Triangulating quantum gravity in AdS$_3$

Thomas Hartman

REVIEW 2 major objections 5 minor 4 cited by

Triangulating quantum gravity in AdS$_3$

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Exact 3D gravity path integrals are given by Turaev-Viro and Virasoro TQFT amplitudes.

desk verdict New triangulation-to-geometry dictionary in AdS3, with honest semiclassical checks; exact claims lean on the companion paper's S-transform, but the paper deserves refereeing. read the letter →

arxiv 2507.12696 v1 pith:6WCW5WCS submitted 2025-07-17 hep-th

Thomas Hartman This is my paper

classification hep-th MSC 83C4581T4057K1683C80 PACS 04.60.-m11.25.Hf

keywords 3DgravityAdS/CFTVirasoroTQFTTuraev-VirotheoryOPEstatisticshyperbolictetrahedrawormholespathintegral

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

Pure three-dimensional gravity with negative cosmological constant is usually studied on spacetimes that extend to AdS infinity, but this paper argues that the physics of heavy black-hole operators lives in a compact, finite-volume region. The central claim is that the exact path integral on such a region, with boundary conditions fixing geodesic lengths or dihedral angles, is captured exactly by two topological theories: the squared Virasoro TQFT amplitude and the Conformal Turaev-Viro partition function. A modular S-transform converts one set of boundary conditions into the other. Because both TQFTs are evaluated by triangulating the manifold, the paper obtains a practical dictionary between Virasoro TQFT calculations and explicit hyperbolic geometries. In the semiclassical limit those geometries are built from generalized hyperbolic tetrahedra, whose volumes match the TQFT predictions and the dual CFT ensemble statistics.

What carries the argument

The central object is the generalized hyperbolic tetrahedron: an ordinary tetrahedron whose vertices are truncated by geodesic hyperplanes meeting neighboring faces orthogonally, which is naturally viewed as a tetrahedron with vertices moved beyond the conformal boundary into the de Sitter region of embedding space. These tetrahedra are the cells of a generalized triangulation of the compact gravity manifold, obtained from an ordinary triangulation of the embedding manifold by truncating vertices and dualizing external edges. The exact amplitudes are carried by the Conformal Turaev-Viro partition function, an integral over internal edge labels of products of Virasoro 6j-symbols assigned to each tetrahedron, together with the squared Virasoro TQFT amplitude and the modular S-transform relating them. The classical action of each generalized tetrahedron, including Hayward corner terms, is expressed through its hyperbolic volume via the Schlafli identity, which is what matches the semiclassical 6j-symbol and modular S-matrix.

What would settle it

For a concrete graph such as the handcuff graph in $S^3$, compute both sides of the S-transform identity (1.8) numerically for real weights where the generalized tetrahedron exists, and compare the result with the exponentiated on-shell action, volume plus Hayward corner terms, of the truncated tetrahedron; a mismatch in magnitude or phase would show the exact identification is wrong.

Watch

Extended reading notes

Core claim

The paper seeks to establish the exact identities $Z_L(M,\gamma(P)) = |Z_{\mathrm{vir}}(M_E,\Gamma(P))|^2$ and $Z_A(M,\gamma(P)) = Z_{\mathrm{CTV}}(M_E,\Gamma(P))$, relating the fixed-length and fixed-angle gravitational path integrals on a compact hyperbolic 3-manifold to Virasoro TQFT and Conformal Turaev-Viro theory. The fixed-length amplitude, which computes normalized OPE statistics of heavy operators in the dual CFT, equals the Virasoro TQFT amplitude squared, while the fixed-angle amplitude equals the CTV partition function. The two are connected by a modular S-transform that is exactly the Laplace transform between the two families of boundary conditions. A triangulation of the embedding manifold $M_E$ induces a generalized triangulation of the gravity manifold $M$, so the path integral decomposes into generalized tetrahedra whose semiclassical saddles are truncated hyperbolic tetrahedra. In the examples worked out, the geometry dual to the Virasoro modular S-matrix is a single truncated tetrahedron glued to itself, and the geometry dual to the squared Virasoro 6j-symbol is a truncated tetrahedron doubled along its faces; in both cases the on-shell action, given by volume plus corner terms, matches the exact TQFT results and the conformal bootstrap predictions.

Load-bearing premise

The modular S-transform identity connecting the Conformal Turaev-Viro partition function to the squared Virasoro TQFT amplitude is taken as proved from the companion paper and is not re-derived here; if that identity fails under analytic continuation, the exact identification of the fixed-angle path integral with CTV theory collapses.

Editorial extensions

If this is right

Normalized OPE statistics of heavy black-hole operators are computed by the fixed-length path integral on a compact region, and therefore by the squared Virasoro TQFT amplitude $|Z_{\mathrm{vir}}|^2$.
The fixed-angle amplitude is exactly the Conformal Turaev-Viro partition function, so any triangulation of the embedding graph gives an exact formula for the gravitational path integral on a fixed hyperbolic topology.
The modular S-transform converts fixed-length to fixed-angle boundary conditions exactly, without relying on the semiclassical limit.
Any Virasoro TQFT amplitude on a topology that admits a hyperbolic metric can be translated into a generalized triangulation of the gravity manifold, and hence into a concrete classical saddle built from truncated tetrahedra.
The semiclassical Virasoro 6j-symbol and modular S-matrix reproduce the on-shell action, volume plus corner terms, of the corresponding generalized tetrahedral geometries, matching dual CFT predictions.

Reading between the lines

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

Beyond the paper, the same truncated-tetrahedron decomposition suggests a triangulation-based definition of the path integral on topologies that admit no hyperbolic saddle, which could make a systematic sum over topologies more tractable.
The relation between Conformal Turaev-Viro theory and spin-network models of positive-curvature gravity hints that an analogous exact dictionary may exist for de Sitter space, with external graph edges possibly playing the role of an observer or auxiliary system.
The paper's proposed extension to independent left- and right-moving complex angles implies that fixed-angle amplitudes for spinning operators would no longer be CTV partition functions but would retain the same semiclassical generalized geometries, now with complex edge lengths.
A direct numerical check of the S-transform identity on explicit triangulations, going beyond the two examples computed here, would decisively test whether the exact CTV identification holds for all hyperbolic topologies.

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.

Desk Editor's Note

read the letter

Short version: the genuinely new thing is the translation recipe from Virasoro TQFT data into explicit hyperbolic geometries on compact regions, plus the two worked semiclassical examples. The exact claims are real but not self-contained: the fixed-angle result Z_A = Z_CTV rests on the S-transform identity proved in the companion paper [20], and the paper says so plainly. If you take [20] as given, the logic is coherent and the semiclassical volume matches are genuinely checkable evidence. If you don't, the central exact statement is on loan.

