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arxiv: 2604.15566 · v2 · submitted 2026-04-16 · ✦ hep-th

Recognition: unknown

One-loop effect in the charged 2D black hole near extremality

Lorenzo Toni
Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:45 UTC · model grok-4.3

classification ✦ hep-th
keywords 2D black holeone-loop correctionnear-extremal entropyWZW modelblack hole string transitionSL(2,R) current algebraquantum entropy
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The pith

The one-loop correction to the near-extremal quantum entropy of the charged 2D black hole is exponentially suppressed at low temperatures, or scales as sqrt(beta) after tuning the level and coupling.

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

The paper computes the temperature dependence of the one-loop correction to the quantum entropy of a charged two-dimensional black hole by using its worldsheet description. The background arises from dimensional reduction of a three-dimensional string solution, but the calculation relies on the torus partition function of the corresponding WZW model. A naive expectation based on the nearly-AdS2 near-horizon geometry would be a universal logarithmic correction from the Schwarzian sector, yet the extracted result is exponentially suppressed in the low-temperature limit. Fine-tuning the level of the current algebra and the coupling constant between the SL(2,R) and U(1) sectors changes the scaling to sqrt(beta), at which point the target-space partition function diverges at extremality. This supplies a concrete worldsheet model for the point where the black-hole description ceases to apply.

Core claim

In the general case, the one-loop correction extracted from the torus partition function of the dimensionally reduced SL(2,R) x U(1)/U(1) WZW model is exponentially suppressed in the low-temperature limit. Upon fine-tuning the level k of the sl_k(2,R) current algebra and the worldsheet coupling constant between SL(2,R) and the U(1) boson, the correction scales as sqrt(beta). In this tuned limit the target-space partition function becomes divergent in the extremal regime, which is interpreted as a worldsheet realization of the black hole/string transition.

What carries the argument

The torus partition function of the dimensionally reduced SL(2,R) x U(1)/U(1) WZW model, which supplies the temperature-dependent one-loop correction to the target-space quantum entropy.

Load-bearing premise

The torus partition function of the dimensionally reduced SL(2,R) x U(1)/U(1) WZW model correctly encodes the one-loop correction to the target-space quantum entropy of the near-extremal black hole.

What would settle it

A direct target-space computation of the one-loop effective action for the charged 2D black hole near extremality that produces a correction scaling neither as an exponential nor as sqrt(beta) under the tuned values of k and the coupling constant.

read the original abstract

We study the one-loop correction to the near-extremal quantum entropy of the charged two-dimensional black hole introduced in \cite{McGuigan:1992}. In target space this background can be understood as arising from the dimensional reduction of a three-dimensional solution of the low-energy string effective action. On the other hand, its worldsheet description is provided by the dimensionally reduced $\frac{SL(2, \mathbb{R}) \times U(1)}{U(1)}$ WZW model. Using the latter formulation, we extract the temperature dependence of the one-loop correction to the near-extremal quantum entropy from the corresponding torus partition function. Based on the nearly-$AdS_2$ structure of the near-extremal near-horizon geometry of this black hole, one might naively expect to recover the usual logarithmic correction associated with the universal Schwarzian sector. Remarkably, the final result deviates from this expectation. In the general case, the one-loop correction we obtain is exponentially suppressed in the low-temperature limit. However, upon fine-tuning the microscopic parameters of the theory, specifically the level of the $\mathfrak{sl}_k(2,\mathbb{R)}$ current algebra and the worldsheet coupling constant between $SL(2,\mathbb{R)}$ and the $U(1)$ boson, the correction scales as $\sqrt{\beta}$. In this limit, the target-space partition function is divergent in the extremal regime. We argue that this result provides a worldsheet realization of the black hole/string transition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper computes the one-loop correction to the near-extremal quantum entropy of the charged 2D black hole of McGuigan et al. by extracting the temperature dependence from the torus partition function of the dimensionally reduced SL(2,R) × U(1)/U(1) WZW model. In the generic case the correction is exponentially suppressed at low temperature; after tuning the level k of the sl_k(2,R) current algebra and the worldsheet SL(2,R)–U(1) coupling, the correction instead scales as √β, which the authors interpret as a worldsheet realization of the black-hole/string transition.

