Anyon Condensation in Virasoro TQFT: Wormhole Factorization

Shunta Takahashi

REVIEW 3 major objections 5 minor 1 cited by

Anyon Condensation in Virasoro TQFT: Wormhole Factorization

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Condensing the continuous diagonal anyon in Virasoro TQFT factorizes two-boundary wormhole amplitudes into products of Liouville CFT partition functions.

desk verdict A substantive and transparent formal computation that establishes wormhole factorization in VTQFT only conditional on a 'condensable anyon' the paper itself admits is not mathematically defined; worth refereeing, but it is a computation, not a theorem. read the letter →

arxiv 2412.11486 v3 pith:AZ6EUXDS submitted 2024-12-16 hep-th cond-mat.str-el

Shunta Takahashi This is my paper · ORCID

classification hep-thcond-mat.str-el

keywords VirasoroTQFTanyoncondensationwormholefactorizationLiouvilleCFTnon-invertiblesymmetry3dAdSquantumgravitycontinuousWilsonlinesmodulartensorcategory

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

Virasoro TQFT is a proposed reformulation of three-dimensional AdS quantum gravity whose boundary Hilbert spaces are Virasoro conformal blocks of Liouville theory. The paper's central claim is that gauging a continuous non-invertible symmetry by condensing the diagonal anyon $\mathcal{A}=\int_0^\infty dp\,L_p\boxtimes\overline{L}_p$ makes the partition functions of two-boundary wormhole geometries factorize into the product of two Liouville CFT partition functions, with no sum over bulk topologies needed. The computation is carried out explicitly for the torus wormhole $\Sigma_{1,1}\times[0,1]$ and the genus-two wormhole $\Sigma_{2,0}\times[0,1]$, each with a Wilson line $L_p\boxtimes\overline{L}_p$ connecting the boundaries. If the paper is right, this supplies one of the first concrete examples of gauging a continuous non-invertible symmetry, and it resolves the wormhole factorization puzzle inside the VTQFT framework.

What carries the argument

The argument is carried by a projector identity (4.8): any condensable anyon inserted on a fine mesh pulls a Wilson-line network down to $\delta(p_x-\overline{p}_x)/\rho_0(p_x)$ times the diagonal component of that line. The projector is derived from the Verlinde-loop identity, the Wilson bubble and Wilson triangle identities, and the Moore-Seiberg consistency equations for the continuous crossing kernels $F$ and $S$; those tools, together with the superposition rule for $\mathcal{A}$ (weighted by $\rho_0$ and $C_0$ factors), convert the complicated link integral into the factorized product of two boundary states.

What would settle it

Compute the condensation of $\mathcal{A}$ on a fixed wormhole using two genuinely different fine meshes and compare the resulting $\langle\tau_1,\tau_2|\Sigma,\mathcal{A}\rangle$; if the answer changes, or if the algebra data of $\mathcal{A}$ (product $m$ and coproduct $\Delta$) fail the separability or Frobenius equations in the continuous limit, the factorization claim collapses.

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Extended reading notes

Core claim

Within Virasoro TQFT, the paper shows that placing the continuous diagonal condensable anyon $\mathcal{A}=\int_0^\infty dp\,L_p\boxtimes\overline{L}_p$ on the fine mesh of the wormhole $\Sigma_{1,1}\times[0,1]$ or $\Sigma_{2,0}\times[0,1]$ collapses the Wilson-line network so that the two-boundary amplitude becomes $$\langle \tau_1,\tau_2|\Sigma,\mathcal{A}\rangle=\frac{\delta(p-p')}{\rho_0(p)}\langle \tau_1|\Sigma,\mathcal{A}\rangle\,\langle \tau_2|\Sigma,\mathcal{A}\rangle .$$ Each boundary factor is identified, via the symmetry-TFT sandwich construction, with a Liouville CFT amplitude on the corresponding boundary Riemann surface. The delta function in the prefactor enforces that only diagonal Wilson lines can terminate on the topological boundary produced by the condensation. The paper concludes that after condensation the partition function no longer depends on the bulk topology, so the factorization puzzle is resolved for these geometries.

Load-bearing premise

The load-bearing premise is that $\mathcal{A}=\int_0^\infty dp\,L_p\boxtimes \overline{L}_p$ genuinely is a condensable anyon, meaning it obeys the algebraic axioms that make condensation independent of how the space is cut into pieces, even though the paper does not construct those axioms for this continuous object.