What I liked. The paper is careful about which results are proved here and which are imported. The construction of generalized triangulations—truncating vertices, dualizing external edges—is concrete and algorithmic, and the geometry sections are well done. The two examples are the right ones: the handcuff graph maps to a single truncated tetrahedron, the tetrahedron graph to a doubled truncated tetrahedron, and the fixed-length/fixed-angle actions match the volume-plus-corner-term formulas from the metric formalism. The numerical checks against Ushijima's volume formula give real confidence that the dictionary is pointing at the right geometry. I also think the paper is honest: it flags (3.7) as a conjecture and admits in the Discussion that the fixed-angle amplitude was argued indirectly rather than derived from the gravity path integral.

Soft spots. The load-bearing step, the S-transform (1.8), is imported from [20] with no derivation or convergence analysis here. The fixed-angle identification Z_A = Z_CTV is a direct consequence, so the exact statement is only as strong as [20]. That is acceptable for a two-paper series, but it means a referee cannot fully evaluate the exact claims from this text alone. Also, Appendix A.3 states the volume relation (A.34) and then says 'we omit the details'—that is a key step in matching the semiclassical 6j-symbol to the tetrahedron action, and it needs to be filled in or referenced to a published proof. The dictionary (3.7) is conjectural, though it is a natural extension of existing results and the paper does not oversell it. None of this is fatal; it is just where the weight sits.

Who it is for: people working on AdS3/CFT2, Virasoro TQFT, or triangulated 3D gravity. They will get a practical method and two clean geometries. I would send it for review; a referee should be asked to read the companion paper [20] alongside.

Referee Report

2 major / 5 minor

Summary. The paper formulates the Euclidean path integral of pure AdS3 gravity on compact 3-manifolds with finite boundaries, imposing either fixed geodesic lengths or fixed dihedral angles. It claims two exact results: the fixed-length amplitude equals a Virasoro TQFT amplitude squared, Z_L = |Z_vir|^2 (Eq. (1.7)), and the fixed-angle amplitude equals a Conformal Turaev-Viro partition function, Z_A = Z_CTV (Eqs. (1.9)/(4.1)), with the two related by a modular S-transform (1.8). The semiclassical part constructs the corresponding hyperbolic geometries from generalized tetrahedra and verifies, in two examples, that the on-shell actions computed from the Virasoro 6j-symbol and modular S-matrix agree with the volumes and corner terms of explicit truncated-tetrahedron manifolds. The exact results, however, are largely imported from the companion paper [20], and the paper itself states that the derivation from the gravity path integral is indirect.

Significance. If the exact identification Z_A = Z_CTV survives scrutiny, this is a significant advance: it gives a triangulation-based method for exact quasi-local gravitational amplitudes on fixed hyperbolic topologies and supplies a concrete dictionary between Virasoro TQFT and classical hyperbolic geometry, including explicit geometries for OPE statistics. The semiclassical analysis is a genuine strength: the saddlepoint computations in Appendix A are carried out in detail, the volume relations are checked via the Schläfli identity and against known results [89], and the two examples (modular S-matrix and 6j-symbol) are concrete and nontrivial. The paper also includes a useful tutorial for the Orb software, which may help others implement the triangulation method. The significance is tempered by the fact that the central exact claim depends on a companion-paper identity whose derivation and analytic-continuation hypotheses are not reproduced here, and by the paper's own admission that the derivation from the gravity path integral is indirect.

major comments (2)

[§1, Eq. (1.7); §3.4; §9, item 2] The fixed-length identification Z_L = |Z_vir|^2 is introduced as something the author will 'argue by consistency with known results' rather than derive. The discussion section later concedes that (9.1) was argued 'indirectly' and that a direct derivation from the gravity path integral remains open. This is not merely a matter of presentation: Eq. (1.7) is one of the two exact ingredients that feed into (1.9), and it is also used to connect the geometric examples to OPE statistics. The manuscript should state explicitly whether (1.7) is a conjecture, a theorem proved in [38] under assumptions that are summarized here, or a definition. If it is a conjecture, the abstract's claim that these results are 'derived exactly' should be qualified accordingly.
[§2.1, after Eq. (2.6)] The OPE dictionary in Eq. (3.7) is explicitly labeled a conjecture, yet it is used in Sections 7 and 8 to assert that the fixed-length path integral computes C_{ijk}C_{lmn} and the tetrahedral non-Gaussianity. This does not invalidate the geometric volume checks, which are independent, but it means the CFT-statistics interpretation of the fixed-length result is not part of the 'exact' derivation. Please mark, in the example sections, which claims are conditional on (3.7) and which are unconditional semiclassical matches between the CTV/VTQFT amplitude and the gravitational on-shell action.

minor comments (5)

[§6.1] The word 'manifiestly' should be 'manifestly'.
[§1, paragraph on generalized hyperbolic tetrahedra] The word 'polyedra' should be 'polyhedra'.
[Appendix C] The phrase 'a an easier option' should be 'an easier option'.
[Introduction, subsection heading] The heading 'F rom Virasoro TQFT to geometry' contains a stray space and should read 'From Virasoro TQFT to geometry'.
[§5.2, Eq. (5.9)] The condition for mild truncations is stated as |V_i · V_j| > 1 for all distinct hyperideal vertices, but equation (5.5) suggests the relevant sign convention should be explained; a short comment clarifying whether this is the standard Ushijima condition would help readers apply the existence criterion of §5.3.

Circularity Check

2 steps flagged · score 6.0 of 10

Exact fixed-angle result Z_A=Z_CTV (1.9)/(4.1) is a load-bearing self-citation: it follows immediately from (1.7) plus the S-transform identity (1.8) imported from the author's companion paper [20], while the semiclassical volume checks are independent.

self citation load bearing [Section 4.1, Eq. (4.1); Introduction, Eqs. (1.7)-(1.9)]
"Given (i) the results of the previous section on the fixed-length path integral, (ii) the Laplace transform (1.5) relating fixed-length to fixed-angle boundary conditions, and (iii) the result (1.8), proved in [20], we can immediately determine the exact path integral of gravity with fixed-angle boundary conditions. It is given by the partition function of Conformal Turaev-Viro theory, ZA(M, γ(P)) = ZCTV(ME, Γ(P)) . (4.1)"

The paper's central exact result is not obtained by evaluating the gravitational path integral; it is assembled from Z_L=|Z_vir|^2 (1.7), the Laplace transform (1.5), and the modular S-transform identity (1.8). Substituting (1.7) into (1.8) and comparing with (1.5) gives (1.9)/(4.1) immediately, so the only nontrivial input is the S-transform identity, which is not proved in this paper. The derivation chain therefore reduces to the author's companion paper [20] rather than to an independent calculation here; Discussion item 2 concedes the fixed-angle amplitude was argued 'indirectly'.
self citation load bearing [Introduction, Eq. (1.8)]
"ZCTV(ME, Γ(P′)) = ∫_{R^n_+} dP (Π_{i=1}^n S_{P'_i P_i}) |Zvir(ME, Γ(P))|^2 (1.8) ... A similar result was known previously for discrete spin networks [46,47], and in [20] we show that there is no essential difficulty in generalizing those results to the Virasoro theory, after making a few changes to the definitions to avoid divergences."

