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REVIEW 3 major objections 4 minor 44 references

A free scalar field in AdS3 is a tower of Wilson loops, one for each multi-trace primary, and the same tower accounts for one-loop determinants on both thermal AdS3 and the BTZ black hole.

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2026-08-01 07:32 UTC pith:METMXC6F

load-bearing objection Thermal AdS3 half is a careful and correct rewrite; the BTZ 'microscopic' claim is a modular-covariance dressed as a derivation and rests on companion paper [9] 'to appear'. the 3 major comments →

arxiv 2607.21397 v1 pith:METMXC6F submitted 2026-07-23 hep-th

Wilson Towers as Local Bulk Fields

Vishal Gayari , Chethan Krishnan This is my paper
classification hep-th
keywords Wilson spoolmulti-trace primariesAdS3/CFT2one-loop partition functionVirasoro charactersBTZ handlebodyVerlinde linesWilson towers
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the Wilson spool—the multi-winding Wilson-loop sum that reproduces one-loop determinants of free fields in three-dimensional anti-de Sitter space—is not the fundamental bookkeeping. Using symmetric-function identities, it rewrites the spool as a tower of single-winding Wilson loops, one per multi-trace primary in the dual CFT: the bulk free field is literally a collection of Wilson lines, one per particle sector. It then shows that boundary gravitons dress each global character into a full Virasoro character, so the tower slots directly into a CFT partition function. On the BTZ black-hole handlebody, the same logic gives a microscopic derivation of the one-loop determinant: a light probe on a heavy primary is a Verlinde line, and summing heavy primaries with the Cardy density exchanges the thermal and spatial cycles. If correct, this unifies local bulk fields with the Wilson-network picture of quantum gravity and supports the view that a smooth horizon is an ensemble of heavy Wilson-line microstates.

Core claim

The paper's central claim is that the one-loop partition function of a free scalar on thermal AdS3—the plethystic exponential of the single-particle global character—admits a reorganization in which every multi-particle sector is a sum over single-winding Wilson loops labeled by multi-trace primaries, with multiplicities fixed by symmetric-function counting. Each such loop evaluates to a global SL(2)-character; multiplying by the universal vacuum Virasoro character promotes each module to the full Virasoro character. On the BTZ handlebody, a light probe Wilson line around a heavy primary is the defect-twined trace of a Verlinde line with eigenvalue S_{aP}/S_{0P}, and summing over the heavy s

What carries the argument

The central object is the Wilson tower: a bulk generalized free field is a sum of single-winding Wilson loops, one per multi-trace primary, instead of multi-winding loops in a fixed representation. The bookkeeping is carried by symmetric-function identities—the cycle-index formula, plethystic exponentiation, and Schur-function (Frobenius–Schur–Weyl) decompositions—which convert the multi-winding trace into a sum over higher-representation single-winding loops. On the BTZ side, the key mechanism is the Verlinde line: a light topological defect D_a inserted on a heavy primary P acts by the modular S-matrix eigenvalue S_{aP}/S_{0P}, and the vacuum row S_{0P} supplies the heavy-primary Cardy den

Load-bearing premise

The BTZ result rests on the assumption, deferred to a companion paper, that the UV-complete bulk description of AdS3/CFT2 is a TQFT with Wilson lines, so that a heavy Virasoro primary is literally a Wilson line and a light probe is a Verlinde line with monodromy eigenvalue S_{aP}/S_{0P}.