Editorial extensions

If this is right

After condensation, the wormhole amplitude is fixed by the product of two boundary Liouville amplitudes, with the bulk topology playing no role.
The explicit computation covers $\Sigma_{1,1}\times[0,1]$ and $\Sigma_{2,0}\times[0,1]$, and the paper argues the same mechanism extends to other hyperbolic $\Sigma_{g,n}\times[0,1]$ and to multi-boundary wormholes with several Wilson lines.
Since $\mathcal{A}$ does not contain the identity line $1\boxtimes 1$, the resulting boundary theory has no vacuum sector, which is consistent with the Liouville vacuum being unnormalizable.
The limit $p,\bar p\to 1$ after condensation yields a well-defined factorization for the pure genus-two wormhole, while the pure torus wormhole remains ill-defined because $\Sigma_{1,0}\times[0,1]$ is not hyperbolic.
The symmetry-TFT sandwich construction identifies each boundary factor with a Liouville CFT amplitude, giving a concrete boundary description of the condensed bulk.

Reading between the lines

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

If the continuous direct integral $\mathcal{A}$ can be given a rigorous categorical meaning, the same projector argument would likely extend trivialization and factorization to all multi-boundary wormholes, making Virasoro TQFT a fully explicit example of ensemble holography without a sum over topologies.
The projector/Verlinde-loop technique could be reused to derive non-rational Verlinde formulae or boundary OPE statistics in the condensed theory, since all crossing kernels used here are explicit.
A direct test of the condensation prescription would be to compute the boundary factor $\langle \tau|\Sigma,\mathcal{A}\rangle$ independently in Liouville CFT and compare it with the VTQFT expression obtained from the projector; agreement would confirm that the formal continuum condensation is the correct gauging operation.
The paper notes that commutative candidates of the form $L_s\boxtimes L_t$ exist for $s^2-t^2\in\mathbb{Z}$; classifying which of these are genuinely condensable could produce non-diagonal boundary CFTs, potentially linking the construction to the non-rational limit of D-series minimal models.

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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

Quick take: this is a real computation, honestly scoped, and the explicit VTQFT factorization is new as far as I can tell — [26] treated finite-Chern-Simons/MTC, [12,29] only suggested the VTQFT version. But the central object is not a condensable anyon in the technical sense, the author knows it, and says so. Read it as a conditional computation, not a theorem.

What earns credit. Section 4 is laborious and internally consistent. The projector (4.8) is a genuinely new building block, and the two wormhole examples are worked in enough detail that a referee can check the crossing-kernel moves. The contrast between the torus case (4.15), which honestly hits δ(0) divergences, and the genus-two p,p→1 limit (4.24), which goes through cleanly, is the right way to handle a non-hyperbolic geometry. The boundary theory being Liouville CFT via the symmetry-TFT sandwich is a clean punchline. Bonus: Section 2's claim that the Hawking-Page transition occurs exactly once for any c>1, not just in the semiclassical limit, is worth noting. Citation pattern looks appropriate.

The weak spot is the one the stress test pinned, and it is load-bearing. A condensable anyon needs a unit 1→A. For A = ∫_{R≥0} dp L_p ⊠ L_p the identity sits at p = ±iQ/2, outside the integration domain, so Hom(1,A) = 0 and the unit axiom fails before we get to separability or Frobenius. No algebra or coproduct for a continuous direct integral in this non-semisimple, non-locally-finite category is constructed — Section 5 concedes this flatly. So (4.8), (4.13) and (4.23) are formal manipulations of an object that does not satisfy the axioms laid out in Section 3. To his credit the author says this himself; it is not hidden. But it means the central claim rests on a gap the paper identifies. The acknowledgments' mention of a referee-flagged genus-2 issue also tells me the computation has already needed one round of correction.

Bottom line: I would send this out. The computation is exactly the kind of explicit check the VTQFT program needs, the gap is stated plainly, and a good referee can push on whether the projector can be derived from a regularized or otherwise well-defined object — or whether the paper should be reframed as a formal argument with open validity. People working on the factorization puzzle and on continuous non-invertible symmetries will want it, with caveats.

Referee Report

3 major / 5 minor

Summary. The paper develops an anyon-condensation formalism for the Virasoro TQFT and claims that condensing the object A = ∫_0^∞ dp L_p ⊠ L_p on two-boundary wormhole geometries makes the partition function factorize. The relevant computations are presented for the torus wormhole Σ_{1,1}×[0,1] and the genus-two wormhole Σ_{2,0}×[0,1], each with a Wilson line L_p ⊠ L_p connecting the two boundaries. The paper first reviews the VTQFT Hilbert space, crossing kernels, Heegaard-splitting rules, and anyon condensation in modular tensor categories, then defines the projector identity that is the computational engine of the factorization. The final section claims, from the sandwich construction, that the resulting boundary theory is Liouville CFT, and it projects the result to arbitrary hyperbolic two-boundary geometries.