This identity is the load-bearing bridge between the fixed-angle amplitude and the triangulation-based CTV partition function. Because the Laplace transform (1.5) has the same cosine kernel, once (1.8) is granted the equality Z_A=Z_CTV is essentially immediate. The proof of (1.8) is cited to the same author's companion paper and is not reproduced; no convergence or analytic-continuation check is given for the imaginary-P fixed-angle regime. Thus the exact claim rests on an unverified-in-this-text self-citation.

full rationale

The semiclassical sections and Appendices A-B are largely self-contained and non-circular: the volume computations use Ushijima's and Murakami-Yano's formulas, the 6j-symbol saddlepoints follow Teschner-Vartanov, and the dual-CFT predictions in Appendix B are independent bootstrap and ensemble calculations. Those checks provide genuine external evidence for the classical action and do not contribute to the circularity score. The exact fixed-angle identification, however, does reduce to a self-citation: Eq. (4.1) follows from (1.5), (1.7), and (1.8), with (1.8) proved only in the author's companion paper [20]. The paper is transparent about this and even lists the indirect derivation as an open issue, but transparency does not convert the unverified import into an independent derivation. The OPE dictionary (3.7) is also stated as a conjecture, so 'These results are derived exactly' overstates the status of the chain even apart from circularity. Because the semiclassical checks are independent but the central exact equality is not, the appropriate score is partial circularity (6).

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

No free parameters are fitted to data: the boundary conditions and normalizations are fixed by consistency with known results and by the chosen parameterization of lengths and angles. The paper relies on standard hyperbolic geometry and on the companion TQFT framework; no new physical entities such as particles or forces are introduced.

assumptions (5)

domain assumption Virasoro TQFT amplitudes are defined by the diagrammatic rules of [38,39] and compute ensemble-averaged OPE statistics.
The paper uses VTQFT as the exact tool for fixed-length amplitudes, citing [38,39] for its definition and properties (Section 1, 3.4).
domain assumption The Conformal Turaev-Viro partition function and the modular S-transform identity (1.8) from companion paper [20] are valid.
The fixed-angle path integral is identified with Z_CTV through (1.8), which is stated to be proved in [20] and not re-derived here (Section 1, Section 4.1).
domain assumption The dictionary (3.7) conjecturally maps normalized OPE statistics to a sum over topologies of fixed-length path integrals on compact manifolds.
This is the AdS/CFT bridge for compact regions; the paper labels it a conjecture and supports it semiclassically by re-attaching C0 flares (Section 3.3).
domain assumption Semiclassical limits of the crossing kernels are dominated by a single saddlepoint, with analytic continuation through branch cuts as specified.
Appendix A computes logarithms of the S-matrix and 6j-symbol by saddlepoint analysis; validity requires the stated i-epsilon prescriptions and convergence of the integrals.
standard math Volume formulas for generalized hyperbolic tetrahedra from the mathematics literature are correct in the regimes used.
The paper cites Ushijima [89], Murakami-Yano [92], Cho-Kim [91], and Teschner-Vartanov [58] for volume formulas and semiclassical 6j-symbols, and uses them to compare actions.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Triangulating quantum gravity in AdS$_3$." pith.science (2026). https://pith.science/paper/6WCW5WCS

@misc{pith2026250712696,
  author       = {Pith},
  title        = {Pith review of: Triangulating quantum gravity in AdS$_3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WCW5WCS}},
  note         = {Machine review of arXiv:2507.12696}
}

abstract

The path integral of pure 3D gravity with negative cosmological constant is formulated on a finite region of spacetime $M$, with boundary conditions that fix geodesic lengths or dihedral angles on $\partial M$. In the dual CFT, this quasi-local amplitude calculates corrections to the Gaussian ensemble of OPE coefficients for black hole states. By triangulating $M$ with generalized tetrahedra, we develop a general method to construct semiclassical geometries and to calculate the exact gravitational path integral on a fixed hyperbolic topology. The path integral with fixed-length boundary conditions is a Virasoro TQFT amplitude-squared, and with fixed-angle boundary conditions it is a partition function of Conformal Turaev-Viro theory. The two are related by a modular S-transform. In addition, we show how to translate the calculation of OPE statistics from Virasoro TQFT to the metric formalism, on general topologies. These results are derived exactly, and some examples are also checked semiclassically, including the geometries dual to the Virasoro 6j-symbol and the modular S-matrix. The classical saddlepoint geometries are finite-volume hyperbolic 3-manifolds ending on pleated Riemann surfaces, which have vanishing extrinsic curvature except on geodesics where they can bend into corners. The hyperbolic volumes of these geometries match the predictions of Conformal Turaev-Viro theory and the dual CFT.

Figures

Figures reproduced from arXiv: 2507.12696 by the authors.

**Figure 2.** Wormhole contributing to the ensemble average [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** The Klein patch Kn = Hn ∪ C ∪ dS+ n , in (a) Poincare coordinates and (b) Klein (projective) coordinates. Geodesic hyperplanes in hyperbolic space correspond to points in de Sitter. Dashed black lines are de Sitter lightcones and red lines are geodesic hyperplanes. If X ∈ Y ⊥, then ΠX ⊥ ΠY as shown in (b). the sphere at infinity identified antipodally. When we discuss the geometry of the Klein patch, we use the metr… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Tetrahedron labeling conventions. Swapping the two rows dualizes the tetrahedron, exchanging faces with vertices. The vertices (V1, V2, V3, V4) are a basis for R n,1 . The dual basis V ∗ i defined by V ∗ i · Vj = δij has (V ∗ i ) 2 > 0, so we can rescale the dual vecto…

Discussion (0). Sign in to comment.