What would settle it

Compute the defect-twined trace Tr_{V_P}(D_a q^{L_0-c/24}) for a light defect line on a heavy primary in an exactly solvable holographic CFT; if the eigenvalue is not S_{aP}/S_{0P}, the BTZ spool interpretation fails. A simpler spectral test: check whether the double-trace primary spectrum at N=2 obeys the claimed parity rule d=1 for m congruent to \bar m modulo 2.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every free bulk field has a precise CFT avatar as a tower of Virasoro-dressed single-winding Wilson loops; the one-loop determinant is a sum of full Virasoro characters, one per multi-trace primary.
  • The multiplicities of multi-trace primaries are fixed by symmetric-function counting, so the generalized-free-field spectrum follows from the single-particle character rather than being an independent input.
  • On the BTZ handlebody, the one-loop determinant is a Cardy-density sum over heavy primaries of defect-twined characters; the spatial loop of the probe emerges from summing thermal-channel monodromies.
  • The smooth BTZ horizon is represented as a coarse-grained ensemble of heavy Wilson-line microstates; the probe one-loop determinant is a sum over that ensemble.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: for a CFT with several light single-trace fields, the same plethystic argument should produce one Wilson tower per field, and cross-field mixings could be checked against the Schur-function channel structure.
  • The BTZ derivation suggests a CFT-level check: the twined partition function of a light defect on a heavy primary should equal the dual-channel light character, a statement that could be probed in exactly solvable holographic CFTs.
  • If the TQFT-skeleton program is completed, this paper implies that coupling matter to gravity in AdS3 is not an added interaction but a choice of which primary towers exist; the equivalence principle becomes a property of Wilson-network junctions.
  • The Virasoro-dressing step is universal, so the tower construction should extend to spinning fields and supersymmetric or higher-spin analogues by replacing the seed global character.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims that the Wilson spool representation of the one-loop scalar partition function on thermal AdS3 can be rewritten as a sum over single-winding Wilson loops, one for each multi-trace primary, and that this 'Wilson tower' is the bulk description of a generalized free field. Sections 2 and 3 develop the symmetric-function decomposition: the fixed-N multi-particle characters are decomposed into global SL(2) characters labeled by multi-trace primaries, and a universal boundary-graviton factor promotes each global module to a Virasoro module. Section 4 extends the picture to the BTZ handlebody, where a light probe on a heavy primary is represented by a Verlinde-line eigenvalue and summing over heavy primaries with the Cardy density yields the cycle-exchanged light character. The paper is explicit that this last step is natural only if one accepts the TQFT-skeleton premise developed in a companion paper 'to appear'.

Significance. The thermal-AdS3 half is a clean and useful piece of bookkeeping. The explicit decomposition of the spool into single-winding loops labeled by multi-trace primaries, with detailed counts and honest primary constructions for N=2,3 in Appendices A and B, is a genuine contribution that will be useful for further work on Wilson-line descriptions of matter in AdS3/CFT2. The BTZ half is suggestive and modular-covariant, but it is not an independent derivation; its value is entirely conditional on the TQFT-skeleton framework promised in the companion paper. If that framework is supplied, the BTZ interpretation could be significant; as the manuscript stands, the advertised 'microscopic description' of BTZ one-loop determinants is a reinterpretation of standard modular S-transforms.

major comments (3)
  1. [§4.3, Eqs. (4.7)–(4.8)] The derivation of the defect-twined sum Z^tw_a(τ) = χ_a(-1/τ) is the modular S-transform identity combined with the Verlinde-line eigenvalue S_{aP}/S_{0P}. The physical identification of a heavy Virasoro primary with a Wilson line and a light probe with a topological Verlinde line is assumed from the TQFT skeleton of the companion paper [9], which is 'to appear'. The manuscript itself concedes in §4.3 that this is mysterious from a bulk EFT point of view and requires granting the TQFT premise. Since the abstract presents the BTZ result as a 'microscopic' description, this load-bearing premise must either be proved here or the BTZ section and abstract must be explicitly reframed as conditional on [9].
  2. [§4.3, 'structure of that step is the same as in thermal AdS3'] The step from the twined character to the full one-loop determinant on BTZ is explicitly skipped. The paper says the plethystic exponentiation has the same structure as in thermal AdS3 and will not be repeated. But for BTZ the geometry, the contractible cycle, and the role of the heavy-state ensemble are different, so the advertised 'microscopic' one-loop determinant is not actually shown. Either provide the details of this step or state that the BTZ one-loop determinant is only obtained by analogy at the level of an interpretation.
  3. [Appendix A, Eq. (A.9)] The general multiplicity formula d^{(N)}_{m,\bar m} is obtained by replacing the true primary condition ker(L^tot_1) ∩ ker(\bar L^tot_1) with the zero-sum subspace V_N, with the claim that the two spaces 'always have the same dimension'. This is asserted, not proved. The explicit construction in Appendix B shows that the zero-sum representative is not the honest primary already at N=3, level (1,1), so the dimension equality is nontrivial. Since Eq. (2.31) is stated for general N, the paper should either supply a representation-theoretic proof or explicitly restrict the general-N claim to the dimension-counting statement and verify it for the cases actually needed.
minor comments (4)
  1. [§3, Eq. (3.4)] The relation between global and Virasoro characters assumes generic weights with no null states. For light multi-trace primaries in the large-c generalized-free limit this is reasonable, but the manuscript should state the genericity assumption explicitly, especially since the abstract says 'correctly reproduce the full Virasoro character of each module'.
  2. [§2.1, Eq. (2.13)] The notation χ1(q2, \bar q2) is introduced in the expansion but the shorthand χ1(q2) is used without the anti-holomorphic argument. This is clear from context but should be made uniform to avoid confusion in a paper whose main point is the chiral/anti-chiral pairing.
  3. [§4.3, grammar] The sentence 'this was our motivation is using the TQFT language' contains a grammatical error and should be rewritten.
  4. [References [9], [10]] The companion papers [9] and [10] are listed as 'to appear'. Since the BTZ argument in Section 4 depends on [9], the manuscript should flag clearly which results are dependent on unpublished work, both in the introduction and at the point of use.