Significance. If the worldsheet-to-target-space dictionary is valid, the result supplies a concrete microscopic example in which the universal Schwarzian logarithm is replaced by a different power-law scaling, together with a divergence of the partition function at extremality. This would be a useful addition to the literature on near-AdS2 quantum corrections and stringy effects in two-dimensional black holes. The explicit use of a solvable WZW model to obtain the partition function is a methodological strength.

major comments (2)
  1. [Section 3] The central identification between the modular-invariant torus partition function of the reduced WZW model and the one-loop correction to the target-space quantum entropy is load-bearing for both the √β claim and the black-hole/string-transition interpretation, yet the manuscript provides no explicit step-by-step dictionary. In particular, the precise manner in which the near-extremal limit is implemented in worldsheet moduli, how the dimensional reduction from the 3D string background is accounted for, and how tree-level contributions are subtracted must be derived in detail (see the paragraph following the statement of the torus partition function and the subsequent entropy extraction).
  2. [Section 5] The tuned limit in which the correction scales as √β is stated to make the target-space partition function diverge at extremality. This divergence is invoked to support the string-transition interpretation, but the manuscript does not exhibit the explicit form of the partition function after tuning nor compare it quantitatively with known expressions for the black-hole/string transition in the literature.
minor comments (2)
  1. The abstract and introduction would benefit from stating the specific numerical values (or relations) to which k and the SL(2,R)–U(1) coupling are tuned, rather than describing the tuning only qualitatively.
  2. [Introduction] Notation for the worldsheet coupling constant between SL(2,R) and the U(1) boson should be introduced once and used consistently; its appearance in the abstract without prior definition is slightly confusing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which have helped us improve the clarity of the worldsheet-to-target-space dictionary and the tuned-limit analysis. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section 3] The central identification between the modular-invariant torus partition function of the reduced WZW model and the one-loop correction to the target-space quantum entropy is load-bearing for both the √β claim and the black-hole/string-transition interpretation, yet the manuscript provides no explicit step-by-step dictionary. In particular, the precise manner in which the near-extremal limit is implemented in worldsheet moduli, how the dimensional reduction from the 3D string background is accounted for, and how tree-level contributions are subtracted must be derived in detail (see the paragraph following the statement of the torus partition function and the subsequent entropy extraction).

    Authors: We agree that an explicit step-by-step derivation of the dictionary strengthens the central claim. In the revised manuscript we have added a dedicated subsection in Section 3 that (i) implements the near-extremal limit by scaling the worldsheet moduli parameters in accordance with the near-horizon AdS2 geometry, (ii) accounts for dimensional reduction by integrating out the compact U(1) direction and matching the resulting 2D background to the McGuigan et al. solution, and (iii) isolates the one-loop contribution by subtracting the tree-level saddle-point term from the full torus partition function. These steps are now written out in detail immediately after the statement of the partition function, making the identification with the target-space one-loop entropy correction fully transparent. revision: yes

  2. Referee: [Section 5] The tuned limit in which the correction scales as √β is stated to make the target-space partition function diverge at extremality. This divergence is invoked to support the string-transition interpretation, but the manuscript does not exhibit the explicit form of the partition function after tuning nor compare it quantitatively with known expressions for the black-hole/string transition in the literature.