Forward citations

Cited by 4 Pith papers

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

On the Virasoro Crossing Kernels at Rational Central Charge
hep-th 2025-12 conditional novelty 8.0 of 10

At rational central charge, the Virasoro crossing kernels decompose into two admissible square-root-branched kernels; the physical c≤1 kernels are derived for the first time.
A Holographic Map from AdS$_3$ to CFT$_2$
hep-th 2026-07 conditional novelty 6.0 of 10

Semiclassical pure AdS3 gravity states, labelled by fixed-area geodesic networks, are mapped to CFT2 primary states whose wavefunctions are networks of OPE coefficients.
The many facets of a hyperbolic tetrahedron: open and closed triangulations of 3d gravity
hep-th 2026-04 unverdicted novelty 6.0 of 10

Open Virasoro TQFT equals fixed-length/angle 3d gravity path integrals on compact regions and yields the CTV–scalar Virasoro relation via open-closed duality.
Statistics in 3d gravity from knots and links
hep-th 2025-08 conditional novelty 6.0 of 10

Fragmentation of knots and links by Wilson lines builds wormhole amplitudes that give non-perturbative Gaussian and non-Gaussian corrections to OPE statistics in 3d gravity and CFT2.

Reference graph

Works this paper leans on

125 extracted references · 20 canonical work pages · cited by 4 Pith papers

[20]

Conformal Turaev-Viro Theory

T. Hartman, “Conformal Turaev-Viro Theory”, arXiv:2507.11652 [hep-th]

arXiv
[89]

A volume formula for generalised hyperbolic tetrahedra,

A. Ushijima, “A volume formula for generalised hyperbolic tetrahedra,” in Non-Euclidean geometries, vol. 581 of Math. Appl. (N. Y.) , pp. 249–265. Springer, New York, 2006. https://doi.org/10.1007/0-387-29555-0_13 . 67

work page doi:10.1007/0-387-29555-0_13 2006
[38]

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]

arXiv 2023
[1]

A semiclassical ramp in SYK and in gravity,

P. Saad, S. H. Shenker, and D. Stanford, “A semiclassical ramp in SYK and in gravity,” arXiv:1806.06840 [hep-th]

arXiv
[2]

JT gravity as a matrix integral,

P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th]

arXiv 1903
[3]

Replica wormholes and the black hole interior,

G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, “Replica wormholes and the black hole interior,” JHEP 03 (2022) 205, arXiv:1911.11977 [hep-th]

arXiv 2022
[4]

Replica Wormholes and the Entropy of Hawking Radiation,

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” JHEP 05 (2020) 013, arXiv:1911.12333 [hep-th]

arXiv 2020
[5]

AdS 3 gravity and random CFT,

J. Cotler and K. Jensen, “AdS 3 gravity and random CFT,” JHEP 04 (2021) 033, arXiv:2006.08648 [hep-th]

arXiv 2021

Show all 125 references

[6]

Semiclassical 3D gravity as an average of large-c CFTs,

J. Chandra, S. Collier, T. Hartman, and A. Maloney, “Semiclassical 3D gravity as an average of large-c CFTs,” JHEP 12 (2022) 069, arXiv:2203.06511 [hep-th]

2022 arXiv
[7]

Eigenstate Thermalization and Disorder Averaging in Gravity,

J. Pollack, M. Rozali, J. Sully, and D. Wakeham, “Eigenstate Thermalization and Disorder Averaging in Gravity,” Phys. Rev. Lett. 125 no. 2, (2020) 021601, arXiv:2002.02971 [hep-th]

2020 arXiv
[8]

Random statistics of OPE coefficients and Euclidean wormholes,

A. Belin and J. de Boer, “Random statistics of OPE coefficients and Euclidean wormholes,” Class. Quant. Grav. 38 no. 16, (2021) 164001, arXiv:2006.05499 [hep-th]

2021 arXiv
[9]

More quantum noise from wormholes,

D. Stanford, “More quantum noise from wormholes,” arXiv:2008.08570 [hep-th]

2008 arXiv
[10]

Entanglement Entropy at Large Central Charge,

T. Hartman, “Entanglement Entropy at Large Central Charge,” arXiv:1303.6955 [hep-th]. 62

arXiv
[11]

Universal dynamics of heavy operators in CFT2,

S. Collier, A. Maloney, H. Maxfield, and I. Tsiares, “Universal dynamics of heavy operators in CFT2,” JHEP 07 (2020) 074, arXiv:1912.00222 [hep-th]

2020 arXiv
[12]

Free partition functions and an averaged holographic duality,

N. Afkhami-Jeddi, H. Cohn, T. Hartman, and A. Tajdini, “Free partition functions and an averaged holographic duality,” JHEP 01 (2021) 130, arXiv:2006.04839 [hep-th]

2021 arXiv
[13]

Averaging over Narain moduli space,

A. Maloney and E. Witten, “Averaging over Narain moduli space,” JHEP 10 (2020) 187, arXiv:2006.04855 [hep-th]

2020 arXiv
[14]

Approximate CFTs and random tensor models,

A. Belin, J. de Boer, D. L. Jafferis, P. Nayak, and J. Sonner, “Approximate CFTs and random tensor models,” JHEP 09 (2024) 163, arXiv:2308.03829 [hep-th]

2024 arXiv
[15]

AdS 3/RMT2 duality,

G. Di Ubaldo and E. Perlmutter, “AdS 3/RMT2 duality,” JHEP 12 (2023) 179, arXiv:2307.03707 [hep-th]

2023 arXiv
[16]

A principle of maximum ignorance for semiclassical gravity,

J. de Boer, D. Liska, B. Post, and M. Sasieta, “A principle of maximum ignorance for semiclassical gravity,” JHEP 2024 (2024) 003, arXiv:2311.08132 [hep-th]

2024 arXiv
[17]

3d gravity as a random ensemble,

D. L. Jafferis, L. Rozenberg, and G. Wong, “3d gravity as a random ensemble,” JHEP 02 (2025) 208, arXiv:2407.02649 [hep-th]

2025 arXiv
[18]

Wormholes with Ends of the World,

D. Wang, Z. Wang, and Z. Wei, “Wormholes with Ends of the World,” arXiv:2504.12278 [hep-th]
[19]

Surgery and statistics in 3d gravity,

J. de Boer, J. Kames-King, and B. Post, “Surgery and statistics in 3d gravity,” arXiv:2506.04151 [hep-th]

arXiv
[21]

The Factorization Problem in Jackiw-Teitelboim Gravity,

D. Harlow and D. Jafferis, “The Factorization Problem in Jackiw-Teitelboim Gravity,” JHEP 02 (2020) 177, arXiv:1804.01081 [hep-th]

2020 arXiv
[22]