Circularity Check

2 steps flagged

Thermal-AdS3 Wilson-tower rewrite is self-contained; the BTZ 'microscopic' claim is partly imported from the authors' companion TQFT-skeleton paper [9] and partly a relabeling of modular S-covariance.

specific steps
  1. self citation load bearing [Section 4.3, after Eq. (4.8)]
    "if one grants that the underlying UV-complete bulk description of AdS3/CFT2 is a TQFT with Wilson lines, the result (and the calculation from the twined character) is natural: this was our motivation is using the TQFT language to derive the eigenvalue of the Verlinde defect. But from a bulk EFT/gravity point of view, it is completely mysterious. This calculation can therefore be viewed as giving evidence that the TQFT skeleton [9] is the fundamental approach to AdS3/CFT2."

    The BTZ half of the central claim - a 'microscopic' description of the one-loop determinant on the smooth BTZ handlebody - depends on the premise that the bulk is a TQFT with Wilson lines, but that framework is relegated to the authors' own companion paper [9], 'to appear', and is not independently checkable here. The calculation performed after granting the premise is just the modular-S identity (4.8), so presenting that calculation as evidence for [9] assumes the very framework it is meant to support. This makes the physical interpretation load-bearing on an unverified self-citation, even though the thermal-AdS3 half is self-contained.

  2. renaming known result [Sections 4.2-4.3, Eqs. (4.6)-(4.8)]
    "Consequently Z tw_a(tau) = integral_0^infty dP S_aP(c) hat-chi_P(tau) = hat-chi_a(-1/tau). (4.8) This is the cycle-exchange statement relevant for the spool. In the original thermal-AdS channel, the light line is a meridian linking the heavy Wilson-line source. After summing over heavy labels with the modular density, the same object is reorganized as the dual-channel character of the light primary."

    The 'microscopic' BTZ result is assembled by construction from the density factor S0P and the Verlinde eigenvalue factor S_aP/S0P introduced in Eqs. (4.6)-(4.7). Their product cancels to S_aP, so Eq. (4.8) is exactly the modular S-kernel transformation already stated in Eq. (4.2). The verbal dressing - a dense family of heavy Polyakov loops, a spatially wound probe, a Verlinde line - is a relabeling of modular covariance rather than an independent microscopic derivation. The BTZ one-loop determinant is the modular image of the thermal-AdS determinant, so the claimed microscopic reading does not add content beyond the S-transform.

full rationale

Sections 2-3 are clean and self-contained: the Wilson spool (2.9) is exactly the plethystic exponent of the global character, the fixed-N decomposition into multi-trace primaries is standard symmetric-function bookkeeping with explicit counting (Eqs. (2.22), (2.27), (A.9)), and the Virasoro dressing (3.4) is a straightforward product identity. Those results are checked against the GMY determinant and involve no fitted parameters, so no circularity there. The circularity is confined to the BTZ claim in Section 4. Equation (4.8) reduces, by construction, to the modular S-kernel: the Cardy density S0P and the Verlinde eigenvalue S_aP/S0P cancel, and the result is just hat-chi_a(-1/tau). Presenting this as a 'microscopic' ensemble derivation is a reinterpretation of modular covariance. Moreover, the physical identification of heavy primaries with Wilson lines and light probes with Verlinde lines is imported from the authors' own companion paper [9], 'to appear', and the paper explicitly conditions on that premise ('if one grants...') before using the calculation as evidence for [9]. That is a load-bearing self-citation loop for the BTZ half. Overall score 4: real self-citation and a partial by-construction reduction for the BTZ 'microscopic' claim, while the thermal-AdS3 Wilson-tower core retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 2 invented entities