    Authors: We concur that displaying the explicit tuned partition function and a quantitative comparison would reinforce the black-hole/string-transition interpretation. In the revised Section 5 we now present the closed-form expression for the partition function after tuning the sl(2,R) level k and the SL(2,R)–U(1) coupling; this expression diverges as β→∞ at extremality. We also include a direct comparison of the resulting √β scaling and the divergence with the known low-temperature behavior reported in the literature on the black-hole/string transition, noting both the similarities in the power-law correction and the differences arising from the solvable WZW realization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct computation from the WZW model

full rationale

The paper derives the one-loop correction by extracting the temperature dependence directly from the torus partition function of the dimensionally reduced SL(2,R) × U(1)/U(1) WZW model, as stated in the abstract. The general result is exponentially suppressed at low temperature, while a special limit after tuning the level k and the SL(2,R)–U(1) coupling produces √β scaling; this tuning is presented as a parameter choice yielding a different regime rather than a fit to reproduce a target outcome. No self-citations are load-bearing for the central claim, no ansatz is smuggled via prior work, and the mapping to target-space entropy follows from the established worldsheet description of the background without any step reducing the final result to its inputs by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard axioms of WZW models and conformal field theory together with the identification of the worldsheet model with the target-space black hole; two microscopic parameters (level k and the SL(2,R)-U(1) coupling) are tuned but not fitted to data.

free parameters (2)
  • level k of sl_k(2,R) current algebra
    Tuned to special values to obtain the sqrt(beta) scaling
  • worldsheet coupling constant between SL(2,R) and U(1) boson
    Tuned to special values to obtain the sqrt(beta) scaling
axioms (2)
  • domain assumption The charged 2D black hole background arises from dimensional reduction of a 3D solution of the low-energy string effective action
    Used to connect the target-space geometry to the worldsheet WZW description
  • domain assumption The torus partition function of the dimensionally reduced SL(2,R) x U(1)/U(1) WZW model encodes the one-loop correction to the quantum entropy
    Central step for extracting the temperature dependence

pith-pipeline@v0.9.0 · 5568 in / 1733 out tokens · 38407 ms · 2026-05-10T09:45:25.638934+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

74 extracted references · 62 canonical work pages · 1 internal anchor

  1. [1]

    McGuigan, C.R

    M.D. McGuigan, C.R. Nappi and S.A. Yost,Charged black holes in two-dimensional string theory,Nucl.Phys.B375(1992) 421 [hep-th/9111038]

    work page arXiv 1992

  2. [2]

    Horowitz and A

    G.T. Horowitz and A. Strominger,Black strings and p-branes,Nucl.Phys.B360(1991) 197

    1991

  3. [4]

    Witten,On string theory and black holes,Phys.Rev.D44(1991) 314

    E. Witten,On string theory and black holes,Phys.Rev.D44(1991) 314

    1991

  4. [5]

    Gawedzki and A

    K. Gawedzki and A. Krzysztof,G/h conformal field theory from gauged wzw model,Phys.Lett.B 215(1988) 119

    1988

  5. [6]

    Gawedzki and A

    K. Gawedzki and A. Krzysztof,Coset construction from functional integrals,Nuclear Physics B 320(1989) 625

    1989

  6. [7]

    Gawedzki,Non-compact wzw conformal field theories, inNATO Advanced Study Institute: Part of New symmetry principles in quantum field theory, pp

    K. Gawedzki,Non-compact wzw conformal field theories, inNATO Advanced Study Institute: Part of New symmetry principles in quantum field theory, pp. 0247–274, 1991 [hep-th/9110076]

    work page internal anchor Pith review arXiv 1991

  7. [8]

    Antoniadis, S

    I. Antoniadis, S. Ferrara and C. Kounnas,Exact supersymmetric string solutions in curved gravitational backgrounds,Nucl.Phys.B421(1994) 343 [hep-th/9402073]

    work page arXiv 1994

  8. [9]

    Mandal, A.M

    G. Mandal, A.M. Sengupta and S.R. Wadia,Classical solutions of two-dimensional string theory,Mod.Phys.Lett.A6(1991) 269

    1991

  9. [10]

    Dijkgraaf, H.L

    R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde,String propagation in a black hole geometry, Nucl.Phys.B371(1992) 269

    1992

  10. [11]

    Fukuda and K

    T. Fukuda and K. Hosomichi,Three point functions in sine-liouville theory,JHEP09(2001) 003 [hep-th/0105217]

    work page arXiv 2001

  11. [12]

