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A spool for every quotient: One-loop partition functions in AdS$_3$ gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single topological formula—the Wilson spool—gives one-loop matter determinants on every smooth, cusp-free hyperbolic AdS3 quotient.

desk verdict Genuinely extends the Wilson spool to all smooth cusp-free hyperbolic quotients and gives the first s≥2 one-loop formulas on them; plausible and well-motivated, but the s≥2 steps rest on quoted rather than verified input. read the letter →

arxiv 2507.05364 v2 pith:6ZW2XSX3 submitted 2025-07-07 hep-th

Robert Bourne , Jackson R. Fliss , Bob Knighton This is my paper · ORCID
classification hep-th
keywords Wilsonspoolone-loopdeterminantsAdS3gravityhyperbolicthree-manifoldsChern-SimonsSelbergtraceformulamassivehigher-spinfieldsquasinormalmodes
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

This paper aims to show that the one-loop determinant of a massive spinning field on any smooth, cusp-free hyperbolic three-manifold—the class that includes multi-boundary wormholes and compact hyperbolic manifolds—is exactly a sum of Wilson-loop observables in the topological Chern-Simons formulation of three-dimensional gravity. The sum runs over all nontrivial free-homotopy classes of loops in the manifold, with each class weighted by one over its multiplicity and by a product of characters of lowest-weight $\mathrm{sl}(2,\mathbb{R})$ representations fixed by the field's mass and spin. On-shell, this "Wilson spool" collapses to a product over primitive geodesics that reproduces the known one-loop determinants for scalars and vectors and gives new determinants for massive spin $s\ge 2$ fields. A sympathetic reader would care because it turns a spectral problem on a complicated quotient geometry into a topological, gauge-invariant object that can also be inserted off-shell into the gravitational path integral.

What carries the argument

The central object is the Wilson spool, a topological line operator defined as a sum over all unoriented nontrivial conjugacy classes $[\gamma]_+$ of the fundamental group, weighted by $1/n_\gamma$, of the product of holonomy traces $\mathrm{Tr}_{R_L} P\exp(\oint_\gamma A_L)\, \mathrm{Tr}_{R_R} P\exp(-\oint_\gamma A_R)$, with $R_L\otimes R_R$ ranging over the lowest-weight representations selected by the mass-shell condition $j_\pm = (\Delta\pm s)/2$. The key mechanism is the length–holonomy correspondence: around a geodesic, the connection holonomy equals $q_\gamma^{L_0}$ with $q_\gamma = e^{-(l_\gamma+i\theta_\gamma)}$, so every geometric quantity in the trace formula becomes a product of representation characters. The multiplicity $n_\gamma$, coming from writing each group element as a power of a primitive element, is reinterpreted in the worldline derivation as the symmetry factor of a loop lifted to the covering space, which makes the sum topological rather than geometric.

What would settle it

Compute the one-loop determinant of a massive spin-2 field on thermal AdS3 by direct summation over the known quasinormal-mode poles in the complex conformal dimension plane and compare the pole locations and multiplicities with the s=2 case of the product formula; any mismatch would falsify the new s>=2 claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is the identity $\log Z^M_{\Delta,s}[g] = W_\Gamma[A_L,A_R]$, where $W_\Gamma = \sum_{[\gamma]_+} \sum_{R^{LW}_{\Delta,s}} \frac{1}{n_\gamma} \mathrm{Tr}_{R_L} P\exp(\oint_\gamma A_L)\, \mathrm{Tr}_{R_R} P\exp(-\oint_\gamma A_R)$, and $A_{L/R}$ are the connections built from the coframe and spin connection. Because the quotient group $\Gamma$ is torsion-free, every nontrivial loop is classified by a conjugacy class $[\gamma]$ and a complex length $\hat{l}_\gamma = l_\gamma + i\theta_\gamma$, and the holonomy traces evaluate to lowest-weight characters. The authors show that on-shell this identity is equivalent to the primitive-loop product $\exp(W_\Gamma) = \prod_{[\gamma_0]_+}\prod_\pm \prod_{\ell,\bar\ell=0}^\infty (1 - q_{\gamma_0}^{\ell+(\Delta\pm s)/2}\bar q_{\gamma_0}^{\bar\ell+(\Delta\mp s)/2})^{-1}$, which for $s=0,1$ matches the known scalar and vector determinants and for $s\ge 2$ is claimed to be new. They support the identity by three derivations: the trace formula, a worldline path integral that is two-loop exact, and the quasinormal-mode method.