No free parameters are fitted: h is the physical scalar weight, c is the central charge, and P is an integration variable; the large-c limit is an approximation rather than a fit. The main implicit inputs are the Wilson-spool identification, the generalized-free-field/multi-trace dictionary, the Virasoro character formulas, the modular S-kernel, and the TQFT-skeleton/Verlinde interpretation. The central thermal-AdS3 identity is self-contained; the BTZ microstate claim is not.

axioms (6)
  • domain assumption Large-c generalized free field duality: multi-particle states of a bulk scalar are dual to towers of multi-trace primaries.
    Used throughout to interpret Sym^N(H1) as the spectrum of light primaries; introduced after Eq. (1.1) and in Section 2.2.
  • domain assumption Wilson spool formula: W_Gamma in Eq. (2.6) equals the GMY one-loop determinant on the quotient.
    Imported from [26,27], not rederived; it is the starting point of the spool-to-tower calculation.
  • standard math Virasoro Verma module character has no null-state subtraction: chiVir_{hP} = q^{hP}(1-q)^{-1} prod_{n>=2}(1-q^n)^{-1}.
    Used in Eqs. (3.2)–(3.3) to dress global characters to Virasoro characters; valid for generic h_P but silently assumes the large-c multi-trace operators are exact Virasoro primaries.
  • standard math Modular S-kernel/Plancherel and Cardy density: bchi_A(-1/tau) = ∫ dP S_AP bchi_P(tau), with the vacuum row S0P as the heavy-primary density.
    Assumed in Section 4.1 and used to collapse the heavy sum in Eq. (4.8); the continuum Virasoro kernel is imported from the Virasoro TQFT literature.
  • domain assumption Verlinde-line eigenvalue on a heavy Virasoro primary: D_a|V_P = (S_aP/S0P) 1.
    Eq. (4.5). Standard Verlinde technology, but its application to heavy Virasoro primaries inside the TQFT skeleton is part of the companion framework [9].
  • domain assumption Smooth BTZ handlebody is represented by an S0P-weighted sum over heavy Virasoro primaries.
    Section 4.1 and the concluding interpretation; taken from [8] and [9], and load-bearing for the 'microscopic' claim.
invented entities (2)
  • Wilson tower (tower of single-winding Wilson lines) no independent evidence
    purpose: Represent a local generalized free bulk field as a sum over Wilson loops, one per multi-trace primary.
    A bookkeeping/organizational object with no new observable handle; it re-expresses the known one-loop determinant.
  • Ensemble-of-heavy-Wilson-lines resolution of the smooth BTZ horizon no independent evidence
    purpose: Give a microstate interpretation of one-loop determinants on BTZ as a Cardy-density sum over heavy primaries.
    Assumed from [8] and companion [9]; no falsifiable prediction outside this framework is provided.

pith-pipeline@v1.3.0-alltime-deepseek · 32249 in / 18872 out tokens · 187890 ms · 2026-08-01T07:32:49.341240+00:00 · methodology

0 comments
read the original abstract

Multi-winding Wilson loops (``Wilson spools'') in 2+1-dimensional gravity can reproduce one-loop partition functions of local free fields in the bulk. Bulk free fields have a second-quantized Fock space, and the AdS/CFT correspondence suggests that the associated multi-particle sectors are dual to multi-trace primaries in the CFT. In this note, we use standard symmetric-function methods to show that the Wilson spool in thermal AdS$_3$ can be recast as a sum over $single$-winding Wilson loops --- one for each multi-trace primary. In a companion paper to appear, we will view Wilson networks and TQFTs as the natural language of non-perturbative bulk quantum gravity. The present note illustrates how this can apply to $local$ bulk fields, and not just defects: a bulk (generalized free) field is to be viewed as a full tower of multi-trace Wilson lines. We further show that the $SL(2)$ descendants of each multi-trace primary, together with the boundary gravitons of the AdS$_3$ background, correctly reproduce the full Virasoro character of each module. In this language, the role of a light insertion on a heavy primary is played by a topological Verlinde line. This allows us to obtain a ``microscopic" description of one-loop determinants on the smooth BTZ handlebody. A spatially wound probe Wilson line on a torus with a contractible $thermal$ cycle can be traded for a Verlinde line inserted on a dense family of heavy Polyakov loops --- with the roles of the two cycles exchanged, so that the $spatial$ cycle is now contractible. The vacuum row of the modular $S$-kernel acts as the (approximate) density of the heavy primaries. This reinforces the case made in arXiv:2601.18775 that a smooth horizon is a stand-in for an ensemble of quantum states, each produced by a heavy Wilson line that appears as a singular horizon in the semi-classical limit.