Significance. If the central claim were established, the paper would be a notable advance: it would provide a concrete example of gauging a continuous non-invertible symmetry, and it would connect wormhole factorization in 3d gravity to anyon condensation and Liouville CFT. The manuscript has real strengths: the crossing-kernel computations are detailed and traceable, the divergence issues are discussed transparently rather than hidden, no parameters are fitted, and the review of Moore-Seiberg consistency conditions is useful. The derivation of the Hawking-Page transition and the BTZ entropy within VTQFT is also a valuable consistency check. However, the load-bearing premise, that the object A is a condensable anyon in the relevant non-semisimple, non-locally-finite category, is not established, and the paper explicitly concedes the missing mathematical construction in Section 5. As a result, the factorization identities are formally derived but not rigorously grounded.

major comments (3)

[Section 4.1 and Section 5] The central object A defined in Eq. (4.1) is never shown to satisfy the definition of a condensable anyon given in Section 3.1. A condensable anyon is required to be a connected commutative separable Frobenius algebra object, which in particular needs a unit morphism η: 1 → A and dim Hom(1,A)=1. In the paper's own conventions the identity line corresponds to p = ±iQ/2 and lies outside the integration domain R_{≥0} of Eq. (4.1); the paper itself notes that A does not contain 1 ⊠ 1. Consequently Hom(1,A)=0 under the paper's rules, and A is not even an algebra object with unit. Section 5 concedes that a direct sum over continuous labels is non-trivial to define and that the statement that this holds in VTQFT has not been proven. Because Eqs. (4.8), (4.13), and (4.23) all rely on treating A as a condensable anyon with a well-defined product, coproduct, and projector, the main factorization claim is not established by the computations.
[Section 4.1 and Section 3.1] The paper assumes that inserting the formal direct integral A on a fine mesh is triangulation-independent and equivalent to a topological boundary, but it does not prove the Pachner-move invariance for the non-semisimple, non-locally-finite category C. In Section 3.1, triangulation independence is justified for modular tensor categories by the defining properties of a condensable anyon, and the argument is imported into VTQFT by analogy. For a continuous label set and a formal direct integral, the standard finiteness and semisimplicity arguments do not apply. The paper acknowledges in Section 5 that this statement has not been proven. The identification of the condensed phase with a topological boundary in the sandwich construction, and hence the Liouville-CFT interpretation of the factorized factors, rests on this unproven equivalence.
[Section 4.2.2] The final paragraph of Section 4.2.2 states that the computation has been carried out for (g,n) = (1,1), (2,0), and (2,1), but the main body explicitly computes only the torus wormhole Σ_{1,1}×[0,1] and the genus-two wormhole Σ_{2,0}×[0,1]. The claimed (2,1) case is not written out, and no argument is supplied that the same projector technique applies without modification. This mismatch between the claim and the displayed computation is a presentation issue, but it should be corrected because the abstract and conclusions imply a broader set of verified examples.

minor comments (5)

[Section 2.2] There are several typographical issues in Section 2.2, such as 'inserts a complete set of state' and 'the VTQFT path-integral introduces' instead of a complete set of states; these should be corrected for clarity.
[Section 2.3, Eq. (2.56)] In Eq. (2.56), the notation 'eq := e^{-2πi/τ}' appears to be a typo for a variable such as q̃ or a similar quantity; as written it introduces an undefined symbol that is never used again.
[Reference list] Reference [33] is listed as 'Introduction to Teichmüller Theory: Lecture Notes, .' with a trailing comma and an apparently incomplete URL field; it should be completed and formatted consistently.
[Abstract and Section 1] The phrase 'among the very few explicit computational examples' should be 'among the first explicit computational examples' or 'one of the very few explicit computational examples'; the current wording is grammatically awkward and also overstates the status given the unproven algebraic premise.
[Section 4.2.2, Eq. (4.15)] The discussion of the divergent pure-torus limit in Eq. (4.15) refers to 'the second line of eq. (4.15)' when the displayed expression has several lines; the exact line reference should be clarified.

Circularity Check

1 steps flagged · score 3.0 of 10

Definitional reliance: A is called 'condensable' by fiat, but the factorization calculation itself is a genuine computation from Moore–Seiberg kernels.

self definitional [Eq. (4.1); Section 5, 'Mathematical formulation']
"we will define A := Z ⊕ R≥0 dp Lp ⊠ Lp (4.1) and still call it the diagonal condensable anyon. ... It is non-trivial to define mathematically a direct sum for continuous labels that appears in the diagonal condensable anyon (4.1)."

