Pith. sign in

REVIEW 3 major objections 5 minor 2 cited by

Strings and near-extremal black holes in theories with large $\mathcal{N}=4$ superconformal symmetry

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A one-loop worldsheet computation on AdS3×S3×S3×S1 recovers the supergravity spectrum and boundary currents, yields a temperature-dependent BPS index for near-extremal BTZ black holes, and suggests non-unique worldsheet CFTs at k=1.

desk verdict Convincing one-loop spectrum match for AdS3 x S3 x S3 x S1, but the near-extremal BTZ results rest on an unproved twist gap that the two-SU(2) spectrum may violate. read the letter →

arxiv 2505.14380 v2 pith:S56L3SFX submitted 2025-05-20 hep-th gr-qc

Sameer Murthy , Mukund Rangamani This is my paper
classification hep-thgr-qc
keywords spectrumblackfunctionholeslargemathbfmathcalpartition
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 studies strings moving in a curved ten-dimensional spacetime made of an AdS3 piece (a negatively curved space with a holographic boundary), two three-spheres, and a circle. The background is special because its low-energy supergravity spectrum contains both the usual short (chiral) multiplets and long (non-chiral) multiplets. The authors compute the one-loop partition function, a sum over string worldsheets, and show that in the semiclassical limit it reproduces the full supergravity spectrum previously derived by harmonic analysis. They also identify how the boundary currents, such as the boundary graviton and the SU(2) currents, appear as zero-mode contributions.

The second part concerns nearly supersymmetric BTZ black holes. Near extremality, quantum effects of nearly-gapless modes localized near the horizon dominate. The authors compute the low-temperature partition function in both the Neveu-Schwarz and Ramond sectors. They recover the known T^{3/2} Schwarzian suppression in the fixed-charge ensemble, plus an exponential damping from the energy carried by R-charge quanta. They then compute the analog of the BPS index for the large N=(4,4) algebra. Unlike smaller supersymmetric algebras, the BPS states in the Ramond sector do not sit at the threshold energy c/12; they carry a spin-dependent excess energy. The resulting index is therefore explicitly temperature dependent, and it predicts the spectral distribution of these states around supersymmetric BTZ black holes.

Finally, the authors examine the limit where the AdS radius equals the string scale (k=1), which is accessible in the RNS formalism only for this compactification.

Extended reading notes

Core claim

The load-bearing claim is that the one-loop worldsheet partition function for strings on AdS3×S3×S3×S1, after modular integration and contour deformation, reproduces the full supergravity spectrum (2.18), including the non-chiral long multiplets, in the semiclassical limit. The key equation is (4.43): f_PF,disc restricted to supergravity equals the sum over ℓ+,ℓ− of |χ^g_L(h_CFT, ℓ+, ℓ−)|^2, which the authors state 'is indeed the expected supergravity spectrum (2.18)'. A second load-bearing claim is that the R-sector trace around BTZ, after soaking up fermion and boson zero modes, yields the temperature-dependent BPS index (5.29), which captures the spectrum of short multiplets around supersymmetric BTZ black holes with energies E_R(j) = 16α/(1+α)^2 (j+1/2)^2 E_gap.

Load-bearing premise

The most fragile load-bearing premise is the vacuum-module dominance of the thermal AdS3 trace for k>1, used throughout Section 5 to drop all non-vacuum states (the ellipsis in (5.5) and (5.11)) and to extract the low-temperature BTZ behavior. This twist-gap assumption is imported from the universal Schwarzian arguments of [13,14], but the large N=4 algebra here has two SU(2) currents and a U(1) current, and the paper does not prove the gap for this specific background. If the twist gap fails, the T^{3/2} suppression, the exponential damping, and the index (5.29) would all need revision. A second structurally distinct fragile premise is the contour deformation and pole-selection rule (4.37), made strict at k=1 in (4.52), which determines which discrete states survive; the non-uniqueness claim for the k=1 theory hinges on this prescription.