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Reference graph

Works this paper leans on

44 extracted references · 31 linked inside Pith

  1. [1]

    Quantization of Teichm¨ uller spaces and the quantum dilogarithm,

    R. M. Kashaev, “Quantization of Teichm¨ uller spaces and the quantum dilogarithm,” Lett. Math. Phys.43(1998) 105–115, q-alg/9705021

    Pith/arXiv arXiv 1998

  2. [2]

    Liouville bootstrap via harmonic analysis on a noncompact quantum group,

    B. Ponsot and J. Teschner, “Liouville bootstrap via harmonic analysis on a noncompact quantum group,” hep-th/9911110

    Pith/arXiv arXiv

  3. [3]

    Clebsch–Gordan and Racah–Wigner coefficients for a con- tinuous series of representations ofU q(sl(2,R)),

    B. Ponsot and J. Teschner, “Clebsch–Gordan and Racah–Wigner coefficients for a con- tinuous series of representations ofU q(sl(2,R)),” Commun. Math. Phys.224(2001) 613–655, math/0007097

    Pith/arXiv arXiv 2001

  4. [4]

    An analog of a modular functor from quantized Teichm¨ uller theory,

    J. Teschner, “An analog of a modular functor from quantized Teichm¨ uller theory,” in Handbook of Teichm¨ uller Theory, Vol. I, EMS (2007), math/0510174

    Pith/arXiv arXiv 2007

  5. [5]

    A TQFT from quantum Teichm¨ uller theory,

    J. E. Andersen and R. Kashaev, “A TQFT from quantum Teichm¨ uller theory,” Com- mun. Math. Phys.330(2014) 887–934, arXiv:1109.6295

    Pith/arXiv arXiv 2014

  6. [6]

    Solving 3d gravity with Virasoro TQFT,

    S. Collier, L. Eberhardt and M. Zhang, “Solving 3d gravity with Virasoro TQFT,” SciPost Phys.15(2023) 151, arXiv:2304.13650

    Pith/arXiv arXiv 2023

  7. [7]

    3d gravity from Virasoro TQFT: Holography, wormholes and knots,

    S. Collier, L. Eberhardt and M. Zhang, “3d gravity from Virasoro TQFT: Holography, wormholes and knots,” SciPost Phys.17(2024) 134, arXiv:2401.13900

    Pith/arXiv arXiv 2024

  8. [8]

    Holography with an Inner Boundary: A Smooth Horizon as a Sum over Horizonless States,

    C. Krishnan and P. S. Pathak, “Holography with an Inner Boundary: A Smooth Horizon as a Sum over Horizonless States,” arXiv:2601.18775 [hep-th]

    arXiv

  9. [9]

    A TQFT Skeleton for AdS 3/CFT2,

    V. Gayari and C. Krishnan, “A TQFT Skeleton for AdS 3/CFT2,” to appear

  10. [10]

    Building a Quantum Black Hole Microstate: A Bulk Path Integral for a Heavy Virasoro Primary,

    C. Krishnan and R. Mitra, “Building a Quantum Black Hole Microstate: A Bulk Path Integral for a Heavy Virasoro Primary,” to appear

  11. [11]

    Three-Dimensional Einstein Gravity: Dynamics of Flat Space,

    S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional Einstein Gravity: Dynamics of Flat Space,” Annals Phys.152(1984) 220

    1984

  12. [12]

    (2+1)-Dimensional Gravity as an Exactly Soluble System,

    E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys. B 311(1988) 46–78

    1988

  13. [13]

    Quantum Mechanics of a Point Particle in (2+1)- Dimensional Gravity,

    H. J. Matschull and M. Welling, “Quantum Mechanics of a Point Particle in (2+1)- Dimensional Gravity,” Class. Quant. Grav.15(1998) 2981, arXiv:gr-qc/9708054

