is given.
  • domain assumption The two boundary CQM copies form a thermofield double with inverse temperature beta = 2 epsilon/a in the tau to -infinity limit.
    Section 5.3, eqs. (5.65)-(5.66); relies on the specific regularization of the stretched boundary states.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $AdS_2/CFT_1$ setup." pith.science (2026). https://pith.science/paper/SF2Q5MUK

@misc{pith2026250201144,
  author       = {Pith},
  title        = {Pith review of: Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $AdS_2/CFT_1$ setup},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SF2Q5MUK}},
  note         = {Machine review of arXiv:2502.01144}
}
abstract

Using recent developments in expressing one-loop partition functions in Euclidean $AdS_2$ space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in $AdS_2$ (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global $AdS_2$ boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the $CFT_1$ (CQM), each living near one of the two boundaries of global $AdS_2$, in a state at global time $\tau \rightarrow - \infty$. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean $AdS_2$ disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the $AdS_2/CFT_1$ version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in $AdS_2$. Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian $AdS_2$ and show that it precisely matches the boundary entanglement entropy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 9 canonical work pages

  1. [10]

    Arzano, A

    M. Arzano, A. D’Alise, and D. Frattulillo, Entanglement entropy in conformal quantum mechanics, JHEP 10 (2023) 165, [ 2306.12291]

    arXiv 2023

  2. [1]

    Anninos, F

    D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions , JHEP 01 (2022) 088, [2009.12464]

    arXiv 2022

  3. [2]

    Sun, AdS one-loop partition functions from bulk and edge characters , JHEP 12 (2021) 064, [2010.15826]

    Z. Sun, AdS one-loop partition functions from bulk and edge characters , JHEP 12 (2021) 064, [2010.15826]

    arXiv 2021

  4. [3]

    Grewal and K

    M. Grewal and K. Parmentier, Characters, quasinormal modes, and Schwinger pairs in dS 2 with flux , JHEP 03 (2022) 165, [ 2112.07630]

    arXiv 2022

  5. [4]

    J. R. David and J. Mukherjee, Partition functions of p-forms from Harish-Chandra characters, JHEP 09 (2021) 094, [ 2105.03662]

    arXiv 2021

  6. [5]

    Banerjee, R

    S. Banerjee, R. K. Gupta, and A. Sen, Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function , JHEP 03 (2011) 147, [ 1005.3044]

    arXiv 2011

  7. [6]

    Banerjee, R

    S. Banerjee, R. K. Gupta, I. Mandal, and A. Sen, Logarithmic Corrections to N=4 and N=8 Black Hole Entropy: A One Loop Test of Quantum Gravity , JHEP 11 (2011) 143, [1106.0080]

    arXiv 2011

  8. [7]

    de Alfaro, S

    V. de Alfaro, S. Fubini, and G. Furlan, Conformal Invariance in Quantum Mechanics , Nuovo Cim. A 34 (1976) 569

    work page 1976

Show all 40 references
  1. [8]

    Chamon, R

    C. Chamon, R. Jackiw, S.-Y. Pi, and L. Santos, Conformal quantum mechanics as the CFT 1 dual to AdS 2, Phys. Lett. B 701 (2011) 503–507, [ 1106.0726]. – 33 –

    2011 arXiv

  2. [9]

    Jackiw and S

    R. Jackiw and S. Y. Pi, Conformal Blocks for the 4-Point Function in Conformal Quantum Mechanics, Phys. Rev. D 86 (2012) 045017, [ 1205.0443]. [Erratum: Phys.Rev.D 86, 089905 (2012)]

    2012 arXiv

  3. [11]

    Sen, State Operator Correspondence and Entanglement in AdS2/CF T1, Entropy 13 (2011) 1305–1323, [ 1101.4254]

    A. Sen, State Operator Correspondence and Entanglement in AdS2/CF T1, Entropy 13 (2011) 1305–1323, [ 1101.4254]

    2011 arXiv

  4. [12]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT , Phys. Rev. Lett. 96 (2006) 181602, [ hep-th/0603001]

    2006 arXiv

  5. [13]

    Ryu and T

    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy , JHEP 08 (2006) 045, [hep-th/0605073]

    2006 arXiv

  6. [14]

    Sen, Quantum Entropy Function from AdS(2)/CFT(1) Correspondence , Int

    A. Sen, Quantum Entropy Function from AdS(2)/CFT(1) Correspondence , Int. J. Mod. Phys. A 24 (2009) 4225–4244, [ 0809.3304]

    2009 arXiv

  7. [15]

    Kitaev, Notes on fSL(2, R) representations, 1711.08169

    A. Kitaev, Notes on fSL(2, R) representations, 1711.08169

    arXiv

  8. [16]

    Sen, Entropy Function and AdS(2) / CFT(1) Correspondence , JHEP 11 (2008) 075, [0805.0095]

    A. Sen, Entropy Function and AdS(2) / CFT(1) Correspondence , JHEP 11 (2008) 075, [0805.0095]

    2008 arXiv

  9. [17]