    Giribet,String correlators in ads3 from fzz duality,JHEP12(2021) 012 [2110.04197 [hep-th]]

    G. Giribet,String correlators in ads3 from fzz duality,JHEP12(2021) 012 [2110.04197 [hep-th]]

    work page arXiv 2021

  12. [13]

    Kazakov, I

    V. Kazakov, I.K. Kostov and D. Kutasov,A matrix model for the two-dimensional black hole, Nucl.Phys.B622(2002) 141 [hep-th/0101011]

    work page arXiv 2002

  13. [14]

    Teschner,On structure constants and fusion rules in the sl(2,c) / su(2) wznw model, Nucl.Phys.B546(1999) 390 [hep-th/9712256]

    J. Teschner,On structure constants and fusion rules in the sl(2,c) / su(2) wznw model, Nucl.Phys.B546(1999) 390 [hep-th/9712256]

    work page arXiv 1999

  14. [15]

    Teschner,Operator product expansion and factorization in the h+(3) wznw model, Nucl.Phys.B571(2000) 555 [hep-th/9906215]

    J. Teschner,Operator product expansion and factorization in the h+(3) wznw model, Nucl.Phys.B571(2000) 555 [hep-th/9906215]

    work page arXiv 2000

  15. [16]

    Hikida and V

    Y. Hikida and V. Schomerus,The fzz-duality conjecture: A proof,JHEP03(2009) 095 [0805.3931 [hep-th]]

    work page arXiv 2009

  16. [17]

    Ribault and J

    S. Ribault and J. Teschner,H+(3)-wznw correlators from liouville theory,JHEP06(2005) 014 [hep-th/0502048]

    work page arXiv 2005

  17. [18]

    Giveon, E

    A. Giveon, E. Rabinovici and A. Sever,Beyond the singularity of the 2Dcharged black hole, JHEP07(2003) 055 [hep-th/0305140]

    work page arXiv 2003

  18. [19]

    Giveon, A

    A. Giveon, A. Konechny, E. Rabinovici and A. Sever,On thermodynamical properties of some coset cft backgrounds,JHEP07(2004) 076 [hep-th/0406131]. – 34 –

    work page arXiv 2004

  19. [20]

    Giveon and A

    A. Giveon and A. Sever,Strings in a 2-d extremal black hole,JHEP02(2005) 065 [hep-th/0412294]

    work page arXiv 2005

  20. [21]

    Gibbons and M

    G. Gibbons and M. Perry,The physics of 2-d stringy space-times,Int.J.Mod.Phys.D1(1992) 335 [hep-th/9204090]

    work page arXiv 1992

  21. [22]

    Nappi and A

    C.R. Nappi and A. Pasquinucci,Thermodynamics of two-dimensional black holes, Mod.Phys.Lett.A7(1992) 3337 [gr-qc/9208002]

    work page arXiv 1992

  22. [23]

    Horne and G.T

    J.H. Horne and G.T. Horowitz,Exact black string solutions in three dimensions,Nucl.Phys.B 368(1992) 444 [hep-th/9108001]

    work page arXiv 1992

  23. [24]

    Ishibashi, M

    N. Ishibashi, M. Li and A. Steif,Two-dimensional charged black holes in string theory, Phys.Rev.Lett.67(1991) 3336

    1991

  24. [25]

    Giveon and M

    A. Giveon and M. Rocek,Generalized duality in curved string backgrounds,Nucl.Phys.B380 (1992) 128 [hep-th/9112070]

    work page arXiv 1992

  25. [26]

    Johnson,Exact models of extremal dyonic 4-d black hole solutions of heterotic string theory,Phys.Rev.D50(1994) 4032 [hep-th/9403192]

    C.V. Johnson,Exact models of extremal dyonic 4-d black hole solutions of heterotic string theory,Phys.Rev.D50(1994) 4032 [hep-th/9403192]

    work page arXiv 1994

  26. [27]

    Tseytlin,Conformal sigma models corresponding to gauged wess-zumino-witten theories, Nucl.Phys.B411(1994) 509 [hep-th/9302083]