The central claim that anyon condensation is applicable to VTQFT rests on A being a condensable anyon. But eq. (4.1) supplies no unit (the identity line sits at p = ±iQ/2, outside the integration domain), no product/coproduct, and no commutative separable Frobenius algebra data for a continuous direct integral; Section 5 concedes the direct sum is not mathematically defined. The label 'condensable' is therefore assumed by fiat rather than derived. The factorization identities (4.13)–(4.23) then inherit that unproven premise. The computation itself is not circular—it uses Moore–Seiberg kernels and Wilson identities—but the applicability of the condensation mechanism is definitionally assumed, making the central premise partially question-begging.

full rationale

The factorization results (4.13) and (4.23) are not fitted predictions: no parameters are matched to data, and the δ(p−p̄)/ρ0(p) prefactors are obtained by evaluating crossing-kernel orthogonality (A.7), the fusion-kernel identity (A.17), and the Wilson bubble/triangle identities (2.22)–(2.23), all taken from the established VTQFT/CFT literature rather than from the present author's prior work. The diagonal choice A = ∫ Lp⊠Lp is what enforces diagonality, but the reduction of the wormhole network to the product of two boundary states is a genuine algebraic calculation. The one load-bearing definitional issue is that 'condensable' is asserted for a continuous direct integral even though Section 3's axioms—unit, connectedness, separability, Frobenius condition, and the finiteness/semisimplicity assumptions of an MTC—are not verified; Section 5 explicitly concedes that the direct sum for continuous labels is not mathematically defined and that generic-topology factorization has not been proven in VTQFT. This is a rigor gap and a partially self-definitional premise, but it does not make the whole derivation circular in the fitting or self-citation sense. Accordingly, the circularity score is moderate rather than severe.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central computation rests on four external inputs: the non-rational Moore-Seiberg consistency equations, the Wilson bubble/triangle/Verlinde-loop identities, the delta-function-normalizable inner product with measure ρ0, and the assumption that the continuous direct integral A behaves like a condensable anyon. The first three are taken from the established VTQFT and CFT literature. The fourth is introduced in this paper and is not proven; it is the main source of residual risk. There are no free parameters fitted to data: b, Q, μ and the functions ρ0, C0 are determined by Liouville theory, not by any optimization.

assumptions (4)

domain assumption Non-rational Moore-Seiberg consistency equations (pentagon, hexagon, invertibility) hold for Virasoro conformal block crossing kernels.
Used throughout Section 4 and Appendix A; equations (A.8)-(A.11) are quoted from [66] and are essential for the projector and factorization computations.
domain assumption The Wilson bubble (2.22), Wilson triangle (2.23), and Verlinde loop (2.24) identities hold in VTQFT.
Borrowed from [12,43]; these identities are essential to the projector computation in Section 4.1 and to the genus-two reduction in Section 4.2.2.
domain assumption The inner product and delta-function normalizability of Virasoro conformal blocks (2.6)-(2.7) are valid, including the measure ρ0 and C0 given by (A.26) and (A.14).
Defines the VTQFT state space and the weights used in the superposition rule for the condensable anyon.
ad hoc to paper The insertion of A on a fine mesh is triangulation-independent and equivalent to a topological boundary (the sandwich construction).
The paper does not prove this for the continuous direct integral; it assumes the MTC mechanism of [56,26] extends to VTQFT. This is the load-bearing gap flagged in Section 5.

invented entities (1)

Diagonal condensable anyon A = ∫_{R≥0} dp L_p ⊠ L_p
purpose: Defines the gauging operation whose condensation is claimed to factorize wormhole partition functions.
The object is a formal continuous direct integral over Wilson lines; no algebra structure or falsifiable prediction is provided, and the paper acknowledges the mathematical definition is incomplete.

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Pith. "Pith review of Anyon Condensation in Virasoro TQFT: Wormhole Factorization." pith.science (2026). https://pith.science/paper/AZ6EUXDS

@misc{pith2026241211486,
  author       = {Pith},
  title        = {Pith review of: Anyon Condensation in Virasoro TQFT: Wormhole Factorization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AZ6EUXDS}},
  note         = {Machine review of arXiv:2412.11486}
}

abstract

Anyon condensation in wormhole geometries is investigated in the Virasoro TQFT (VTQFT) formulation, a proposed reformulation of 3d AdS quantum gravity. We first review some elementary techniques of VTQFT and summarize a gauging scheme for non-invertible symmetries referred to as anyon condensation. We then exhibit that anyon condensation is applicable to VTQFT even though the category of Wilson lines associated with it is not strictly a modular tensor category (MTC) due to the continuously infinite label $p\in\mathbb{R}_+$. More specifically, it is shown that the partition function of the wormhole factorizes upon condensing the so-called diagonal condensable anyon $\mathcal{A}=\int_{0}^{\infty}dp\,L_p\boxtimes\overline{L}_p$ in VTQFT. The resulting $2$d boundary theory is Liouville CFT by symmetry TFT construction, and to our knowledge, this is among the very few explicit computational examples of gauging \textit{continuous non-invertible} symmetries in the literature.