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 / 5 minor

Summary. The paper studies the one-loop worldsheet partition function for Type II superstrings on AdS3 × S3 × S3 × S1, a background preserving large N = (4,4) superconformal symmetry. In Section 4 the authors decompose the worldsheet result into discrete and continuous parts and show that the discrete part, restricted to supergravity states, equals the full supergravity spectrum of [9], including its non-chiral long multiplets; the key identity is Eq. (4.43), which reproduces Eq. (2.18). They also show that the zero-mode contribution encodes the boundary supercurrents and that a spectral-flow average recovers the vacuum character of the large N = 4 algebra. In Section 5, by an S-modular transformation to the BTZ background, they derive low-temperature NS and R sector traces, obtaining T^{3/2} canonical suppression, exponential activation factors, and a temperature-dependent BPS index (5.29) for R-sector states with energies E_R(j) = 16α/(1+α)^2 (j+1/2)^2 E_gap. Finally, Section 4.4 analyzes the string-scale limit k=1, finding a spectrum that differs from the tensionless-string spectrum of [18,19], and the authors argue that there may be two distinct consistent worldsheet CFTs at that value of parameters. The paper explicitly assumes that the tree-level contribution S_tree equals the semiclassical gravity answer, and it assumes a twist gap separating the vacuum module from other states in the k>1 thermal-AdS trace.

Significance. If the central claims hold, this is a significant advance: it provides the first one-loop worldsheet derivation of the supergravity spectrum for AdS3 × S3 × S3 × S1, including the unusual non-chiral long multiplets, and it extends near-extremal black hole thermodynamics to theories with large N = 4 superconformal symmetry. The derivation of Section 4 is explicit and the match (4.43) = (2.18) is a strong, non-trivially checked consistency result. The paper also makes concrete falsifiable predictions: the temperature-dependent BPS index (5.29), the energy formula (5.27), and the claimed non-uniqueness of the k=1 worldsheet CFT. However, the black-hole part of the paper rests on assumptions that are stated but not proved, most notably the twist-gap premise and the value of S_tree, so the near-extremal results are conditional on those inputs.

major comments (3)
  1. [§5.1–5.3, Eqs. (5.5), (5.11), (2.17)] The low-temperature near-extremal analysis assumes vacuum-module dominance of the thermal-AdS trace: in Eqs. (5.5) and (5.11) all non-vacuum states are dropped with an ellipsis, justified by 'assuming a twist gap' and by citing the universal Schwarzian arguments of [13,14]. No gap theorem is proved for the large N=4 algebra with two SU(2) currents and a U(1) current. This is load-bearing because the supergravity spectrum contains long multiplets with dimensions (2.17); for example, ℓ_+ = 1/2, ℓ_- = 0 gives h = sqrt(k/k_+ + k/(4k_-)) - 1/2, which can be O(1) or smaller for admissible large-c parameters with k_-/k_+ small. Such states would contribute at temperatures T ~ T_gap, potentially spoiling the T^5 grand-canonical suppression, the canonical T^{3/2} prefactor, and the index (5.29). The paper should either prove the twist gap for this specific background or provide a quantitative estimate showing that the lightest non-vacuum states are exponentially subleading in the relevant regime.
  2. [§4.4, Eq. (4.52)] The pole-selection rule at k=1 is imposed by making the inequalities (4.37) strict, as stated in (4.52), rather than derived from a regulator-independent contour prescription. The conclusion that the k=1 RNS partition function differs from the tensionless string of [18,19] depends on which discrete states survive: with the strict inequalities, the w=0 and w=1 sectors contain no discrete states, while higher w sectors do. The paper notes that the would-be vacuum pole lies on the shifted contour, signalling a merger with the continuum, but this makes the resulting spectrum sensitive to the chosen resolution of the coincidence. An independent justification, such as a limiting procedure k→1^+ with a fixed regulator, or a modular-invariance argument, is needed before the non-uniqueness claim can be considered established.
  3. [§4.2, Eq. (4.18)] The tree-level contribution S_tree is not computed; the paper states 'We will not attempt to evaluate it here, but assume that it is given by the semiclassical gravity answer when needed.' This assumption enters as an overall factor e^{S_tree} in all the Section 5 partition functions, including (5.6), (5.9), (5.20), and (5.29). While the temperature dependence of the results is independent of this factor, the absolute normalization and the claimed consistency with the Schwarzian near-horizon analysis rely on this input. Since this is explicitly admitted, the near-extremal results should be framed as conditional on the semiclassical value of S_tree, and the paper should state clearly which of its conclusions depend on this assumption.
minor comments (5)
  1. [Eq. (3.14)] The S_- spectral-flow image appears to contain a typo: the expression ending 'k_-/2 - ℓ_-+' has a dangling plus sign and likely should read 'k_-/2 - ℓ_- + 1/2' or similar.
  2. [Eq. (5.6)] The exponent in the first line of (5.6) contains an unexplained '+β+' term; either a term is missing after the second 'β' or the symbol should be removed.
  3. [Eq. (5.10)] The quantity E_1 appearing in the exponent is not defined before its use; please define it explicitly or refer to the zero-point-energy contribution described in the text.
  4. [After Eq. (4.48)] The spectral-flow average (4.48) is described as recovering the full vacuum character, but the summation range and convergence properties of the integers n,m are not specified; please clarify whether the sum is over all of Z^2 and whether it converges uniformly for the relevant chemical potentials.
  5. [Eq. (5.3)] The definition T = 1/(iπ) 1/(τ - τ̄) is nonstandard; since T appears throughout Section 5, a cleaner definition, together with an explicit statement of the τ→i∞, τ̄→-i0 limit, would help the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the worldsheet partition function reproduces the supergravity spectrum from the WZW data and pole structure, not by fitting; self-citations are methodological and non-load-bearing.