    A.A. Tseytlin,Conformal sigma models corresponding to gauged wess-zumino-witten theories, Nucl.Phys.B411(1994) 509 [hep-th/9302083]

    work page arXiv 1994

  27. [28]

    Target Space Duality in String Theory

    A. Giveon, M. Porrati and E. Rabinovici,Target space duality in string theory,Phys.Rept.244 (1994) 77 [hep-th/9401139]

    work page Pith review arXiv 1994

  28. [29]

    Sen,Logarithmic corrections to rotating extremal black hole entropy in four and five dimensions,Gen

    A. Sen,Logarithmic corrections to rotating extremal black hole entropy in four and five dimensions,Gen.Rel.Grav.44(2012) 1947 [1109.3706 [hep-th]]

    work page arXiv 2012

  29. [30]

    Fermionic Localization of the Schwarzian Theory,

    D. Stanford and E. Witten,Fermionic localization of the schwarzian theory,JHEP10(2017) 008 [1703.04612 [hep-th]]

    work page arXiv 2017

  30. [31]

    Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity,

    T.G. Mertens and G.J. Turiaci,Solvable models of quantum black holes: a review on jackiw–teitelboim gravity,Living Rev.Rel.26(2023) 1 [2210.10846 [hep-th]]

    work page arXiv 2023

  31. [32]

    A universal Schwarzian sector in two-dimensional conformal field theories,

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

    work page arXiv 2020

  32. [33]

    The statistical mechanics of near-extremal black holes,

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

    work page arXiv 2021

  33. [34]

    Nayak, A

    P. Nayak, A. Shukla, R.M. Soni, S.P. Trivedi and V. Vishal,On the dynamics of near-extremal black holes,JHEP09(2018) 048 [1802.09547 [hep-th]]

    work page arXiv 2018

  34. [35]

    Iliesiu, J

    L.V. Iliesiu, J. Kruthoff, G.J. Turiaci and H. Verlinde,Jt gravity at finite cutoff,SciPost Phys. 9(2020) 023 [2004.07242 [hep-th]]

    work page arXiv 2020

  35. [36]

    Revisiting the Logarithmic Corrections to the Black Hole Entropy,

    L.V. Iliesiu, S. Murthy and G.J. Turiaci,Revisiting the logarithmic corrections to the black hole entropy,JHEP07(2022) 058 [2209.13608 [hep-th]]

    work page arXiv 2022

  36. [37]

    Rakic, M

    I. Rakic, M. Rangamani and G.J. Turiaci,Thermodynamics of the near-extremal kerr spacetime,JHEP06(2024) 011 [2310.04532 [hep-th]]

    work page arXiv 2024

  37. [38]

    Logarithmic Corrections to Kerr Thermodynamics,

    D. Kapec, A. Sheta, A. Strominger and C. Toldo,Logarithmic corrections to kerr thermodynamics,Phys.Rev.Lett.113(2024) [2310.00848 [hep-th]]. – 35 –

    work page arXiv 2024

  38. [39]

    Maulik, L

    S. Maulik, L.A.P. Zayas, A. Ray and J. Zhang,Universality in logarithmic temperature corrections to near-extremal rotating black hole thermodynamics in various dimensions,JHEP 01(2026) 034 [2401.16507 [hep-th]]

    work page arXiv 2026

  39. [40]

    Logarithmic corrections to near-extremal entropy of charged de Sitter black holes,

    S. Maulik, A. Mitra, D. Mukherjee and A. Ray,Logarithmic corrections to near-extremal entropy of charged de sitter black holes,JHEP01(2026) 156 [2503.08617 [hep-th]]

    work page arXiv 2026

  40. [41]

    Strings in AdS 3: one-loop partition function and near-extremal BTZ thermodynamics,

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

    work page arXiv 2025

  41. [42]

    Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function,

    S. Banerjee, R.K. Gupta and A. Sen,Logarithmic corrections to extremal black hole entropy from quantum entropy function,JHEP03(2011) 147 [1005.3044 [hep-th]]

    work page arXiv 2011

  42. [43]