Figures

Figures reproduced from arXiv: 2412.11486 by the authors.

**Figure 2.** Two compression bodies that constitute the boundary of the wormhole wormhole geometry with Wilson lines non-trivially linking in the bulk as seen in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** The red wall is the topological interface induced from a condensable anyon A. (a) The red line is contained in A and therefore “condensed”, meaning that it can end on the topological interface, leaving a point-like operator on the boundary theory. The blue line is outside A and braids nontrivially with A so that it is “confined”, meaning it is restricted on the topological boundary. Only those colored in green can b… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The condenable anyon on a fine mesh splits the wormhole geometry into two parts Slab1, Slab2. one of them. The former lines pass through the topological interface (green line), while the latter are confined on it, meaning they create extra excitations on the interface …

**Figure 5.** Figure 5: The trinity relation among Lagrangian condensable anyons, topological boundaries, and boundary modular invariant WZW model in SU(2)k Chern-Simons theory. Each of them obeys the ADE classification. so we conclude that m ◦ ∆(A) = 4A (3.25) Parenthetically, we note the li…

**Figure 6.** Figure 6: The torus wormhole with the Wilson line insertion connecting the two boundaries. The dashed lines are the inner boundary Σ1,1. The graph in the integrand is further modified by the fusion transformation as Lp1 ⊠ Lp1 Lp2 ⊠ Lp2 Lpx ⊠ Lpx Lpx ⊠ Lpx Lpa ⊠ Lpa = Z ∞ 0 dpbdp…

**Figure 7.** Figure 7: Σ2,0 × [0, 1] wormhole with A inserted along its fine mesh. The dashed lines are the inner boundary Σ2,0. even when we ignore the factor δ(p − p) ρ0(p) : lim p,p→1 | Lp ⊠ Lp ⟩ ∝ Z ∞ 0 dpxdpα δ(px − px) | {z } =δ(0) δ(pα − pα) | {z } =δ(0) × | Lpx ⊠ Lpx ⟩ ⊗ | Lpα ⊠ Lpα …

**Figure 8.** Figure 8: The left half of the link in eq. (4.18). The container manifold Σ2,0 is omitted for simplicity and the X mark represents the location of the left genus. strategy is to attempt by some means to create the Wilson bubble to apply the formula (2.22). The procedure breaks …

**Figure 9.** Figure 9: Inclusion relation for various categorical definitions. Surrounded by the red rectangle is the family of tensor categories. The fusion categories are marked inside the blue rectangle and categories in the olive region satisfying the modularity condition are MTC. The gr…

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Works this paper leans on

69 extracted references · 13 canonical work pages · cited by 1 Pith paper

[26]

Benini, C

F. Benini, C. Copetti, and L. Di Pietro, Factorization and Global Symmetries in Holography , SciPost Phys. 14 (2023), no. 2 019, [ arXiv:2203.09537]

arXiv 2023
[1]

Ach´ ucarro and P

A. Ach´ ucarro and P. K. Townsend,A Chern-Simons Action for Three-Dimensional Anti-De Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89

work page 1986
[2]

Witten, Quantum Field Theory and the Jones Polynomial , Commun

E. Witten, Quantum Field Theory and the Jones Polynomial , Commun. Math. Phys. 121 (1989) 351–399

1989
[3]

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

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

1988
[4]

H. L. Verlinde, Conformal Field Theory, Two-Dimensional Quantum Gravity and Quantization of Teichm¨ uller Space, Nucl. Phys. B 337 (1990) 652–680

work page 1990
[5]

Quantum Liouville theory versus quantized Teichm\"uller spaces

J. Teschner, Quantum Liouville Theory Versus Quantized Teichm¨ uller Spaces, Fortsch. Phys. 51 (2003) 865–872, [ hep-th/0212243]

work page Pith review arXiv 2003
[6]

Teschner, From Liouville Theory to the Quantum Geometry of Riemann Surfaces , in 14th International Congress on Mathematical Physics, 8, 2003

J. Teschner, From Liouville Theory to the Quantum Geometry of Riemann Surfaces , in 14th International Congress on Mathematical Physics, 8, 2003. hep-th/0308031

arXiv 2003
[7]

Teschner, On the Relation Between Quantum Liouville Theory and the Quantized Teichm¨ uller Spaces, Int

J. Teschner, On the Relation Between Quantum Liouville Theory and the Quantized Teichm¨ uller Spaces, Int. J. Mod. Phys. A 19S2 (2004) 459–477, [ hep-th/0303149]

arXiv 2004

Show all 69 references

[8]