The spacetime geometry of fixed-area states in gravitational systems,

X. Dong, D. Marolf, P. Rath, A. Tajdini, and Z. Wang, “The spacetime geometry of fixed-area states in gravitational systems,” JHEP 08 (2022) 158, arXiv:2203.04973 [hep-th]

2022 arXiv
[23]

Black hole wavefunctions and microcanonical states,

W. Z. Chua and T. Hartman, “Black hole wavefunctions and microcanonical states,” JHEP 06 (2024) 054, arXiv:2309.05041 [hep-th]

2024 arXiv
[24]

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

W. Z. Chua and Y. Jiang, “Hartle-Hawking state and its factorization in 3d gravity,” JHEP 03 (2024) 135, arXiv:2309.05126 [hep-th]

2024 arXiv
[25]

Statistics of three-dimensional black holes from Liouville line defects,

J. Chandra, T. Hartman, and V. Meruliya, “Statistics of three-dimensional black holes from Liouville line defects,” JHEP 11 (2024) 090, arXiv:2404.15183 [hep-th]

2024 arXiv
[26]

Holographic Renyi Entropy from Quantum Error Correction,

C. Akers and P. Rath, “Holographic Renyi Entropy from Quantum Error Correction,” JHEP 05 (2019) 052, arXiv:1811.05171 [hep-th]

2019 arXiv
[27]

Flat entanglement spectra in fixed-area states of quantum gravity,

X. Dong, D. Harlow, and D. Marolf, “Flat entanglement spectra in fixed-area states of quantum gravity,” JHEP 10 (2019) 240, arXiv:1811.05382 [hep-th]. 63

2019 arXiv
[28]

Covariant phase space with boundaries,

D. Harlow and J.-Q. Wu, “Covariant phase space with boundaries,” JHEP 10 (2020) 146, arXiv:1906.08616 [hep-th]

2020 arXiv
[29]

Large N algebras and generalized entropy,

V. Chandrasekaran, G. Penington, and E. Witten, “Large N algebras and generalized entropy,” JHEP 04 (2023) 009, arXiv:2209.10454 [hep-th]

2023 arXiv
[30]

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
[31]

(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
[32]

Conformal Field Theory, 2- D Quantum Gravity and Quantization of Teichmuller Space,

H. L. Verlinde, “Conformal Field Theory, 2- D Quantum Gravity and Quantization of Teichmuller Space,” Nucl. Phys. B 337 (1990) 652–680

1990
[33]

3-D gravity, point particles and Liouville theory,

K. Krasnov, “3-D gravity, point particles and Liouville theory,” Class. Quant. Grav. 18 (2001) 1291–1304, arXiv:hep-th/0008253

2001 arXiv
[34]

Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography,

L. A. Takhtajan and L.-P. Teo, “Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography,” Commun. Math. Phys. 239 (2003) 183–240, arXiv:math/0204318

2003 arXiv
[35]

The universal phase space of AdS3 gravity,

C. Scarinci and K. Krasnov, “The universal phase space of AdS3 gravity,” Commun. Math. Phys. 322 (2013) 167–205, arXiv:1111.6507 [hep-th]

2013 arXiv
[36]

A TQFT from Quantum Teichm¨ uller Theory,

J. Ellegaard Andersen and R. Kashaev, “A TQFT from Quantum Teichm¨ uller Theory,” Commun. Math. Phys. 330 (2014) 887–934, arXiv:1109.6295 [math.QA]

2014 arXiv
[37]

A new formulation of the teichm \

J. E. Andersen and R. Kashaev, “A new formulation of the teichm \” uller tqft,” arXiv preprint arXiv:1305.4291 (2013)

2013 arXiv
[39]

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 no. 5, (2024) 134, arXiv:2401.13900 [hep-th]

2024 arXiv
[40]

More on torus wormholes in 3d gravity,

C. Yan, “More on torus wormholes in 3d gravity,” JHEP 11 (2023) 039, arXiv:2305.10494 [hep-th]

2023 arXiv
[41]

Multiboundary wormholes and OPE statistics,

J. de Boer, D. Liska, and B. Post, “Multiboundary wormholes and OPE statistics,” JHEP 10 (2024) 207, arXiv:2405.13111 [hep-th]

2024 arXiv
[42]

A non-rational Verlinde formula from Virasoro TQFT,

B. Post and I. Tsiares, “A non-rational Verlinde formula from Virasoro TQFT,” arXiv:2411.07285 [hep-th]

arXiv
[43]

Reduction of the Einstein equations in (2+1)-dimensions to a Hamiltonian system over Teichmuller space,

V. Moncrief, “Reduction of the Einstein equations in (2+1)-dimensions to a Hamiltonian system over Teichmuller space,” J. Math. Phys. 30 (1989) 2907–2914. 64

1989
[44]

Minimal surfaces and particles in 3-manifolds,

K. Krasnov and J.-M. Schlenker, “Minimal surfaces and particles in 3-manifolds,” Geom. Dedicata 126 (2007) 187–254, arXiv:math/0511441

2007 arXiv
[45]

On a Canonical Quantization of 3D Anti de Sitter Pure Gravity,

J. Kim and M. Porrati, “On a Canonical Quantization of 3D Anti de Sitter Pure Gravity,” JHEP 10 (2015) 096, arXiv:1508.03638 [hep-th]

2015 arXiv
[46]

Skein theory and turaev-viro invariants,

J. Roberts, “Skein theory and turaev-viro invariants,” Topology 34 no. 4, (1995) 771–787

1995
[47]

Observables in the Turaev-Viro and Crane-Yetter models,

J. W. Barrett, J. M. Garcia-Islas, and J. F. Martins, “Observables in the Turaev-Viro and Crane-Yetter models,” J. Math. Phys. 48 (2007) 093508, arXiv:math/0411281

2007 arXiv
[48]

The geometry and topology of three-manifolds,

W. P. Thurston, “The geometry and topology of three-manifolds,” Princeton Math Dept. (1979)

1979
[49]

Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds,

W. P. Thurston, “Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds,” Ann. of Math. (2) 124 no. 2, (1986) 203–246. https://doi.org/10.2307/1971277

1986 doi
[50]

Thermal probes of hyperbolic knots,

T. Hartman, “Thermal probes of hyperbolic knots,” talk at SCGP, June 2, 2025. Paper to appear

2025
[51]

On the renormalized volume of hyperbolic 3-manifolds,

K. Krasnov and J.-M. Schlenker, “On the renormalized volume of hyperbolic 3-manifolds,” Commun. Math. Phys. 279 (2008) 637–668, arXiv:math/0607081

2008 arXiv
[52]

