Conformal Turaev–Viro Theory

• Conformal Turaev–Viro theory is defined as a continuous, Virasoro-based state sum model for 3-manifolds using tetrahedral triangulations and Cardy density measures.
• It constructs a dual formulation of Virasoro TQFT, where the resulting partition function is the modular S-transform of the squared Virasoro amplitude.
• Key features include a chain-mail derivation, explicit use of Virasoro 6j-symbols, and restricted triangulation moves that ensure finiteness and controlled topological invariance.

Searching arXiv for the directly relevant and supporting papers on Conformal Turaev–Viro theory and its foundational context. Conformal Turaev–Viro theory is a triangulation-based state-sum theory defined as a dual formulation of Virasoro TQFT. In its explicit 2025 formulation, a closed $3$-manifold is triangulated by tetrahedra, edges are labeled by continuous conformal weights, tetrahedra are weighted by Virasoro $6j$-symbols, and internal edges are glued with the Cardy density of states. Its central structural statement is that the resulting partition function is the modular $S$-transform of the squared Virasoro TQFT amplitude, $|Z_{\mathrm{Vir}}|^2$, providing a continuous-spectrum analogue of the relation between ordinary Turaev–Viro theory and discrete spin-network TQFT (Hartman, 15 Jul 2025).

1. Definition and conceptual position

In the 2025 paper that introduces the subject by name, Conformal Turaev–Viro (CTV) theory is defined as a tetrahedral state sum attached to a triangulated $3$-manifold with an embedded framed trivalent graph. Its local labels are not discrete simple objects of a fusion category, but continuous Virasoro momenta $P\in \mathbb R_+$. The construction is therefore an irrational, continuous-spectrum analogue of ordinary Turaev–Viro theory rather than a reformulation of the spherical-fusion-category state sum (Hartman, 15 Jul 2025).

This explicit definition sits within a longer development of structures that, while not themselves called “Conformal Turaev–Viro theory,” supplied much of its topological and categorical background. Extended Turaev–Viro theory had already been identified with Reshetikhin–Turaev theory of the Drinfeld center at the level of surface state spaces and $(3\text{-}2\text{-}1)$-TQFT structure, thereby linking Turaev–Viro theory to modular-functor data (Balsam, 2010). Boundary and excitation sectors had likewise been identified with Turaev–Viro state spaces for surfaces with boundary labeled by objects of the Drinfeld center, furnishing a precise topological framework for punctures and boundary sectors (Balsam et al., 2012). In a different direction, a non-compact analytic theory “of Turaev–Viro type” on shaped triangulations had already replaced discrete edge sums by absolutely convergent integrals over real edge variables, with tetrahedral weights built from hyperbolic gamma functions; this established a hyperbolic and Teichmüller-flavored precedent for continuous-label state sums (Kashaev et al., 2012).

Taken together, these earlier developments suggest that the phrase “conformal Turaev–Viro” had a broader anticipatory meaning before receiving a direct definition. The 2025 CTV construction turns that suggestion into a concrete theory: a Virasoro-based, triangulated, continuous-label state sum (Hartman, 15 Jul 2025).

2. State-sum data and local amplitudes

The basic kinematic parameterization is Virasoro-theoretic. Conformal weights are written as

$h=\frac{c-1}{24}+P^2,\qquad c=1+6(b+b^{-1})^2.$

The chiral Cardy density of states is

$$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P).$$

This $\rho_0(P)$ replaces the quantum dimension measure familiar from ordinary Turaev–Viro theory (Hartman, 15 Jul 2025).

Let $6j$0 be a closed $6j$1-manifold with an embedded framed trivalent graph $6j$2, and choose a triangulation $6j$3 such that $6j$4. The graph edges are the external edges of the triangulation, labeled by prescribed $6j$5, while the remaining triangulation edges are internal and carry integration variables $6j$6. The intended physical domain is $6j$7 (Hartman, 15 Jul 2025).