Load-bearing premise

The load-bearing premise is that, for every spin, the massive field's auxiliary component fields reduce exactly to the spectrum of the symmetric transverse-traceless Laplacian that enters the quoted trace formula, with the stated normalization and constant-mode counting; if the reduction or normalization fails for s>=2, the new product formula for spinning fields fails even though the s=0,1 checks pass.

Editorial extensions

If this is right

  • On every smooth cusp-free hyperbolic three-manifold, the one-loop determinant of a massive scalar or vector is a topological object: it depends on the background only through the holonomies of A_L and A_R, so it is invariant under metric deformations that preserve those holonomies.
  • The product formula provides one-loop determinants for massive spin s>=2 fields on quotients where no such determinants were previously known, including multi-boundary wormholes and compact hyperbolic manifolds.
  • Because the spool is expressed in terms of gauge-invariant Wilson loops, it can be promoted to an off-shell operator in the gravitational path integral about a saddle, meaning matter one-loop effects can be included inside diffeomorphism-invariant observables without fixing a particular metric.
  • For quotients with a single primitive generator, the sum over conjugacy classes collapses to the earlier spool results on the torus/black-hole background, and the integral form with the contour wraps the positive real axis to reproduce the known spool representation.
  • If the identity survives quantization of the gravitational sector, the spool gives a route to coupling matter to the fully quantized topological formulation of three-dimensional gravity, since the spool's Wilson loops are the natural extended operators of that theory.

Reading between the lines

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

  • Editorial inference: the on-shell product is structurally a zeta function for the symmetric transverse-traceless spin-s Laplacian; if the identity holds, the Wilson spool could be analytically continued in the conformal dimension and used to extract spectral data, such as determinant ratios or zeta-regularized volumes, from geodesic length spectra alone.
  • Editorial inference: the worldline derivation suggests a concrete off-shell test: compute the scalar determinant on a handlebody with a non-hyperbolic, perturbatively off-shell metric by worldline methods and check that the spool, evaluated on the same holonomies, agrees to the two-loop order shown.
  • Editorial inference: for s>=2 the missing check is the auxiliary-field reduction; a direct comparison on the thermal AdS3 background between the pole spectrum of the massive spin-2 determinant and the s=2 product would settle the most exposed step before moving to more complicated quotients.
  • Editorial inference: the divergence of parabolic characters noted in the discussion suggests that a regularized limit of the spool might capture cusp contributions, potentially extending the formula to finite-volume quotients with cusps, where the trace formula has a continuous spectrum.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a Wilson-spool prescription, Eqs. (3.1)-(3.2), expressing the one-loop determinant of a massive spin-s field on any smooth cusp-free hyperbolic three-manifold M = H^3/Γ as a sum over conjugacy classes of Γ of products of Wilson loops in lowest-weight sl(2,R) representations. On-shell the prescription gives the product formula (3.10), which matches Giombi-Maloney-Yin for s=0,1 and is claimed to be new for s≥2. The authors offer three supporting derivations: a Selberg trace formula argument (Sec. 3.2.1), a worldline path integral (Sec. 3.2.2), and a quasinormal-mode argument (Sec. 3.2.3), and they conjecture an off-shell extension within diffeomorphism-invariant correlators.