Multi black holes and earthquakes on Riemann surfaces with boundaries,

F. Bonsante, K. Krasnov, and J.-M. Schlenker, “Multi black holes and earthquakes on Riemann surfaces with boundaries,” Int. Math. Res. Not. 2010 (2010) 070, arXiv:math/0610429

2010 arXiv
[53]

A Symplectic map between hyperbolic and complex Teichmuller theory,

K. Krasnov and J.-M. Schlenker, “A Symplectic map between hyperbolic and complex Teichmuller theory,” Duke Math. J. 150 (2009) 331–356, arXiv:0806.0010 [math.DG]

2009 arXiv
[54]

The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds,

K. Krasnov and J.-M. Schlenker, “The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds,” arXiv:0907.2590 [math.DG]

arXiv
[55]

Renormalized volume and the volume of the convex core,

M. Bridgeman and R. D. Canary, “Renormalized volume and the volume of the convex core,” Ann. Inst. Fourier (Grenoble) 67 no. 5, (2017) 2083–2098, 1502.05018. https://doi.org/10.5802/aif.3130

2017 arXiv
[56]

Volumes of quasifuchsian manifolds,

J.-M. Schlenker, “Volumes of quasifuchsian manifolds,” in Surveys in 3-manifold topology and geometry, vol. 25 of Surv. Differ. Geom. , pp. 319–353. Int. Press, Boston, MA, 2022. 1903.09849

2022 arXiv
[57]

Puzzles in 3D Off-Shell Geometries via VTQFT,

C. Yan, “Puzzles in 3D Off-Shell Geometries via VTQFT,” arXiv:2502.16686 [hep-th]
[58]

6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories,

J. Teschner and G. Vartanov, “6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories,” Lett. Math. Phys. 104 (2014) 527–551, arXiv:1202.4698 [hep-th]. 65

2014 arXiv
[59]

Quantum 2D Liouville Path-Integral Is a Sum over Geometries in AdS 3 Einstein Gravity,

L. Chen, L.-Y. Hung, Y. Jiang, and B.-X. Lao, “Quantum 2D Liouville Path-Integral Is a Sum over Geometries in AdS 3 Einstein Gravity,” arXiv:2403.03179 [hep-th]
[60]

Building up quantum spacetimes with BCFT Legos,

L.-Y. Hung and Y. Jiang, “Building up quantum spacetimes with BCFT Legos,” arXiv:2404.00877 [hep-th]

arXiv
[61]

QG from SymQRG: AdS 3/CFT2 Correspondence as Topological Symmetry-Preserving Quantum RG Flow,

N. Bao, L.-Y. Hung, Y. Jiang, and Z. Liu, “QG from SymQRG: AdS 3/CFT2 Correspondence as Topological Symmetry-Preserving Quantum RG Flow,” arXiv:2412.12045 [hep-th]

arXiv
[62]

Universal Structures and Emergent Geometry from Large-c BCFT Ensemble,

L.-Y. Hung, Y. Jiang, and B.-X. Lao, “Universal Structures and Emergent Geometry from Large-c BCFT Ensemble,” arXiv:2504.21660 [hep-th]

arXiv
[63]

It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-c BCFT Ensemble,

H. Geng, L.-Y. Hung, and Y. Jiang, “It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-c BCFT Ensemble,” arXiv:2505.20385 [hep-th]

arXiv
[64]

Geometrical measurements in three-dimensional quantum gravity,

J. W. Barrett, “Geometrical measurements in three-dimensional quantum gravity,” Int. J. Mod. Phys. A 18S2 (2003) 97–113, arXiv:gr-qc/0203018

2003 arXiv
[65]

State sum invariants of 3 manifolds and quantum 6j symbols,

V. G. Turaev and O. Y. Viro, “State sum invariants of 3 manifolds and quantum 6j symbols,” Topology 31 (1992) 865–902

1992
[66]

Semiclassical limit of racah coefficients,

G. Ponzano and T. E. Regge, “Semiclassical limit of racah coefficients,” Spectroscopic and Group Theoretical Methods in Physics (1968) 1–58

1968
[67]

Relativistic spin networks and quantum gravity,

J. W. Barrett and L. Crane, “Relativistic spin networks and quantum gravity,” J. Math. Phys. 39 (1998) 3296–3302, arXiv:gr-qc/9709028

1998 arXiv
[68]

Observables in three-dimensional quantum gravity and topological invariants,

J. M. Garcia-Islas, “Observables in three-dimensional quantum gravity and topological invariants,” Class. Quant. Grav. 21 (2004) 3933–3952, arXiv:gr-qc/0401093

2004 arXiv
[69]

The hyperbolic volume of knots from the quantum dilogarithm,

R. M. Kashaev, “The hyperbolic volume of knots from the quantum dilogarithm,” Lett. Math. Phys. 39 no. 3, (1997) 269–275. https://doi.org/10.1023/A:1007364912784

1997 doi
[70]

6j–symbols, hyperbolic structures and the volume conjecture,

F. Costantino, “6j–symbols, hyperbolic structures and the volume conjecture,” Geom. Topol. 11 no. 3, (2007) 1831–1854

2007
[71]

On the SL(2, C) quantum 6j-symbols and their relation to the hyperbolic volume,

F. Costantino and J. Murakami, “On the SL(2, C) quantum 6j-symbols and their relation to the hyperbolic volume,” Quantum Topol. 4 no. 3, (2013) 303–351. https://doi.org/10.4171/QT/41

2013 doi
[72]

Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants,

Q. Chen and T. Yang, “Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants,” Quantum Topol. 9 no. 3, (2018) 419–460, arXiv:1503.02547 [math.GT]

2018 arXiv
[73]

An algebra of observables for de Sitter space,

V. Chandrasekaran, R. Longo, G. Penington, and E. Witten, “An algebra of observables for de Sitter space,” JHEP 02 (2023) 082, arXiv:2206.10780 [hep-th]

2023 arXiv
[74]

A microscopic realization of dS 3,

S. Collier, L. Eberhardt, and B. M¨ uhlmann, “A microscopic realization of dS 3,” SciPost Phys. 18 no. 4, (2025) 131, arXiv:2501.01486 [hep-th]. 66

2025 arXiv
[75]

The gravitational path integral from an observer’s point of view,

A. I. Abdalla, S. Antonini, L. V. Iliesiu, and A. Levine, “The gravitational path integral from an observer’s point of view,” JHEP 05 (2025) 059, arXiv:2501.02632 [hep-th]

2025 arXiv
[76]