Editor’s term: it is convenient to write $6j$8 for the state sum itself. With that notation, the defining partition function is

$6j$9

where each tetrahedron $S$0 is weighted by the Virasoro $S$1-symbol

$S$2

The paper identifies this $S$3-symbol with the Virasoro fusion kernel in Racah–Wigner normalization (Hartman, 15 Jul 2025).

Formally, this has the same architecture as ordinary Turaev–Viro theory: local tetrahedral amplitudes, edge gluing, and a global product-integral over internal labels. The decisive difference is that the spectrum is continuous and the global measure is Cardy-theoretic rather than categorical.

3. Relation to Virasoro TQFT and the modular $S$4-transform

The principal theorem identifies CTV with a modular transform of Virasoro TQFT. If $S$5 denotes the Virasoro TQFT amplitude, then

$S$6

where the vacuum modular $S$7-kernel is

$S$8

This is the paper’s central identity and the sense in which CTV is “dual” to Virasoro TQFT (Hartman, 15 Jul 2025).

The derivation uses a Virasoro-adapted chain-mail formalism. A chain-mail graph $S$9 is associated to the triangulated graph complement, and the same chain-mail object can be evaluated in two different ways. One evaluation reduces tetrahedral pieces to Virasoro $|Z_{\mathrm{Vir}}|^2$0-symbols, reproducing the CTV state sum. The other evaluation, after Fourier transforming external edge data with the modular kernel, reduces to the squared Virasoro amplitude. The equality of the two evaluations yields the transform formula (Hartman, 15 Jul 2025).

Several local rules drive this derivation. The formal $|Z_{\mathrm{Vir}}|^2$1-line integrates against the Cardy measure, bubble removal produces delta-function orthogonality, and $|Z_{\mathrm{Vir}}|^2$2-loops act as projectors. The key modular-loop identity has the form

$|Z_{\mathrm{Vir}}|^2$3

and the tetrahedral reduction is controlled by the Virasoro $|Z_{\mathrm{Vir}}|^2$4-symbol. The pentagon identity provides the continuous analogue of the discrete $|Z_{\mathrm{Vir}}|^2$5-$|Z_{\mathrm{Vir}}|^2$6 Pachner-move algebra (Hartman, 15 Jul 2025).

The resulting picture is explicitly not a heuristic analogy. The modular $|Z_{\mathrm{Vir}}|^2$7-transform of $|Z_{\mathrm{Vir}}|^2$8 is the CTV partition function, with chain mail supplying the topological proof.

4. Triangulations, Pachner moves, and finiteness

Because the spectrum is continuous and unbounded, CTV does not admit unrestricted triangulation refinement. The paper defines a large triangulation by the condition

$|Z_{\mathrm{Vir}}|^2$9

where $3$0 denotes the internal vertices of the triangulation. It conjectures that, when restricted to large triangulations and to graphs that are finite in Virasoro TQFT, the CTV partition function is finite and independent of the triangulation (Hartman, 15 Jul 2025).

The restriction is tied to the divergence of the total state density: $3$1 In ordinary discrete Turaev–Viro theory, $3$2-$3$3 Pachner moves are normalized by the finite total quantum dimension. In CTV, the analogous contractible $3$4-loops diverge. Accordingly, the paper states that large triangulations allow $3$5-$3$6 and $3$7-$3$8 Pachner moves, but not $3$9-$P\in \mathbb R_+$0 moves (Hartman, 15 Jul 2025).

This feature sharply distinguishes CTV from compact, semisimple state-sum TQFTs. Its topological invariance is expected only in a restricted triangulation class, and the usual global normalization by total quantum dimension is absent for the same reason. The paper relates this limitation to the fact that Virasoro and Teichmüller TQFT are not standard TQFTs in the Atiyah–Segal sense; in particular, the summary explicitly notes that they do not have a well-defined Hilbert space on a $P\in \mathbb R_+$1-sphere (Hartman, 15 Jul 2025).