Significance. If fully established, the result would be a substantial advance: it expresses matter one-loop determinants on all these geometries as topological data (holonomies of the Chern-Simons connections), and extends the spool program beyond single-cycle geometries and beyond s≤1. The paper's checks against GMY for s=0,1 are explicit, and the worldline derivation provides a non-circular route to the scalar formula. The genuinely new s≥2 on-shell formula, however, rests on two unverified inputs — the spin-s Selberg trace formula in Eq. (3.11) and the Stückelberg reduction in Eq. (2.35) — and the off-shell claim is explicitly conjectural. These gaps are load-bearing for the advertised new content, so the paper is promising but not yet conclusive.

major comments (4)
  1. [3.2.1, Eq. (3.11)] The Selberg trace formula for the symmetric transverse-traceless spin-s Laplacian is quoted with a specific normalization, including the factor (1+δ_{s,0}), the parametrization λ_m^{(s)} = (t_m^{(s)})^2 + s + 1, the volume term (s^2 H(0) − H''(0)), and the geometric term cos(sθ)/(cosh l − cos θ). For s=0,1 this is standard, but for s≥2 no derivation or independent verification is supplied. Since the central new product (3.10) for s≥2 follows directly from this formula, the authors should either derive the formula in their normalization or check it against an independent result (e.g., a known BTZ or thermal-AdS determinant at fixed s), and clarify which parts of [26] apply for all s.
  2. [2.3, Eq. (2.35)] The path integral of the massive spin-s system, including the Stückelberg tower, is asserted to reduce to det(−∇²_(s) + m̄_s²ℓ²)^{-1/2}. This reduction is standard for s=0 and 1 but is not demonstrated for s≥2; the ghost fields, measure factors, and any additional determinants are not specified. This is load-bearing because the eigenvalues of −∇²_(s) are exactly the input to the trace formula (3.11); a different mode count or normalization would invalidate the s≥2 product (3.10) while leaving the s=0,1 checks intact.
  3. [3.2.1, first paragraph (scope)] The main result (3.1)-(3.2) is stated for all smooth, cusp-free hyperbolic three-manifolds, which includes non-compact examples such as thermal AdS and multi-boundary wormholes, but the Selberg trace formula (3.11) is formulated only for compact quotients, where the spectrum is discrete. The paper acknowledges this and says the non-compact case 'suggests' an extension, but for s≥2 no derivation for non-compact quotients is given. The theorem should either be restricted to compact quotients or supplemented with a proof or a precise continuity argument covering the non-compact cusp-free case.
  4. [3.2, second paragraph (off-shell claim)] The off-shell promotion of (3.2), namely that the spool equals the one-loop determinant also off-shell inside expectation values of diffeomorphism-invariant operators, is explicitly posited rather than proven. The worldline derivation in Sec. 3.2.2 is performed for a scalar and is matched to the spool only on-shell; the spinning and off-shell generalizations are asserted. This does not invalidate the on-shell result, but the abstract and introduction should clearly separate the proven on-shell statement from the conjectural off-shell one.
minor comments (5)
  1. [3.2.2, Eqs. (3.39)-(3.42)] The diagrammatic equations are not labeled in the text; in the provided manuscript the left-hand sides appear blank, so the claimed two-loop exactness cannot be checked from the text. Please ensure the diagrams are rendered and clearly identified.
  2. [2.2, Eq. (2.32)] The notation [Γ]+ and [Γ0]+ is clear, but the relation between the quotient by Z2 and the choice of positive geodesic length could be stated more explicitly to avoid confusion about why only one orientation appears in the spool sum.
  3. [3.2.1, Eqs. (3.16)-(3.17)] The renormalization prescription for the volume divergence is not specified; since the final determinant depends on the subtraction, the authors should state the scheme (e.g., minimal subtraction) used to obtain (3.17).
  4. [1, Eq. (1.4)] The contour C is described qualitatively as 'wrapping tightly clockwise' around the positive Re(α) axis; a precise definition of the contour segments would make the integral representation unambiguous.
  5. [Throughout] There are a few typographical slips (e.g., 'with with either' in Section 2.2) and the Stückelberg name is garbled by the LaTeX encoding; these should be cleaned up in the final version.

Circularity Check

1 steps flagged · score 4.0 of 10

Quasinormal-mode section derives the spool from the spool; on-shell result itself is independently grounded.