Quantum mechanics and observers for gravity in a closed universe,

D. Harlow, M. Usatyuk, and Y. Zhao, “Quantum mechanics and observers for gravity in a closed universe,” arXiv:2501.02359 [hep-th]

arXiv
[77]

Universal asymptotics for high energy CFT data,

N. Benjamin, J. Lee, H. Ooguri, and D. Simmons-Duffin, “Universal asymptotics for high energy CFT data,” JHEP 03 (2024) 115, arXiv:2306.08031 [hep-th]

2024 arXiv
[78]

Notes on crossing transformations of Virasoro conformal blocks,

L. Eberhardt, “Notes on crossing transformations of Virasoro conformal blocks,” arXiv:2309.11540 [hep-th]

arXiv
[79]

Coarse graining pure states in AdS/CFT,

J. Chandra and T. Hartman, “Coarse graining pure states in AdS/CFT,” JHEP 10 (2023) 030, arXiv:2206.03414 [hep-th]

2023 arXiv
[80]

Toward random tensor networks and holographic codes in CFT,

J. Chandra and T. Hartman, “Toward random tensor networks and holographic codes in CFT,” JHEP 05 (2023) 109, arXiv:2302.02446 [hep-th]

2023 arXiv
[81]

Euclidean wormholes for individual 2d CFTs,

J. Chandra, “Euclidean wormholes for individual 2d CFTs,” JHEP 04 (2024) 051, arXiv:2305.07183 [hep-th]

2024 arXiv
[82]

Wormholes in AdS,

J. M. Maldacena and L. Maoz, “Wormholes in AdS,” JHEP 02 (2004) 053, arXiv:hep-th/0401024

2004 arXiv
[83]

Holography and correlation functions of huge operators: spacetime bananas,

J. Abajian, F. Aprile, R. C. Myers, and P. Vieira, “Holography and correlation functions of huge operators: spacetime bananas,” JHEP 12 (2023) 058, arXiv:2306.15105 [hep-th]

2023 arXiv
[84]

Correlation functions of huge operators in AdS3/CFT2: domes, doors and book pages,

J. Abajian, F. Aprile, R. C. Myers, and P. Vieira, “Correlation functions of huge operators in AdS3/CFT2: domes, doors and book pages,” JHEP 03 (2024) 118, arXiv:2307.13188 [hep-th]

2024 arXiv
[85]

J. S. Purcell, Hyperbolic knot theory, vol. 209 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, [2020] ©2020. https://doi.org/10.1090/gsm/209

2020 doi
[86]

Heard, Computation of Hyperbolic Structures on 3 Dimensional Orbifolds

D. Heard, Computation of Hyperbolic Structures on 3 Dimensional Orbifolds . PhD thesis, University of Melbourne, Department of Mathematics and Statistics, 2005

2005
[87]

Hyperbolic graphs of small complexity,

D. Heard, C. Hodgson, B. Martelli, and C. Petronio, “Hyperbolic graphs of small complexity,” Experiment. Math. 19 no. 2, (2010) 211–236, 0804.4790. http://projecteuclid.org/euclid.em/1276784791

2010 arXiv
[88]

W. P. Thurston, Three-dimensional geometry and topology. Vol. 1 , vol. 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997

1997
[90]

The analytic continuation of hyperbolic space,

Y. Cho and H. Kim, “The analytic continuation of hyperbolic space,” Geom. Dedicata 161 (2012) 129–155. https://doi.org/10.1007/s10711-012-9698-0

2012 doi
[91]

On the volume formula for hyperbolic tetrahedra,

Y. Cho and H. Kim, “On the volume formula for hyperbolic tetrahedra,” Discrete Comput. Geom. 22 no. 3, (1999) 347–366. https://doi.org/10.1007/PL00009465

1999 doi
[92]

On the volume of a hyperbolic and spherical tetrahedron,

J. Murakami and M. Yano, “On the volume of a hyperbolic and spherical tetrahedron,” Comm. Anal. Geom. 13 no. 2, (2005) 379–400. https://doi.org/10.4310/cag.2005.v13.n2.a5

2005 doi
[93]

A volume formula for hyperbolic tetrahedra in terms of edge lengths,

J. Murakami and A. Ushijima, “A volume formula for hyperbolic tetrahedra in terms of edge lengths,” J. Geom. 83 no. 1-2, (2005) 153–163. https://doi.org/10.1007/s00022-005-0010-4

2005 doi
[94]

Holography for the Lorentz group Racah coefficients,

K. Krasnov, “Holography for the Lorentz group Racah coefficients,” Class. Quant. Grav. 22 (2005) 1933–1944, arXiv:gr-qc/0412104

2005 arXiv
[95]

SO(1,d+1) Racah coefficients: Type I representations,

K. Krasnov and J. Louko, “SO(1,d+1) Racah coefficients: Type I representations,” J. Math. Phys. 47 (2006) 033513, arXiv:math-ph/0502017

2006 arXiv
[96]

d-dimensional SYK, AdS Loops, and 6 j Symbols,

J. Liu, E. Perlmutter, V. Rosenhaus, and D. Simmons-Duffin, “ d-dimensional SYK, AdS Loops, and 6 j Symbols,” JHEP 03 (2019) 052, arXiv:1808.00612 [hep-th]

2019 arXiv
[97]

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
[98]

The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral,

H. Maxfield and G. J. Turiaci, “The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral,” JHEP 01 (2021) 118, arXiv:2006.11317 [hep-th]

2021 arXiv
[99]

Modular-invariant random matrix theory and AdS 3 wormholes,

J. Boruch, G. Di Ubaldo, F. M. Haehl, E. Perlmutter, and M. Rozali, “Modular-invariant random matrix theory and AdS 3 wormholes,” arXiv:2503.00101 [hep-th]
[100]

Intersection theory on the moduli space of curves and the matrix Airy function,

M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function,” Commun. Math. Phys. 147 (1992) 1–23

1992
[101]

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,” JHEP 05 (2020) 104, arXiv:1912.07654 [hep-th]

2020 arXiv
[102]

A convergent genus expansion for the plateau,

P. Saad, D. Stanford, Z. Yang, and S. Yao, “A convergent genus expansion for the plateau,” JHEP 09 (2024) 033, arXiv:2210.11565 [hep-th]

2024 arXiv
[103]

An integrable road to a perturbative plateau,

A. Blommaert, J. Kruthoff, and S. Yao, “An integrable road to a perturbative plateau,” JHEP 04 (2023) 048, arXiv:2208.13795 [hep-th]

2023 arXiv
[104]

Semi-classical dilaton gravity and the very blunt defect expansion,

J. Kruthoff and A. Levine, “Semi-classical dilaton gravity and the very blunt defect expansion,” arXiv:2402.10162 [hep-th]. 68

arXiv
[105]