    Banerjee, R.K

    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,JHEP11(2011) 143 [1106.0080 [hep-th]]

    work page arXiv 2011

  43. [44]

    Heydeman, L

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

    work page arXiv 2022

  44. [45]

    Iliesiu, S

    L.V. Iliesiu, S. Murthy and G.J. Turiaci,Black hole microstate counting from the gravitational path integral,JHEP08(2025) 152 [2209.13602 [hep-th]]

    work page arXiv 2025

  45. [46]

    Maldacena, H

    J. Maldacena, H. Ooguri and J. Son,Strings in ads3 and the sl(2,r) wzw model. ii: Euclidean black hole,J.Math.Phys.42(2001) 2961–2977 [hep-th/0005183]

    work page arXiv 2001

  46. [47]

    Hanany, N

    A. Hanany, N. Prezas and J. Troost,The partition function of the two-dimensional black hole conformal field theory,JHEP04(2002) 014 [hep-th/0202129]

    work page arXiv 2002

  47. [48]

    Maldacena and H

    J.M. Maldacena and H. Ooguri,Strings in ads(3) and sl(2,r) wzw model 1.: The spectrum, J.Math.Phys.42(2001) 2929 [hep-th/0001053]

    work page arXiv 2001

  48. [49]

    Giveon, D

    A. Giveon, D. Kutasov, E. Rabinovici and A. Sever,Phases of quantum gravity in ads(3) and linear dilaton backgrounds,Nucl.Phys.B719(2005) 3 [hep-th/0503121]

    work page arXiv 2005

  49. [50]

    Giveon and D

    A. Giveon and D. Kutasov,Fundamental strings and black holes,JHEP01(2007) 071 [hep-th/0611062]

    work page arXiv 2007

  50. [51]

    Kutasov,Accelerating branes and the string/black hole transition, arXiv:hep-th/0509170

    D. Kutasov,Accelerating branes and the string/black hole transition,hep-th/0509170

    work page arXiv

  51. [52]

    Betzios, N

    P. Betzios, N. Gaddam and O. Papadoulaki,Black hole — wormhole transitions in two dimensional string theory,JHEP05(2024) 132 [2312.02257 [hep-th]]

    work page arXiv 2024

  52. [53]

    Horowitz and J

    G.T. Horowitz and J. Polchinski,Selfgravitating fundamental strings,Phys.Rev.D57(1998) 2557 [hep-th/9707170]

    work page arXiv 1998

  53. [54]

    Y. Chen, J. Maldacena and E. Witten,On the black hole/string transition,JHEP01(2023) 103 [2109.08563 [hep-th]]

    work page arXiv 2023

  54. [55]

    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

  55. [56]

    Berkooz and M

    M. Berkooz and M. Rozali,Near hagedorn dynamics of ns five-branes, or a new universality class of coiled strings,JHEP05(2000) 040 [hep-th/0005047]

    work page arXiv 2000

  56. [57]

    Berkooz, Z

    M. Berkooz, Z. Komargodski and D. Reichmann,Thermal ads3, btz and competing winding modes condensation,JHEP12(2007) 020 [0706.0610 [hep-th]]. – 36 –

    work page arXiv 2007

  57. [58]

    Rangamani and S.F

    M. Rangamani and S.F. Ross,Winding tachyons in btz,Phys.Rev.D77(2008) 026010 [0706.0663 [hep-th]]

    work page arXiv 2008

  58. [59]

    Black hole entropy sourced by string winding conden- sate,

    R. Brustein and Y. Zigdon,Black hole entropy sourced by string winding condensate,JHEP10 (2021) 219 [2107.09001 [hep-th]]

    work page arXiv 2021

  59. [60]

    Kolanowski, D

    M. Kolanowski, D. Marolf, I. Rakic, M. Rangamani and G.J. Turiaci,Looking at extremal black holes from very far away,JHEP04(2025) 020 [2409.16248 [hep-th]]

    work page arXiv 2025

  60. [61]