full rationale

The central derivation is self-contained against external benchmarks. The supergravity spectrum (2.18) is reproduced in (4.43) via the worldsheet-derived conformal weight (4.40), which follows from the sl(2) and su(2) WZW levels and the discrete-pole condition (4.37), not from inserting the supergravity dimension formula (2.17). The character identity (4.42) is representation theory, not an input of the target spectrum. The near-extremal section assumes a twist gap for k>1, citing the external universal Schwarzian results [13,14]; this is a substantive assumption and a correctness risk, but it is not circular, because the cited gap is not established by the present authors and the low-temperature partition function is then computed from the vacuum character rather than fitted. The temperature-dependent R-sector index (5.29) and the energies E_R(j) in (5.27) are extracted from the lattice-sum exponent after the S-transform, not imposed from the unitarity bound (3.3); the bound is used only to interpret the result. The k=1 non-uniqueness discussion is a suggested resolution with external supporting evidence [23], and the main methodological overlap is the companion paper [1] by the same authors, whose techniques are reused but whose results are not assumed. The match to the supergravity spectrum is therefore an independent consistency check, not a reduction of the prediction to its inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The analysis introduces no fitted parameters: the levels k±, the ratio α=k−/k+, and N1 are inputs of the compactification, and the scale E_gap=3/c is defined rather than fitted. The central claims rest on standard domain assumptions of AdS3 string theory, on a contour-deformation prescription inherited from [1,2], and on two admitted ad hoc inputs: the uncomputed tree-level term S_tree and the vacuum-module dominance (twist gap) in Section 5. No new particles, forces, or dimensions are postulated; the suggested second k=1 worldsheet CFT is a theory variant whose independent evidence is cited as concurrent work [23].

assumptions (7)
  • domain assumption The worldsheet is described by the RNS formalism with sl(2) and su(2) super-WZW models, with fermions decoupled into free fermions.
    Standard AdS3 string description; invoked in Section 4.1 to build the torus partition function.
  • domain assumption The spacetime CFT free energy around thermal AdS3 is the exponential of the multi-string free energy.
    Equation (4.18); standard in AdS3/CFT2 but not derived.
  • ad hoc to paper The tree-level contribution S_tree equals the semiclassical gravity answer.
    Admitted in Section 4.2: 'We will not attempt to evaluate it here, but assume that it is given by the semiclassical gravity answer when needed.' Controls the BTZ Boltzmann factor.
  • ad hoc to paper Discrete states are extracted by shifting the ζ-contour and keeping poles satisfying (4.37); for k=1 the inequalities are made strict.
    Sections 4.2 and 4.4; the pole selection determines which discrete states survive and drives the k=1 non-uniqueness claim.
  • domain assumption For k>1 the vacuum module of the large N=4 algebra dominates the thermal AdS3 trace (twist gap).
    Section 5, based on [13,14]; the ellipsis in (5.5) and (5.11) drops all non-vacuum states, and the low-temperature results depend on it.
  • ad hoc to paper The spectral flow average (4.48) recovers the full superconformal character from the thermal AdS3 piece.
    Section 4.3, motivated by [21]; footnote 10 admits this average does not fully justify the Tρ^2 factors.
  • standard math Standard Jacobi theta function identities, Poisson resummation, and modular properties of characters.
    Used throughout, especially Appendix C for the Ramond lattice sums.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Strings and near-extremal black holes in theories with large $\mathcal{N}=4$ superconformal symmetry." pith.science (2026). https://pith.science/paper/S56L3SFX