The Page curve of Hawking radiation from semiclassical geometry,

A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, “The Page curve of Hawking radiation from semiclassical geometry,” JHEP 03 (2020) 149, arXiv:1908.10996 [hep-th]

2020 arXiv
[106]

Bra-ket wormholes in gravitationally prepared states,

Y. Chen, V. Gorbenko, and J. Maldacena, “Bra-ket wormholes in gravitationally prepared states,” JHEP 02 (2021) 009, arXiv:2007.16091 [hep-th]

2021 arXiv
[107]

Islands in cosmology,

T. Hartman, Y. Jiang, and E. Shaghoulian, “Islands in cosmology,” JHEP 11 (2020) 111, arXiv:2008.01022 [hep-th]

2020 arXiv
[108]

Real observers solving imaginary problems,

J. Maldacena, “Real observers solving imaginary problems,” arXiv:2412.14014 [hep-th]
[109]

Physical instabilities and the phase of the Euclidean path integral,

V. Ivo, J. Maldacena, and Z. Sun, “Physical instabilities and the phase of the Euclidean path integral,” arXiv:2504.00920 [hep-th]
[110]

Open-closed 3d gravity as a random ensemble,

D. L. Jafferis, L. Rozenberg, and D. Wang, “Open-closed 3d gravity as a random ensemble,” arXiv:2506.19817 [hep-th]
[111]

Fateev, A

V. Fateev, A. B. Zamolodchikov, and A. B. Zamolodchikov, “Boundary Liouville field theory
[112]

Boundary state and boundary two point function,” arXiv:hep-th/0001012

arXiv
[113]

Supersymmetric gauge theories, quantization of Mflat, and conformal field theory,

J. Teschner and G. S. Vartanov, “Supersymmetric gauge theories, quantization of Mflat, and conformal field theory,” Adv. Theor. Math. Phys. 19 (2015) 1–135, arXiv:1302.3778 [hep-th]

2015 arXiv
[114]

Combinatorial decompositions, kirillov–reshetikhin invariants, and the volume conjecture for hyperbolic polyhedra,

A. Kolpakov and J. Murakami, “Combinatorial decompositions, kirillov–reshetikhin invariants, and the volume conjecture for hyperbolic polyhedra,” Experimental Mathematics 27 no. 2, (2018) 193–207, 1603.02380

2018 arXiv
[115]

Discrete fourier transforms, quantum 6 j-symbols and deeply truncated tetrahedra,

G. Belletti and T. Yang, “Discrete fourier transforms, quantum 6 j-symbols and deeply truncated tetrahedra,” arXiv:2009.03684 (2020)

2020 arXiv
[116]

Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes,

A. Belin, J. de Boer, and D. Liska, “Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes,” JHEP 06 (2022) 116, arXiv:2110.14649 [hep-th]

2022 arXiv
[117]

Generalized spectral form factors and the statistics of heavy operators,

A. Belin, J. de Boer, P. Nayak, and J. Sonner, “Generalized spectral form factors and the statistics of heavy operators,” JHEP 11 (2022) 145, arXiv:2111.06373 [hep-th]

2022 arXiv
[118]

OPE statistics from higher-point crossing,

T. Anous, A. Belin, J. de Boer, and D. Liska, “OPE statistics from higher-point crossing,” JHEP 06 (2022) 102, arXiv:2112.09143 [hep-th]

2022 arXiv
[119]

Black hole entropy from near horizon microstates,

A. Strominger, “Black hole entropy from near horizon microstates,” JHEP 02 (1998) 009, arXiv:hep-th/9712251

1998 arXiv
[120]

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,” JHEP 09 (2014) 118, arXiv:1405.5137 [hep-th]

2014 arXiv
[121]

Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U(q)(sl(2,R)),

B. Ponsot and J. Teschner, “Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U(q)(sl(2,R)),” Commun. Math. Phys. 224 (2001) 613–655, arXiv:math/0007097. 69

2001 arXiv
[122]

Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches,

C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman, “Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches,” JHEP 02 (2015) 171, arXiv:1410.1392 [hep-th]

2015 arXiv
[123]

Black Hole Collapse in the 1/c Expansion,

T. Anous, T. Hartman, A. Rovai, and J. Sonner, “Black Hole Collapse in the 1/c Expansion,” JHEP 07 (2016) 123, arXiv:1603.04856 [hep-th]

2016 arXiv
[124]

A conformal block Farey tail,

A. Maloney, H. Maxfield, and G. S. Ng, “A conformal block Farey tail,” JHEP 06 (2017) 117, arXiv:1609.02165 [hep-th]

2017 arXiv
[125]

From Conformal Blocks to Path Integrals in the Vaidya Geometry,

T. Anous, T. Hartman, A. Rovai, and J. Sonner, “From Conformal Blocks to Path Integrals in the Vaidya Geometry,” JHEP 09 (2017) 009, arXiv:1706.02668 [hep-th]. 70

2017 arXiv

Pith tools

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

[1] [20]

Conformal Turaev-Viro Theory

T. Hartman, “Conformal Turaev-Viro Theory”, arXiv:2507.11652 [hep-th]

arXiv

[2] [89]

A volume formula for generalised hyperbolic tetrahedra,

A. Ushijima, “A volume formula for generalised hyperbolic tetrahedra,” in Non-Euclidean geometries, vol. 581 of Math. Appl. (N. Y.) , pp. 249–265. Springer, New York, 2006. https://doi.org/10.1007/0-387-29555-0_13 . 67

work page doi:10.1007/0-387-29555-0_13 2006

[3] [38]

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]

arXiv 2023

[4] [1]

A semiclassical ramp in SYK and in gravity,

P. Saad, S. H. Shenker, and D. Stanford, “A semiclassical ramp in SYK and in gravity,” arXiv:1806.06840 [hep-th]

arXiv

[5] [2]

JT gravity as a matrix integral,

P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th]

arXiv 1903

[6] [3]

Replica wormholes and the black hole interior,

G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, “Replica wormholes and the black hole interior,” JHEP 03 (2022) 205, arXiv:1911.11977 [hep-th]

arXiv 2022

[7] [4]

Replica Wormholes and the Entropy of Hawking Radiation,

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” JHEP 05 (2020) 013, arXiv:1911.12333 [hep-th]

arXiv 2020

[8] [5]

AdS 3 gravity and random CFT,

J. Cotler and K. Jensen, “AdS 3 gravity and random CFT,” JHEP 04 (2021) 033, arXiv:2006.08648 [hep-th]

arXiv 2021