Teschner, An Analog of a Modular Functor from Quantized Teichm¨ uller Theory, math/0510174

J. Teschner, An Analog of a Modular Functor from Quantized Teichm¨ uller Theory, math/0510174

arXiv
[9]

R. M. Kashaev, Quantization of Teichm¨ uller Spaces and the Quantum Dilogarithm, Letters in Mathematical Physics 43 (1998) 105–115, [ q-alg/9705021]

1998 arXiv
[10]

J. E. Andersen and R. Kashaev, A New Formulation of the Teich ¨ muller TQFT, arXiv:1305.4291

arXiv
[11]

Mikhaylov, Teichm¨ uller TQFT vs

V. Mikhaylov, Teichm¨ uller TQFT vs. Chern-Simons theory, JHEP 04 (2018) 085, [arXiv:1710.04354]

2018 arXiv
[12]

Collier, L

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

2023 arXiv
[13]

Collier, L

S. Collier, L. Eberhardt, and M. Zhang, 3d Gravity from Virasoro TQFT: Holography, Wormholes and Knots , SciPost Phys. 17 (2024) 134, [ arXiv:2401.13900]

2024 arXiv
[14]

Maloney and E

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

2010 arXiv
[15]

Cotler and K

J. Cotler and K. Jensen, AdS3 Gravity and Random CFT , JHEP 04 (2021) 033, [arXiv:2006.08648]

2021 arXiv
[16]

Chandra, S

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]

2022 arXiv
[17]

Belin, J

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]

2024 arXiv
[18]

D. L. Jafferis, L. Rozenberg, and G. Wong, 3d Gravity as a Random Ensemble , arXiv:2407.02649

arXiv
[19]

Pelliconi, J

P. Pelliconi, J. Sonner, and H. Verlinde, Gravity as a Mesoscopic System , arXiv:2409.13808. – 42 –

arXiv
[20]

J. M. Maldacena and L. Maoz, Wormholes in AdS , JHEP 02 (2004) 053, [ hep-th/0401024]

2004 arXiv
[21]

P. Saad, S. H. Shenker, and S. Yao, Comments on Wormholes and Factorization , JHEP 10 (2024) 076, [ arXiv:2107.13130]

2024 arXiv
[22]

Blommaert, L

A. Blommaert, L. V. Iliesiu, and J. Kruthoff, Gravity Factorized, JHEP 09 (2022) 080, [arXiv:2111.07863]

2022 arXiv
[23]

Gaiotto, A

D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [ arXiv:1412.5148]

2015 arXiv
[24]

Bhardwaj and Y

L. Bhardwaj and Y. Tachikawa, On Finite Symmetries and Their Gauging in Two Dimensions, JHEP 03 (2018) 189, [ arXiv:1704.02330]

2018 arXiv
[25]

Chang, Y.-H

C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, Topological Defect Lines and Renormalization Group Flows in Two Dimensions , JHEP 01 (2019) 026, [arXiv:1802.04445]

2019 arXiv
[27]

Harlow and H

D. Harlow and H. Ooguri, Constraints on Symmetries from Holography , Phys. Rev. Lett. 122 (2019), no. 19 191601, [ arXiv:1810.05337]

2019 arXiv
[28]

Harlow and H

D. Harlow and H. Ooguri, Symmetries in Quantum Field Theory and Quantum Gravity , Commun. Math. Phys. 383 (2021), no. 3 1669–1804, [ arXiv:1810.05338]

2021 arXiv
[29]

Dymarsky and A

A. Dymarsky and A. Shapere, TQFT Gravity and Ensemble Holography , JHEP 02 (2025) 091, [arXiv:2405.20366]

2025 arXiv
[30]

Gaiotto and J

D. Gaiotto and J. Kulp, Orbifold Groupoids, JHEP 02 (2021) 132, [ arXiv:2008.05960]

2021 arXiv
[31]

Kaidi, K

J. Kaidi, K. Ohmori, and Y. Zheng, Symmetry TFTs for Non-invertible Defects , Commun. Math. Phys. 404 (2023), no. 2 1021–1124, [ arXiv:2209.11062]

2023 arXiv
[32]

L. W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, vol. 275. Springer, 2017

2017
[33]

Petri, Introduction to Teichm¨ uller Theory: Lecture Notes,

B. Petri, Introduction to Teichm¨ uller Theory: Lecture Notes, . https://webusers.imj-prg.fr/~bram.petri/t_tt/notes_240212.pdf
[34]

M. F. Atiyah and R. Bott, The Yang-Mills Equations over Riemann Surfaces , Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 308 (1983), no. 1505 523–615

1983
[35]

Scarinci and K

C. Scarinci and K. Krasnov, The Universal Phase Space of AdS 3 Gravity, Commun. Math. Phys. 322 (2013) 167–205, [ arXiv:1111.6507]