  1. ansatz smuggled in via citation [Section 3.2.3, Eqs. (3.49)-(3.50)]
    "Within each subgroup, this reduces the DHS problem again to that on a torus with just one cycle. We can now utilize the same set of representations, R^{LW/HW}_{Δ,s} for each centralizer subgroup. Thus when Γ is purely loxodromic or hyperbolic we again write Z^{H3/Γ}_{Δ,s} = ∏_{R^{LW}_{Δ,s}} ∏_{[Γ0]+} Det_{R_L⊗R_R}(1 - Pexp∮_{γ0}A_L Pexp∮_{γ0}A_R)^{-1}."

    Equation (3.49) is already the exponentiated Wilson spool: expanding its log gives (3.50), which is exactly the main claimed result (3.2). The preceding argument does not show that the one-loop determinant of the STT Laplacian equals this product for non-elementary quotients; it asserts that the thermal-AdS/BTZ quasinormal-mode construction of [7] extends per centralizer subgroup. Thus the quasinormal-mode 'derivation' builds the answer in product form and then recovers it, relying on an unverified extrapolation of the authors' own prior work.

full rationale

The central on-shell claim (3.10) is not circular: it follows from the Selberg trace formula (3.11), quoted from the independent mathematical paper [26], via the heat-kernel and character manipulations (3.14)-(3.18), and for s=0,1 it matches the independent GMY result [14]. The worldline path integral in Section 3.2.2 is likewise an independent computation that reproduces the same on-shell determinant. The only genuinely circular piece is the quasinormal-mode 'derivation' (3.49)-(3.50): (3.49) is the spool written as a product over holonomies, so expanding it returns the main result, while the step from Conditions 0, I, II to (3.49) for arbitrary Kleinian groups is asserted rather than derived, leaning on the authors' own prior work [7]. The off-shell promotion is also an explicitly labeled posit/assumption supported by self-citations ([5], [7]), not a derivation. Because the main on-shell result has independent support and is benchmarked against GMY, the circularity is partial and confined to one of the three advertised derivations.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim loads on standard spectral geometry (the Selberg trace formula, quoted externally) plus the representation-theoretic dictionary of the Wilson spool program. No data fitting is involved; the only by-hand choices are the renormalization scheme and the contour/convention choices. The off-shell equality and the s≥2 trace-formula application are the two unproved load-bearing premises, while the worldline two-loop exactness is an asserted technical lemma.

free parameters (2)
  • renormalization prescription = subtraction of the identity-class UV-divergent term proportional to vol(H3/Γ) in eq. (3.17)
    The β→0 divergent volume term is subtracted by hand to define the renormalized one-loop determinant; the choice is non-universal but does not affect the topology-dependent spool sum.
  • contour C in the integral form (3.3) = wraps the positive real α axis clockwise, crossing just to the right of the origin
    The contour prescription fixes which poles of cot(α/2) contribute; it is an analytic-continuation choice rather than a fitted number, but it is part of the definition of W_Γ.
assumptions (5)
  • standard math Selberg trace formula for the STT spin-s Laplacian on hyperbolic 3-manifolds, in the normalization of eq. (3.11), including the (1+δ_{s,0}) factor and the absorption of trivial-representation contributions into the eigenvalue sum.
    Quoted from [26] (Bonifacio-Mazac-Pal) at Section 3.2.1; it is the main engine of the on-shell derivation. The footnote 14 caveat about absorbing the trivial representation is what makes the application to s≥2 delicate.
  • domain assumption The one-loop determinant of the massive spin-s field, with its Stückelberg tower, equals the determinant of −∇²_(s) + m̄_s²ℓ² acting on STT s-tensors, with eigenvalues parameterized by λ_m^{(s)} = (t_m^{(s)})² + s + 1.
    Section 2.3, eqs. (2.35)-(2.38) and Section 3.2.1; for s≥2 the reduction of the Stückelberg tower to this spectrum is cited to [21],[22] but not verified here.
  • domain assumption Holonomy-length correspondence for on-shell connections: P exp ∮_γ A_L ∼ q^{L0}, P exp(−∮_γ A_R) ∼ q̄^{L̄0} with q = e^{−l̂}, l̂ = l + iθ.
    Derived in Section 2.1.1 via Fermi normal coordinates; standard in the Chern-Simons formulation, but it fixes the convention used to convert the spool into characters.
  • ad hoc to paper The Wilson spool (3.2) equals the one-loop determinant also off-shell, at least inside expectation values of diffeomorphism-invariant operators.
    Stated in Section 3.2 (paragraph 2) as a posit: 'we will posit that (1.2) holds off-shell in a weak sense'; this underlies the abstract's 'promoted to an off-shell operator' claim and is not derived.
  • domain assumption The worldline path integral (3.28) is two-loop exact, with the stated diagram values (3.39)-(3.42).
    Section 3.2.2; the cancellation of all higher loops is asserted with a remark about the symmetry of the AdS3 background, and the diagram values are listed without derivation.