    Pith/arXiv arXiv 1998

  14. [14]

    Wilson Lines and Entanglement Entropy in Higher Spin Gravity,

    M. Ammon, A. Castro and N. Iqbal, “Wilson Lines and Entanglement Entropy in Higher Spin Gravity,” JHEP10(2013) 110, arXiv:1306.4338 [hep-th]. 42

    Pith/arXiv arXiv 2013

  15. [15]

    Entanglement Entropy and Higher Spin Holography in AdS3,

    J. de Boer and J. I. Jottar, “Entanglement Entropy and Higher Spin Holography in AdS3,” JHEP04(2014) 089, arXiv:1306.4347 [hep-th]

    Pith/arXiv arXiv 2014

  16. [16]

    Wilson Lines and Ishibashi States in AdS3/CFT2,

    A. Castro, N. Iqbal and E. Llabr´ es, “Wilson Lines and Ishibashi States in AdS3/CFT2,” JHEP09(2018) 066, arXiv:1805.05398 [hep-th]

    Pith/arXiv arXiv 2018

  17. [17]

    Generalized Free Fields and Models of Local Field Theory,

    O. W. Greenberg, “Generalized Free Fields and Models of Local Field Theory,” Annals Phys.16(1961) 158

    1961

  18. [18]

    Holography from Conformal Field Theory,

    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, “Holography from Conformal Field Theory,” JHEP10(2009) 079, arXiv:0907.0151 [hep-th]

    Pith/arXiv arXiv 2009

  19. [19]

    Effective Conformal Theory and the Flat-Space Limit of AdS,

    A. L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, “Effective Conformal Theory and the Flat-Space Limit of AdS,” JHEP07(2011) 023, arXiv:1007.2412 [hep- th]

    Pith/arXiv arXiv 2011

  20. [20]

    Emergent Spacetime and Holographic CFTs,

    S. El-Showk and K. Papadodimas, “Emergent Spacetime and Holographic CFTs,” JHEP10(2012) 106, arXiv:1101.4163 [hep-th]

    Pith/arXiv arXiv 2012

  21. [21]

    Multiple trace operators and nonlocal string theories,

    O. Aharony, M. Berkooz and E. Silverstein, “Multiple trace operators and nonlocal string theories,” JHEP08(2001) 006, arXiv:hep-th/0105309

    Pith/arXiv arXiv 2001

  22. [22]

    Multi-Trace Operators, Boundary Conditions, And AdS/CFT Correspon- dence,

    E. Witten, “Multi-Trace Operators, Boundary Conditions, And AdS/CFT Correspon- dence,” arXiv:hep-th/0112258 [hep-th]

    Pith/arXiv arXiv

  23. [23]

    The Hagedorn Transition, Deconfinement and N=4 SYM Theory,

    B. Sundborg, “The Hagedorn Transition, Deconfinement and N=4 SYM Theory,” Nucl. Phys. B573(2000) 349, arXiv:hep-th/9908001

    Pith/arXiv arXiv 2000

  24. [24]

    The Hagedorn – Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theo- ries,

    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, “The Hagedorn – Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theo- ries,” Adv. Theor. Math. Phys.8(2004) 603, arXiv:hep-th/0310285

    Pith/arXiv arXiv 2004

  25. [25]

    One-loop Partition Functions of 3D Gravity,

    S. Giombi, A. Maloney and X. Yin, “One-loop Partition Functions of 3D Gravity,” arXiv:0804.1773 [hep-th]

    Pith/arXiv arXiv

  26. [26]

    Coupling Fields to 3D Quan- tum Gravity via Chern-Simons Theory,

    A. Castro, I. Coman, J. R. Fliss and C. Zukowski, “Coupling Fields to 3D Quan- tum Gravity via Chern-Simons Theory,” Phys. Rev. Lett.131(2023) 171602, arXiv:2306.00990 [hep-th]

    Pith/arXiv arXiv 2023

  27. [27]

    A spool for every quotient: One-loop partition functions in AdS3 gravity,

    R. Bourne and J. R. Fliss, “A spool for every quotient: One-loop partition functions in AdS3 gravity,” arXiv:2507.05364 [hep-th]

    Pith/arXiv arXiv

  28. [28]