2013 arXiv
[36]

Bhattacharyya, S

A. Bhattacharyya, S. Ghosh, P. Nandi, and S. Pal, 3D N = 1 Supergravity from Virasoro TQFT: Gravitational Partition Function and Out-of-Time-Order Correlator , JHEP 02 (2025) 027, [ arXiv:2408.01538]

2025 arXiv
[37]

Collier, A

S. Collier, A. Maloney, H. Maxfield, and I. Tsiares, Universal Dynamics of Heavy Operators in CFT 2, JHEP 07 (2020) 074, [ arXiv:1912.00222]

2020 arXiv
[38]

Dorn and H

H. Dorn and H. J. Otto, Two and Three Point Functions in Liouville Theory , Nucl. Phys. B 429 (1994) 375–388, [ hep-th/9403141]

1994 arXiv
[39]

A. B. Zamolodchikov and A. B. Zamolodchikov, Structure Constants and Conformal Bootstrap in Liouville Field Theory , Nucl. Phys. B 477 (1996) 577–605, [ hep-th/9506136]

1996 arXiv
[40]

Ribault, Conformal Field Theory on the Plane , arXiv:1406.4290

S. Ribault, Conformal Field Theory on the Plane , arXiv:1406.4290

arXiv
[41]

Ribault, Exactly Solvable Conformal Field Theories , arXiv:2411.17262

S. Ribault, Exactly Solvable Conformal Field Theories , arXiv:2411.17262. – 43 –

arXiv
[42]

G. W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory , Commun. Math. Phys. 123 (1989) 177

1989
[43]

Post and I

B. Post and I. Tsiares, A Non-rational Verlinde Formula from Virasoro TQFT , JHEP 04 (2024) 015, [ arXiv:2411.07285]

2024 arXiv
[44]

de Boer, D

J. de Boer, D. Liska, and B. Post, Multiboundary Wormholes and OPE Statistics , JHEP 10 (2024) 207, [ arXiv:2405.13111]

2024 arXiv
[45]

Ba˜ nados, C

M. Ba˜ nados, C. Teitelboim, and J. Zanelli, The Black Hole in Three Dimensional Spacetime , Phys. Rev. Lett. 69 (1992) 1849–1851, [ hep-th/9204099]

1992 arXiv
[46]

Carlip and C

S. Carlip and C. Teitelboim, Aspects of Black Hole Quantum Mechanics and Thermodynamics in (2 + 1)-Dimensions, Phys. Rev. D 51 (1995) 622–631, [ gr-qc/9405070]

1995 arXiv
[47]

S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space , Commun. Math. Phys. 87 (1983) 577

1983
[48]

Kurita, CFT Description of Three-Dimensional Hawking-Page Phase Transition ,

Y. Kurita, CFT Description of Three-Dimensional Hawking-Page Phase Transition , . https://www2.yukawa.kyoto-u.ac.jp/~masashi.hamanaka/kurita_y04.pdf
[49]

T. G. Mertens, J. Sim´ on, and G. Wong, A Proposal for 3d Quantum Gravity and Its Bulk Factorization, JHEP 06 (2023) 134, [ arXiv:2210.14196]

2023 arXiv
[50]

Kong, Anyon Condensation and Tensor Categories , Nucl

L. Kong, Anyon Condensation and Tensor Categories , Nucl. Phys. B 886 (2014) 436–482, [arXiv:1307.8244]

2014 arXiv
[51]

Bhardwaj, L

L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre, and H. Tillim, Lectures on Generalized Symmetries, Phys. Rept. 1051 (2024) 1–87, [arXiv:2307.07547]

2024 arXiv
[52]

Kaidi, Z

J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, Higher Central Charges and Topological Boundaries in 2+1-Dimensional TQFTs , SciPost Phys. 13 (2022), no. 3 067, [arXiv:2107.13091]

2022 arXiv
[53]

Roumpedakis, S

K. Roumpedakis, S. Seifnashri, and S.-H. Shao, Higher Gauging and Non-invertible Condensation Defects, Commun. Math. Phys. 401 (2023), no. 3 3043–3107, [arXiv:2204.02407]

2023 arXiv
[54]

C´ ordova and D

C. C´ ordova and D. Garc ´ ıa-Sep´ ulveda,Non-Invertible Anyon Condensation and Level-Rank Dualities, arXiv:2312.16317

arXiv
[55]

C´ ordova and D

C. C´ ordova and D. Garc ´ ıa-Sep´ ulveda,Topological Cosets via Anyon Condensation and Applications to Gapped QCD2, arXiv:2412.01877

arXiv
[56]

Etingof, S

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, vol. 205. American Mathematical Soc. Am. Math. Soc., 2015