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Cite this review

Pith. "Pith review of A spool for every quotient: One-loop partition functions in AdS$_3$ gravity." pith.science (2026). https://pith.science/paper/6ZW2XSX3

@misc{pith2026250705364,
  author       = {Pith},
  title        = {Pith review of: A spool for every quotient: One-loop partition functions in AdS$_3$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZW2XSX3}},
  note         = {Machine review of arXiv:2507.05364}
}
read the original abstract

The Wilson spool is a prescription for expressing one-loop determinants as topological line operators in three-dimensional gravity. We extend this program to describe massive spinning fields on all smooth, cusp-free, solutions of Euclidean gravity with a negative cosmological constant. Our prescription makes use of the expression of such solutions as a quotients of hyperbolic space. The result is a gauge-invariant topological operator, which can be promoted to an off-shell operator in the gravitational path integral about a given saddle-point. When evaluated on-shell, the Wilson spool reproduces and extends the known results of one-loop determinants on hyperbolic quotients. We motivate our construction of the Wilson spool from multiple perspectives: the Selberg trace formula, worldline quantum mechanics, and the quasinormal mode method.

Figures

Figures reproduced from arXiv: 2507.05364 by the authors.

Figure 1
Figure 1. Example Euclidean hyperbolic manifolds to which our prescrip￾tion applies. (Left) A solid handlebody whose asymptotic boundary is the genus two Riemann surface. Depending on the detail of the quotient, this handlebody arises as Euclidean rotation of the Lorentzian spacetime whose time-symmetric slice has three asymptotic boundaries, as well as a Lorentzian spacetime with a single asymptotic boundary and a torus hidd… view at source ↗
Figure 2
Figure 2. (Left) The frame ˜ea(s) picks up a rotation of θγ under parallel transport along γ. (Right) We define a new frame, ea(s), which is gradu￾ally rotated from ˜ea(s) to undo this holonomy. While ea(s) is not parallel transported, it does remain periodic along γ. This frame maintains that −e0(s) = t(s),5 and satisfies the geodesic equation of mo￾tion: e ν 0∇νe µ 0 = 0 . (2.11) Treating T γ as a subspace of TM|γ, the cofr… view at source ↗
Figure 3
Figure 3. Every element γ ∈ Γ induces a non-contractable loop γ ∈ π1(M, p) in the quotient space M = H3/Γ. base point, p, the conjugacy class, [γ], is associated to a free loop in M up to homotopy. Equivalently, since M is hyperbolic, each conjugacy class [γ] is associated to the homotopy class of a closed geodesic in M, where a geodesic is constructed with respect to the hyperbolic metric on M. Group elements of PSL(2, C) ar… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: , runs clockwise below and above the positive Re(α) axis, crossing just to right of the origin. Note that by construction Γ only contains hyperbolic or loxodromic elements and so |qγ| ̸= 1. Thus given the characters (2.44), on-shell Wilson loops contribute poles sittin…
Figure 5
Figure 5. Figure 5: The simplest example of an elliptic element taken along the same axis as a hyperbolic element resulting in a thermal AdS3 with a conical defect (depicted as the dashed line) as its quotient. In this case there are two nontrivial cycles in the geometry. On the integral …