@misc{pith2026250514380,
  author       = {Pith},
  title        = {Pith review of: Strings and near-extremal black holes in theories with large $\mathcalN=4$ superconformal symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S56L3SFX}},
  note         = {Machine review of arXiv:2505.14380}
}
abstract

We study the one-loop partition function of superstrings in the $\mathrm{AdS}_3 \times \mathbf{S}^3 \times \mathbf{S}^3 \times \mathbf{S}^1$ background. Specifically, we show that the supergravity spectrum, which contains non-chiral primary states unlike other similar $\mathrm{AdS}_3$ backgrounds, can be recovered by this partition function in the semiclassical limit. We also show how the boundary currents are encoded in the string spectrum. Furthermore, we discuss the effect of these boundary currents in the quantum partition function of near-extremal black holes in theories with large $\mathcal{N} = (4,4)$ supersymmetry, recovering results consistent with the analysis of the near-horizon (super-)Schwarzian theory. In particular, we show how the BPS index of the large $\mathcal{N} = (4,4)$ theory, which turns out to be temperature dependent, captures the spectrum of excitations around supersymmetric BTZ black holes. Finally, we comment on the limit when the AdS curvature radius is string scale, which is directly accessible within the RNS formalism for this compactification. The spectrum of the limiting theory we obtain differs from the tensionless string spectrum derived in the literature, suggesting that there is not a unique worldsheet CFT for this value of parameters.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A New Term in Type II Effective Action

    hep-th 2026-07 unverdicted novelty 8.0 of 10

    One-loop type II effective actions contain a dilaton times Euler-density term from superconformal ghost zero modes, resolving a black-hole index puzzle on Calabi-Yau spaces.

  2. Harmonic Superspace for Ali-Ilahi's ADHM Instanton Sigma Model

    hep-th 2025-07 conditional novelty 4.0 of 10

    The authors build a dual harmonic superspace and write the off-shell (0,4) superspace actions, interactions, and ADHM instanton gauge field for the complementary ADHM instanton sigma model.

Reference graph

Works this paper leans on

49 extracted references · 10 canonical work pages · cited by 2 Pith papers

  1. [1]

    Ferko, S

    C. Ferko, S. Murthy and M. Rangamani,Strings in AdS 3: one-loop partition function and near-extremal BTZ thermodynamics,JHEP05(2025) 010 [2408.14567]

    arXiv 2025

  2. [9]

    Tong,The holographic dual ofAdS 3×S 3×S 3×S 1,JHEP04(2014) 193 [1402.5135]

    D. Tong,The holographic dual ofAdS 3×S 3×S 3×S 1,JHEP04(2014) 193 [1402.5135]

    arXiv 2014

  3. [2]

    Maldacena, H

    J.M. Maldacena, H. Ooguri and J. Son,Strings in AdS(3) and the SL(2,R) WZW model. Part

  4. [3]

    Euclidean black hole,J. Math. Phys.42(2001) 2961 [hep-th/0005183]

    arXiv 2001

  5. [4]

    Boonstra, B

    H.J. Boonstra, B. Peeters and K. Skenderis,Brane intersections, anti-de Sitter space-times and dual superconformal theories,Nucl. Phys. B533(1998) 127 [hep-th/9803231]

    arXiv 1998

  6. [5]

    Elitzur, O

    S. Elitzur, O. Feinerman, A. Giveon and D. Tsabar,String theory on AdS(3) x S**3 x S**3 x S**1,Phys. Lett. B449(1999) 180 [hep-th/9811245]

    arXiv 1999

  7. [6]

    de Boer, A

    J. de Boer, A. Pasquinucci and K. Skenderis,AdS / CFT dualities involving large 2-D N=4 superconformal symmetry,Adv. Theor. Math. Phys.3(1999) 577 [hep-th/9904073]

    arXiv 1999

  8. [7]

    Gukov, E

    S. Gukov, E. Martinec, G.W. Moore and A. Strominger,The Search for a holographic dual to AdS(3) x S**3 x S**3 x S**1,Adv. Theor. Math. Phys.9(2005) 435 [hep-th/0403090]. – 42 –

    arXiv 2005

Show all 49 references
  1. [8]

    Gaberdiel and R

    M.R. Gaberdiel and R. Gopakumar,Large N=4 Holography,JHEP09(2013) 036 [1305.4181]

    2013 arXiv

  2. [10]

    Eberhardt, M.R

    L. Eberhardt, M.R. Gaberdiel, R. Gopakumar and W. Li,BPS spectrum on AdS 3×S3×S3×S1, JHEP03(2017) 124 [1701.03552]

    2017 arXiv

  3. [11]