5. Boundary, defects, and the extended Turaev–Viro background

CTV is formulated for closed embedding manifolds with embedded graphs, but its architecture is closely aligned with the boundary-sensitive language of extended Turaev–Viro theory. Earlier work had shown that Turaev–Viro state spaces on closed surfaces coincide with the Reshetikhin–Turaev state spaces of the Drinfeld center,

$P\in \mathbb R_+$2

thereby identifying the surface sector of Turaev–Viro theory with a modular-functor sector (Balsam, 2010). In the lattice-model realization, protected excitation sectors are Turaev–Viro state spaces for surfaces with boundary, with boundary labels in

$P\in \mathbb R_+$3

so excitations become punctures or boundary components carrying center labels (Balsam et al., 2012).

A recent boundary characterization sharpened this bulk-boundary perspective further. It showed that explicit boundary locality conditions on a $P\in \mathbb R_+$4-dimensional TQFT with free boundaries and boundary defects force a Turaev–Viro state-sum construction, and that Turaev–Viro theories indeed satisfy these locality conditions (Steffen et al., 19 Aug 2025). This provides a precise “bulk from boundary” principle for ordinary Turaev–Viro theory.

These results do not define CTV. They do, however, provide the structural background in which a conformal variant becomes intelligible: state spaces arise from cutting and gluing, bulk amplitudes are reconstructed from boundary data, and defects or punctures are controlled by categorical local rules. CTV replaces the discrete center-based input by Virasoro fusion, modular kernels, and the Cardy density, but it retains the characteristic Turaev–Viro logic of assembling $P\in \mathbb R_+$5-dimensional amplitudes from local tetrahedral data.

6. Examples, applications, and scope

The paper works out two model examples. For the tetrahedron graph in $P\in \mathbb R_+$6, the Virasoro TQFT amplitude is a single $P\in \mathbb R_+$7-symbol, while the CTV triangulation uses two tetrahedra and yields the square of that $P\in \mathbb R_+$8-symbol. The resulting transform formula is an $P\in \mathbb R_+$9-self-duality statement for the squared Virasoro $(3\text{-}2\text{-}1)$0-symbol (Hartman, 15 Jul 2025).

For the knotted handcuff graph, the Virasoro amplitude is a modular kernel, and the CTV triangulation gives a tetrahedral $(3\text{-}2\text{-}1)$1-symbol. The paper derives an explicit formula expressing the $(3\text{-}2\text{-}1)$2-transform of the squared modular kernel in terms of that tetrahedral amplitude. More generally, it derives formulas for $(3\text{-}2\text{-}1)$3-transforms of squared Virasoro crossing kernels, which it presents as CFT applications of the topological construction (Hartman, 15 Jul 2025).

The gravitational motivation is AdS$(3\text{-}2\text{-}1)$4 quantum gravity. The paper proposes a relation

$(3\text{-}2\text{-}1)$5

with

$(3\text{-}2\text{-}1)$6

and boundary geodesics $(3\text{-}2\text{-}1)$7 identified with meridians of the graph. In this interpretation, the boundary conditions fix dihedral angles

$(3\text{-}2\text{-}1)$8

The paper presents its results as laying the topological foundation for triangulating the exact path integral of pure AdS$(3\text{-}2\text{-}1)$9 quantum gravity, while treating the full gravity equality as a later step rather than an established theorem (Hartman, 15 Jul 2025).

The theory therefore has a sharply delimited status. What is directly established is a continuous-label tetrahedral state sum, a chain-mail derivation, and an exact relation to the modular $h=\frac{c-1}{24}+P^2,\qquad c=1+6(b+b^{-1})^2.$0-transform of $h=\frac{c-1}{24}+P^2,\qquad c=1+6(b+b^{-1})^2.$1. What remains conjectural includes full finiteness and triangulation-independence on the intended class of manifolds, the precise relation to Teichmüller TQFT, and the complete gravitational interpretation. In that sense, Conformal Turaev–Viro theory is already a defined and technically developed object, but still an open program at the level of global topological control and physical application (Hartman, 15 Jul 2025).