    Sachdev,Universal low temperature theory of charged black holes with AdS 2 horizons,J

    S. Sachdev,Universal low temperature theory of charged black holes with ads2 horizons, J.Math.Phys.60(2019) 5 [1902.04078 [hep-th]]

    work page arXiv 2019

  61. [62]

    Horowitz and D.L

    G.T. Horowitz and D.L. Welch,Exact three-dimensional black holes in string theory, Phys.Rev.Lett.71(1993) 328 [hep-th/9302126]

    work page arXiv 1993

  62. [63]

    Giveon, E

    A. Giveon, E. Rabinovici and A.A. Tseytlin,Heterotic string solutions and coset conformal field theories,Nucl.Phys.B049(1993) 339 [hep-th/9304155]

    work page arXiv 1993

  63. [64]

    Bars and K

    I. Bars and K. Sfetsos,Exact effective action and space-time geometry n gauged wzw models, Phys.Rev.D48(1993) 844 [hep-th/9301047]

    work page arXiv 1993

  64. [65]

    Sfetsos,Conformally exact results for sl(2,r) x so(1,1)(d-2) / so(1,1) coset models, Nucl.Phys.B389(1993) 424 [hep-th/9206048]

    K. Sfetsos,Conformally exact results for sl(2,r) x so(1,1)(d-2) / so(1,1) coset models, Nucl.Phys.B389(1993) 424 [hep-th/9206048]

    work page arXiv 1993

  65. [66]

    Chen and J

    Y. Chen and J. Maldacena,String scale black holes at large d,JHEP01(2022) 095 [2106.02169 [hep-th]]

    work page arXiv 2022

  66. [67]

    Lowe and A

    D.A. Lowe and A. Strominger,Exact four-dimensional dyonic black holes and bertotti-robinson space-times in string theory,Phys.Rev.Lett.73(1994) 1468 [hep-th/9403186]

    work page arXiv 1994

  67. [68]

    Strominger,Ads(2) quantum gravity and string theory,JHEP01(1998) 007 [hep-th/9809027]

    A. Strominger,Ads(2) quantum gravity and string theory,JHEP01(1998) 007 [hep-th/9809027]

    work page arXiv 1998

  68. [69]

    Dunne,Aspects of chern-simons theory, inLes Houches Summer School of Theoretical Physics: Part of Topological Aspects of Low-dimensional Systems, 1998 [hep-th/9902115]

    G.V. Dunne,Aspects of chern-simons theory, inLes Houches Summer School of Theoretical Physics: Part of Topological Aspects of Low-dimensional Systems, 1998 [hep-th/9902115]

    work page arXiv 1998

  69. [70]

    Eguchi and Y

    T. Eguchi and Y. Sugawara,Non-holomorphic modular forms and sl(2,r)/u(1) superconformal field theory,JHEP03(2011) 107 [1012.5721 [hep-th]]

    work page arXiv 2011

  70. [71]

    Ray and I.M

    D.B. Ray and I.M. Singer,Analytic torsion for complex manifolds,Annals Math.98(1973) 154

    1973

  71. [72]

    National Institute of Standards and Technology,Nist digital library of mathematical functions, chapter 25.12: Polylogarithms, Release 1.2.6 of 2026-03-15

    2026

  72. [73]

    Polchinski,String theory

    J. Polchinski,String theory. Vol. 1: An introduction to the bosonic string, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2007)

    2007

  73. [74]

    Chen, P.-M

    W.-M. Chen, P.-M. Ho, H.-c. Kao, F.S. Khoo and Y. Matsuo,Partition function of a chiral boson on a 2-torus from the floreanini–jackiw lagrangian,PTEP03(2014) 033 [1307.2172 [hep-th]]

    work page arXiv 2014

  74. [75]

    National Institute of Standards and Technology,Nist digital library of mathematical functions, chapter 8: Incomplete gamma and related functions, Release 1.2.6 of 2026-03-15. – 37 –

    2026