    Eberhardt, M.R

    L. Eberhardt, M.R. Gaberdiel and W. Li,A holographic dual for string theory on AdS3×S3×S3×S1,JHEP08(2017) 111 [1707.02705]

    2017 arXiv

  4. [12]

    Eberhardt and M.R

    L. Eberhardt and M.R. Gaberdiel,Strings on AdS 3×S 3×S 3×S 1,JHEP06(2019) 035 [1904.01585]

    2019 arXiv

  5. [13]

    Witten,Instantons and the largeN= 4algebra,J

    E. Witten,Instantons and the largeN= 4algebra,J. Phys. A58(2025) 035403 [2407.20964]

    2025 arXiv

  6. [14]

    Ghosh, H

    A. Ghosh, H. Maxfield and G.J. Turiaci,A universal Schwarzian sector in two-dimensional conformal field theories,JHEP05(2020) 104 [1912.07654]

    2020 arXiv

  7. [15]

    Iliesiu and G.J

    L.V. Iliesiu and G.J. Turiaci,The statistical mechanics of near-extremal black holes,JHEP05 (2021) 145 [2003.02860]

    2021 arXiv

  8. [16]

    Heydeman, L.V

    M. Heydeman, L.V. Iliesiu, G.J. Turiaci and W. Zhao,The statistical mechanics of near-BPS black holes,J. Phys. A55(2022) 014004 [2011.01953]

    2022 arXiv

  9. [17]

    Heydeman, X

    M. Heydeman, X. Shi and G.J. Turiaci,Can black holes preserveN >4supersymmetry?, 2504.20146

  10. [18]

    and Turiaci, Gustavo J.,Spectrum of BPS black holes in AdS 3 ×S3×S 3×S 1,to appear(2025)

    Heydeman, Matthew and Shi, X. and Turiaci, Gustavo J.,Spectrum of BPS black holes in AdS 3 ×S3×S 3×S 1,to appear(2025)

    2025

  11. [19]

    Gaberdiel and R

    M.R. Gaberdiel and R. Gopakumar,Tensionless string spectra on AdS 3,JHEP05(2018) 085 [1803.04423]

    2018 arXiv

  12. [20]

    Gaberdiel and V

    M.R. Gaberdiel and V. Sriprachyakul,Tensionless strings on AdS 3×S 3×S 3×S 1,JHEP05 (2025) 003 [2411.16848]

    2025 arXiv

  13. [21]

    Eberhardt and I.G

    L. Eberhardt and I.G. Zadeh,N= (3,3)holography onAdS 3×(S 3×S 3×S 1)/Z2,JHEP07 (2018) 143 [1805.09832]

    2018 arXiv

  14. [22]

    Kraus and F

    P. Kraus and F. Larsen,Partition functions and elliptic genera from supergravity,JHEP01 (2007) 002 [hep-th/0607138]

    2007 arXiv

  15. [23]

    Eberhardt, M.R

    L. Eberhardt, M.R. Gaberdiel and R. Gopakumar,The Worldsheet Dual of the Symmetric Product CFT,JHEP04(2019) 103 [1812.01007]

    2019 arXiv

  16. [24]

    Eberhardt and M.R

    L. Eberhardt and M.R. Gaberdiel,A localising AdS 3 sigma model,2505.09226

  17. [25]

    Runkel and G.M.T

    I. Runkel and G.M.T. Watts,A Nonrational CFT with c = 1 as a limit of minimal models, JHEP09(2001) 006 [hep-th/0107118]

    2001 arXiv

  18. [26]

    Roggenkamp and K

    D. Roggenkamp and K. Wendland,Limits and degenerations of unitary conformal field theories, Commun. Math. Phys.251(2004) 589 [hep-th/0308143]

    2004 arXiv

  19. [27]

    Chakraborty, A

    S. Chakraborty, A. Giveon and D. Kutasov,Effective AdS 3/CFT2,JHEP03(2025) 030 [2501.09119]

    2025 arXiv

  20. [28]

    Gukov, E

    S. Gukov, E. Martinec, G.W. Moore and A. Strominger,An Index for 2-D field theories with large N = 4 superconformal symmetry,hep-th/0404023. – 43 –

    arXiv

  21. [29]

    Cecotti, P

    S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa,A New supersymmetric index,Nucl. Phys. B386(1992) 405 [hep-th/9204102]

    1992 arXiv

  22. [30]