    Maldacena, D

    J. Maldacena, D. Stanford, and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space , PTEP 2016 (2016), no. 12 12C104, [ 1606.01857]

    2016 arXiv

  10. [18]

    Kitaev and S

    A. Kitaev and S. J. Suh, Statistical mechanics of a two-dimensional black hole , JHEP 05 (2019) 198, [ 1808.07032]

    2019 arXiv

  11. [19]

    Yang, The Quantum Gravity Dynamics of Near Extremal Black Holes , JHEP 05 (2019) 205, [1809.08647]

    Z. Yang, The Quantum Gravity Dynamics of Near Extremal Black Holes , JHEP 05 (2019) 205, [1809.08647]

    2019 arXiv

  12. [20]

    P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral , 1903.11115

    1903 arXiv

  13. [21]

    Mefford and K

    E. Mefford and K. Suzuki, Jackiw-Teitelboim quantum gravity with defects and the Aharonov-Bohm effect, JHEP 05 (2021) 026, [ 2011.04695]

    2021 arXiv

  14. [22]

    Penington and E

    G. Penington and E. Witten, Algebras and States in JT Gravity , 2301.07257

    arXiv

  15. [23]

    Penington and E

    G. Penington and E. Witten, Algebras and states in super-JT gravity , 2412.15549

    arXiv

  16. [24]

    J. W. van Holten, D = 1 supergravity and spinning particles , hep-th/9510021

    arXiv

  17. [25]

    Brink, P

    L. Brink, P. Di Vecchia, and P. S. Howe, A Lagrangian Formulation of the Classical and Quantum Dynamics of Spinning Particles , Nucl. Phys. B 118 (1977) 76–94

    1977

  18. [26]

    Comtet and P

    A. Comtet and P. J. Houston, Effective action on the hyperbolic plane in a constant external field, Journal of Mathematical Physics 26 (01, 1985) 185–191

    1985

  19. [27]

    Camporesi and A

    R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces , Journal of Mathematical Physics 35 (08, 1994) 4217–4246

    1994

  20. [28]

    Pioline and J

    B. Pioline and J. Troost, Schwinger pair production in AdS(2) , JHEP 03 (2005) 043, [hep-th/0501169]

    2005 arXiv

  21. [29]

    Camporesi and A

    R. Camporesi and A. Higuchi, On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1–18, [ gr-qc/9505009]

    1996 arXiv

  22. [30]

    Gonz´ alez Lezcano, I

    A. Gonz´ alez Lezcano, I. Jeon, and A. Ray, Supersymmetry and complexified spectrum on Euclidean AdS2, Phys. Rev. D 108 (2023), no. 4 045018, [ 2305.12925]. – 34 –

    2023 arXiv

  23. [31]

    Calabrese and J

    P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory , J. Stat. Mech. 0406 (2004) P06002, [ hep-th/0405152]

    2004 arXiv

  24. [32]

    Ban, Su(1,1) lie algebraic approach to linear dissipative processes in quantum optics , Journal of Mathematical Physics 33 (09, 1992) 3213–3228

    M. Ban, Su(1,1) lie algebraic approach to linear dissipative processes in quantum optics , Journal of Mathematical Physics 33 (09, 1992) 3213–3228

    1992

  25. [33]

    Banerjee, J

    S. Banerjee, J. Erdmenger, and J. Karl, Non-Locality induces Isometry and Factorisation in Holography, 2411.09616

    arXiv

  26. [34]

    Witten, Introduction to Black Hole Thermodynamics , 2412.16795

    E. Witten, Introduction to Black Hole Thermodynamics , 2412.16795

    arXiv

  27. [35]

    Azeyanagi, T

    T. Azeyanagi, T. Nishioka, and T. Takayanagi, Near Extremal Black Hole Entropy as Entanglement Entropy via AdS(2)/CFT(1) , Phys. Rev. D 77 (2008) 064005, [ 0710.2956]

    2008 arXiv

  28. [36]

    Lechtenfeld and S

    O. Lechtenfeld and S. Nampuri, A Calogero formulation for four-dimensional black-hole microstates, Phys. Lett. B 753 (2016) 263–267, [ 1509.03256]

    2016 arXiv

  29. [37]

    Sen, Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates, Gen

    A. Sen, Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates, Gen. Rel. Grav. 44 (2012), no. 5 1207–1266, [ 1108.3842]

    2012 arXiv

  30. [38]

    G. L. Cardoso, A. Kidambi, S. Nampuri, V. Reys, and M. Rossell´ o, The Gravitational Path Integral for N = 4 BPS Black Holes from Black Hole Microstate Counting , Annales Henri Poincare 24 (2023), no. 10 3305–3346, [ 2211.06873]

    2023 arXiv

  31. [39]

    D. V. Vassilevich, Heat kernel expansion: User’s manual , Phys. Rept. 388 (2003) 279–360, [hep-th/0306138]

    2003 arXiv

  32. [40]

    Digital Library of Mathematical Functions. – 35 –

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.