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Reference graph

Works this paper leans on

31 extracted references · 10 canonical work pages · cited by 3 Pith papers

  1. [26]

    Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula,

    J. Bonifacio, D. Mazac, and S. Pal, “Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula,” Commun. Math. Phys. 406 (2025) 51, arXiv:2308.11174 [math.SP]

    arXiv 2025

  2. [1]

    Lower Dimensional Gravity,

    R. Jackiw, “Lower Dimensional Gravity,” Nucl. Phys. B 252 (1985) 343–356

    work page 1985

  3. [2]

    Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,

    C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,” Phys. Lett. B 126 (1983) 41–45

    work page 1983

  4. [3]

    A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories,

    A. Achucarro and P. K. Townsend, “A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories,” Phys. Lett. B 180 (1986) 89

    1986

  5. [4]

    (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

    1988

  6. [5]

    Keeping matter in the loop in dS3 quantum gravity,

    A. Castro, I. Coman, J. R. Fliss, and C. Zukowski, “Keeping matter in the loop in dS3 quantum gravity,” JHEP 07 (2023) 120, arXiv:2302.12281 [hep-th]. [Erratum: JHEP 09, 004 (2024)]

    arXiv 2023

  7. [6]

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

    A. Castro, I. Coman, J. R. Fliss, and C. Zukowski, “Coupling Fields to 3D Quantum Gravity via Chern-Simons Theory,” Phys. Rev. Lett. 131 no. 17, (2023) 171602, arXiv:2304.02668 [hep-th]

    arXiv 2023

  8. [7]

    Spinning up the spool: massive spinning fields in 3d quantum gravity,

    R. Bourne, A. Castro, and J. R. Fliss, “Spinning up the spool: massive spinning fields in 3d quantum gravity,” J. Phys. A 58 no. 2, (2025) 025402, arXiv:2407.09608 [hep-th]

    arXiv 2025

Show all 31 references
  1. [8]

    Massive fields and Wilson spools in JT gravity,

    J. R. Fliss, “Massive fields and Wilson spools in JT gravity,” arXiv:2503.08657 [hep-th]

    arXiv

  2. [9]

    Multi - black hole geometries in (2+1)-dimensional gravity,

    D. R. Brill, “Multi - black hole geometries in (2+1)-dimensional gravity,” Phys. Rev. D 53 (1996) 4133–4176, arXiv:gr-qc/9511022

    1996 arXiv

  3. [10]

    Black holes and wormholes in (2+1)-dimensions,

    S. Aminneborg, I. Bengtsson, D. Brill, S. Holst, and P. Peldan, “Black holes and wormholes in (2+1)-dimensions,” Class. Quant. Grav. 15 (1998) 627–644, arXiv:gr-qc/9707036. 33

    1998 arXiv

  4. [11]

    Holography and Riemann surfaces,

    K. Krasnov, “Holography and Riemann surfaces,” Adv. Theor. Math. Phys. 4 (2000) 929–979, arXiv:hep-th/0005106

    2000 arXiv

  5. [12]

    Holography and wormholes in 2+1 dimensions,

    K. Skenderis and B. C. van Rees, “Holography and wormholes in 2+1 dimensions,” Commun. Math. Phys. 301 (2011) 583–626, arXiv:0912.2090 [hep-th]

    2011 arXiv

  6. [13]

    Multiboundary Wormholes and Holographic Entanglement,

    V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf, and S. F. Ross, “Multiboundary Wormholes and Holographic Entanglement,” Class. Quant. Grav. 31 (2014) 185015, arXiv:1406.2663 [hep-th]

    2014 arXiv

  7. [14]

    One-loop Partition Functions of 3D Gravity,

    S. Giombi, A. Maloney, and X. Yin, “One-loop Partition Functions of 3D Gravity,” JHEP 08 (2008) 007, arXiv:0804.1773 [hep-th]