    Brown and M

    J.D. Brown and M. Henneaux,Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,Commun. Math. Phys.104(1986) 207

    1986

  23. [31]

    Goddard and A

    P. Goddard and A. Schwimmer,Factoring Out Free Fermions and Superconformal Algebras, Phys. Lett. B214(1988) 209

    1988

  24. [32]

    Henneaux, L

    M. Henneaux, L. Maoz and A. Schwimmer,Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity,Annals Phys.282(2000) 31 [hep-th/9910013]

    2000 arXiv

  25. [33]

    Gunaydin, J.L

    M. Gunaydin, J.L. Petersen, A. Taormina and A. Van Proeyen,On the Unitary Representations of a Class ofN= 4Superconformal Algebras,Nucl. Phys. B322(1989) 402

    1989

  26. [34]

    Petersen and A

    J.L. Petersen and A. Taormina,Characters of theN= 4Superconformal Algebra With Two Central Extensions,Nucl. Phys. B331(1990) 556

    1990

  27. [35]

    Petersen and A

    J.L. Petersen and A. Taormina,Characters of theN= 4Superconformal Algebra With Two Central Extensions: 2. Massless Representations,Nucl. Phys. B333(1990) 833

    1990

  28. [36]

    Giombi, A

    S. Giombi, A. Maloney and X. Yin,One-loop Partition Functions of 3D Gravity,JHEP08 (2008) 007 [0804.1773]

    2008 arXiv

  29. [37]

    David, M.R

    J.R. David, M.R. Gaberdiel and R. Gopakumar,The Heat Kernel on AdS(3) and its Applications,JHEP04(2010) 125 [0911.5085]

    2010 arXiv

  30. [38]

    Maldacena, G.W

    J.M. Maldacena, G.W. Moore and A. Strominger,Counting BPS black holes in toroidal Type II string theory,hep-th/9903163

    arXiv

  31. [39]

    Di Vecchia, V.G

    P. Di Vecchia, V.G. Knizhnik, J.L. Petersen and P. Rossi,A Supersymmetric Wess-Zumino Lagrangian in Two-Dimensions,Nucl. Phys. B253(1985) 701

    1985

  32. [40]

    Gawedzki,Noncompact WZW conformal field theories, inNATO Advanced Study Institute: New Symmetry Principles in Quantum Field Theory, pp

    K. Gawedzki,Noncompact WZW conformal field theories, inNATO Advanced Study Institute: New Symmetry Principles in Quantum Field Theory, pp. 0247–274, 10, 1991 [hep-th/9110076]

    1991 arXiv

  33. [41]

    Atick and E

    J.J. Atick and E. Witten,The Hagedorn Transition and the Number of Degrees of Freedom of String Theory,Nucl. Phys. B310(1988) 291

    1988

  34. [42]

    Zagier,The Rankin-Selberg method for automorphic functions which are not of rapid decay, J

    D. Zagier,The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math.3(1982) 415–

    1982

  35. [43]

    Polchinski,Evaluation of the One Loop String Path Integral,Commun

    J. Polchinski,Evaluation of the One Loop String Path Integral,Commun. Math. Phys.104 (1986) 37

    1986

  36. [44]

    Mertens and G.J

    T.G. Mertens and G.J. Turiaci,Defects in Jackiw-Teitelboim Quantum Gravity,JHEP08 (2019) 127 [1904.05228]

    2019 arXiv

  37. [45]

    Pal and J

    S. Pal and J. Qiao,Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-Like Formula for Near-Extremal Black Holes,Annales Henri Poincare26(2025) 787 [2307.02587]

    2025 arXiv

  38. [46]

    Defever, S

    F. Defever, S. Schrans and K. Thielmans,Moding of Superconformal Algebras,Phys. Lett. B 212(1988) 467

    1988

  39. [47]

    Giveon, D

    A. Giveon, D. Kutasov and N. Seiberg,Comments on string theory on AdS(3),Adv. Theor. Math. Phys.2(1998) 733 [hep-th/9806194]. – 44 –

    1998 arXiv

  40. [48]

    de Boer, H

    J. de Boer, H. Ooguri, H. Robins and J. Tannenhauser,String theory on AdS(3),JHEP12 (1998) 026 [hep-th/9812046]

    1998 arXiv

  41. [49]

    Kutasov and N

    D. Kutasov and N. Seiberg,More comments on string theory on AdS(3),JHEP04(1999) 008 [hep-th/9903219]. – 45 –

    1999 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.