    Operator Content of Two-Dimensional Conformally Invariant Theories,

    J. L. Cardy, “Operator Content of Two-Dimensional Conformally Invariant Theories,” Nucl. Phys. B270(1986) 186. 43

    1986

  29. [29]

    Universal Spectrum of 2d Conformal Field Theory in the Large c Limit,

    T. Hartman, C. A. Keller and B. Stoica, “Universal Spectrum of 2d Conformal Field Theory in the Large c Limit,” JHEP09(2014) 118, arXiv:1405.5137 [hep-th]

    Pith/arXiv arXiv 2014

  30. [30]

    A Cardy Formula for Three-Point Coefficients or How the Black Hole Got its Spots,

    P. Kraus and A. Maloney, “A Cardy Formula for Three-Point Coefficients or How the Black Hole Got its Spots,” JHEP05(2017) 160, arXiv:1608.03284 [hep-th]

    Pith/arXiv arXiv 2017

  31. [31]

    Fusion Rules and Modular Transformations in 2D Conformal Field The- ory,

    E. Verlinde, “Fusion Rules and Modular Transformations in 2D Conformal Field The- ory,” Nucl. Phys. B300(1988) 360

    1988

  32. [32]

    Generalized twisted partition functions,

    V. B. Petkova and J.-B. Zuber, “Generalized twisted partition functions,” Phys. Lett. B504(2001) 157, arXiv:hep-th/0011021

    Pith/arXiv arXiv 2001

  33. [33]

    Anyonic Chains, Topological Defects, and Conformal Field Theory,

    M. Buican and A. Gromov, “Anyonic Chains, Topological Defects, and Conformal Field Theory,” Commun. Math. Phys. 356 (2017) 1017, arXiv:1701.02800 [hep-th]

    Pith/arXiv arXiv 2017

  34. [34]

    A smooth horizon without a smooth horizon,

    V. Burman, S. Das and C. Krishnan, “A smooth horizon without a smooth horizon,” JHEP03(2024) 014, arXiv:2312.14108 [hep-th]

    Pith/arXiv arXiv 2024

  35. [35]

    A Bottom-Up Approach to Black Hole Microstates,

    V. Burman and C. Krishnan, “A Bottom-Up Approach to Black Hole Microstates,” arXiv:2409.05850 [hep-th]

    Pith/arXiv arXiv

  36. [36]

    What’s the Matter with 3D Gravity?,

    R. Bourne, J. R. Fliss and B. Knighton, “What’s the Matter with 3D Gravity?,” [arXiv:2607.01319 [hep-th]]

    Pith/arXiv arXiv

  37. [37]

    Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity,

    A. Alekseev and S. L. Shatashvili, “Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity,” Nucl. Phys. B323(1989) 719

    1989

  38. [38]

    A theory of reparameterizations for AdS 3 gravity,

    J. Cotler and K. Jensen, “A theory of reparameterizations for AdS 3 gravity,” JHEP02 (2019) 079, arXiv:1808.03263 [hep-th]

    Pith/arXiv arXiv 2019

  39. [39]

    Hartle-Hawking state and its factorization in 3d gravity,

    W. Z. Chua and Y. Jiang, “Hartle-Hawking state and its factorization in 3d gravity,” JHEP03, 135 (2024) doi:10.1007/JHEP03(2024)135 [arXiv:2309.05126 [hep-th]]

    Pith/arXiv arXiv 2024

  40. [40]

    Quantum Field Theory and the Jones Polynomial,

    E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math. Phys. 121(1989) 351

    1989

  41. [41]

    Quantum statistical mechanics in a closed system,

    J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A43, no.4, 2046 (1991) doi:10.1103/PhysRevA.43.2046

    doi:10.1103/physreva.43.2046 2046

  42. [42]

    Chaos and Quantum Thermalization,

    M. Srednicki, “Chaos and Quantum Thermalization,” Phys. Rev. E50, 888 doi:10.1103/PhysRevE.50.888 [arXiv:cond-mat/9403051 [cond-mat]]

    Pith/arXiv arXiv

  43. [43]

    Group Theory and Its Application to Physical Problems,

    M. Hamermesh, “Group Theory and Its Application to Physical Problems,” Dover Publications, New York (1989). 44

    1989

  44. [44]

    Symmetric Functions and Hall Polynomials,

    I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” 2nd edition, Clarendon Press, Oxford, 1995. 45

    1995