    2008 arXiv

  8. [15]

    to appear,

    S. Haupfear, V. Martin, A. Svesko, and C. Zukowski, “to appear,”

  9. [16]

    Solving 3d gravity with Virasoro TQFT,

    S. Collier, L. Eberhardt, and M. Zhang, “Solving 3d gravity with Virasoro TQFT,” SciPost Phys. 15 no. 4, (2023) 151, arXiv:2304.13650 [hep-th]

    2023 arXiv

  10. [17]

    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 [hep-th]

    2024 arXiv

  11. [18]

    Quantum Gravity Partition Functions in Three Dimensions,

    A. Maloney and E. Witten, “Quantum Gravity Partition Functions in Three Dimensions,” JHEP 02 (2010) 029, arXiv:0712.0155 [hep-th]

    2010 arXiv

  12. [19]

    Pure Gravity and Conical Defects,

    N. Benjamin, S. Collier, and A. Maloney, “Pure Gravity and Conical Defects,” JHEP 09 (2020) 034, arXiv:2004.14428 [hep-th]

    2020 arXiv

  13. [20]

    Spinning probes and helices in AdS3,

    P. Fonda, D. Liska, and A. V´ eliz-Osorio, “Spinning probes and helices in AdS3,” Class. Quant. Grav. 35 no. 18, (2018) 185002, arXiv:1803.05283 [hep-th]

    2018 arXiv

  14. [21]

    On massive high spin particles in AdS,

    Y. M. Zinoviev, “On massive high spin particles in AdS,” arXiv:hep-th/0108192

    arXiv

  15. [22]

    Higher Spin Quasinormal Modes and One-Loop Determinants in the BTZ black Hole,

    S. Datta and J. R. David, “Higher Spin Quasinormal Modes and One-Loop Determinants in the BTZ black Hole,” JHEP 03 (2012) 079, arXiv:1112.4619 [hep-th]

    2012 arXiv

  16. [23]

    Metric-like Methods in Higher Spin Holography,

    C. Sleight, “Metric-like Methods in Higher Spin Holography,” PoS Modave2016 (2017) 003, arXiv:1701.08360 [hep-th]

    2017 arXiv

  17. [24]

    Black hole determinants and quasinormal modes,

    F. Denef, S. A. Hartnoll, and S. Sachdev, “Black hole determinants and quasinormal modes,” Class. Quant. Grav. 27 (2010) 125001, arXiv:0908.2657 [hep-th]

    2010 arXiv

  18. [25]

    J. M. J¨ urgen Elstrodt, Fritz Grunewald,Groups Acting on Hyperbolic Space . Springer Berlin, Heidelberg, 1998

    1998

  19. [27]

    A view of the bulk from the worldline,

    H. Maxfield, “A view of the bulk from the worldline,” arXiv:1712.00885 [hep-th]

    arXiv

  20. [28]

    Path integrals in curved space and the worldline formalism,

    F. Bastianelli, “Path integrals in curved space and the worldline formalism,” in 8th International Conference on Path Integrals from Quantum Information to Cosmology. 8, 2005. arXiv:hep-th/0508205. 34

    2005 arXiv

  21. [29]

    Selberg zeta function and trace formula for the BTZ black hole,

    P. A. Perry and F. L. Williams, “Selberg zeta function and trace formula for the BTZ black hole,” International Journal of Pure and Applied Mathematics 9 (2003) 1–22

    2003

  22. [30]

    A generalized Selberg zeta function for flat space cosmologies,

    A. Bagchi, C. Keeler, V. Martin, and R. Poddar, “A generalized Selberg zeta function for flat space cosmologies,” JHEP 04 (2024) 066, arXiv:2312.06770 [hep-th]

    2024 arXiv

  23. [31]

    Notes on fSL(2, R) representations,

    A. Kitaev, “Notes on fSL(2, R) representations,” arXiv:1711.08169 [hep-th]. 35

    arXiv

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Reviewed August 6, 2026 · model on record in the stance chip above.