2015
[57]

Cappelli, C

A. Cappelli, C. Itzykson, and J. B. Zuber, Modular Invariant Partition Functions in Two Dimensions, Nucl. Phys. B 280 (1987) 445–465

1987
[58]

Cappelli, C

A. Cappelli, C. Itzykson, and J. B. Zuber, The A-D-E Classification of Minimal and A(1) 1 Conformal Invariant Theories , Commun. Math. Phys. 113 (1987) 1

1987
[59]

Kirillov Jr and V

A. Kirillov Jr and V. Ostrik, On a q-Analogue of the McKay Correspondence and the ADE Classification of bsl2 Conformal Field Theories , Advances in Mathematics 171 (2002), no. 2 183–227, [math/0101219]

2002 arXiv
[60]

Fuchs, I

J. Fuchs, I. Runkel, and C. Schweigert, TFT Construction of RCFT Correlators I. Partition Functions, Nucl. Phys. B 646 (2002) 353–497, [ hep-th/0204148]

2002 arXiv
[61]

Marden, Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions

A. Marden, Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions. Cambridge University Press, 2016. – 44 –

2016
[62]

Chang, Q

L. Chang, Q. T. Kolt, Z. Wang, and Q. Zhang, Modular Data of Non-semisimple Modular Categories, arXiv:2404.09314

arXiv
[63]

Ribault, The Non-rational Limit of D-Series Minimal Models , SciPost Phys

S. Ribault, The Non-rational Limit of D-Series Minimal Models , SciPost Phys. Core 3 (2020) 002, [ arXiv:1909.10784]

2020 arXiv
[64]

M. H. Freedman, M. Larsen, and Z. Wang, A Modular Functor Which is Universal for Quantum Computation , Commun. Math. Phys. 227 (2002), no. 3 605–622, [quant-ph/0001108]

2002 arXiv
[65]

A. Y. Kitaev, Fault Tolerant Quantum Computation by Anyons , Annals Phys. 303 (2003) 2–30, [quant-ph/9707021]

2003 arXiv
[66]

Eberhardt, Notes on Crossing Transformations of Virasoro Conformal Blocks , arXiv:2309.11540

L. Eberhardt, Notes on Crossing Transformations of Virasoro Conformal Blocks , arXiv:2309.11540

arXiv
[67]

Ponsot and J

B. Ponsot and J. Teschner, Liouville Bootstrap via Harmonic Analysis on a Noncompact Quantum Group, hep-th/9911110

arXiv
[68]

Ponsot and J

B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner Coefficients for a Continuous Series of Representations of Uq(sl(2, R)), Commun. Math. Phys. 224 (2001) 613–655, [math/0007097]

2001 arXiv
[69]

Teschner and G

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]. – 45 –

2015 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [26]

Benini, C

F. Benini, C. Copetti, and L. Di Pietro, Factorization and Global Symmetries in Holography , SciPost Phys. 14 (2023), no. 2 019, [ arXiv:2203.09537]

arXiv 2023

[2] [1]

Ach´ ucarro and P

A. Ach´ ucarro and P. K. Townsend,A Chern-Simons Action for Three-Dimensional Anti-De Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89

work page 1986

[3] [2]

Witten, Quantum Field Theory and the Jones Polynomial , Commun

E. Witten, Quantum Field Theory and the Jones Polynomial , Commun. Math. Phys. 121 (1989) 351–399

1989

[4] [3]

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

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

1988

[5] [4]

H. L. Verlinde, Conformal Field Theory, Two-Dimensional Quantum Gravity and Quantization of Teichm¨ uller Space, Nucl. Phys. B 337 (1990) 652–680

work page 1990

[6] [5]

Quantum Liouville theory versus quantized Teichm\"uller spaces

J. Teschner, Quantum Liouville Theory Versus Quantized Teichm¨ uller Spaces, Fortsch. Phys. 51 (2003) 865–872, [ hep-th/0212243]

work page Pith review arXiv 2003

[7] [6]

Teschner, From Liouville Theory to the Quantum Geometry of Riemann Surfaces , in 14th International Congress on Mathematical Physics, 8, 2003

J. Teschner, From Liouville Theory to the Quantum Geometry of Riemann Surfaces , in 14th International Congress on Mathematical Physics, 8, 2003. hep-th/0308031

arXiv 2003

[8] [7]

Teschner, On the Relation Between Quantum Liouville Theory and the Quantized Teichm¨ uller Spaces, Int

J. Teschner, On the Relation Between Quantum Liouville Theory and the Quantized Teichm¨ uller Spaces, Int. J. Mod. Phys. A 19S2 (2004) 459–477, [ hep-th/0303149]

arXiv 2004