Open Virasoro TQFT Overview

• Open Virasoro TQFT is a three-dimensional topological quantum field theory that quantizes Teichmüller spaces, yielding boundary Hilbert spaces of Virasoro conformal blocks.
• It employs advanced techniques such as surgery, modular kernels, and Virasoro 6j-symbols to accommodate mapping-class-group actions and crossing symmetries in a continuous-spectrum setting.
• It also forms an open–closed framework that bridges 3D gravity with intersection theory, offering insights into boundary CFTs and ensemble interpretations.

Searching arXiv for papers on “Open Virasoro TQFT” and closely related Virasoro TQFT literature. arXiv search query: "Open Virasoro TQFT Virasoro TQFT boundary Liouville open-closed 3d gravity" Open Virasoro TQFT usually denotes the with-boundary sector of Virasoro TQFT, the three-dimensional topological quantum field theory obtained by quantizing the Teichmüller space of Riemann surfaces, in which boundary Hilbert spaces are spaces of normalizable Virasoro conformal blocks and bulk amplitudes are assembled by surgery, crossing kernels, and mapping-class-group sums (Collier et al., 2023). In closely related formulations, the same structure is realized through an analytically continued Chern–Simons path integral with a non-standard integration cycle selected by invertibility of the vielbein on hyperbolic three-manifolds (Mikhaylov, 2017). A second, distinct usage appears in open intersection theory, where “open Virasoro” refers to the descendent theory on moduli of stable pointed disks and its open KdV/Virasoro constraints (Pandharipande et al., 2014). The common thread is an open/closed formalism controlled by Virasoro symmetry, but the underlying objects—hyperbolic three-manifolds in one case, moduli of bordered Riemann surfaces in the other—are different.

1. Virasoro TQFT and the Teichmüller quantization picture

In the three-dimensional gravity literature, Virasoro TQFT is defined by assigning to a Riemann surface $\Sigma$ a Hilbert space $H(\Sigma)$ consisting of normalizable Virasoro conformal blocks on $\Sigma$. Internal states are labeled by Liouville momenta

$$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$

with

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

The universal Plancherel density is

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

and the block inner product is diagonal in the continuous labels, with density built from $\rho_0$ and the universal three-point factor $C_0$ (Collier et al., 2023).

This formulation is presented as an exact reformulation of Euclidean $\mathrm{AdS}_3$ quantum gravity at fixed topology. The classical relation to $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$ Chern–Simons theory is sharpened by restricting the quantum phase space to the Teichmüller component, rather than integrating over all flat bundles. In particular, the naive Chern–Simons quantization includes flat connections that do not always define nondegenerate metrics and does not correctly account for large diffeomorphisms, whereas Virasoro TQFT restricts to the geometric, hyperbolic sector and restores the effect of mapping class groups at the quantum level (Collier et al., 2023).

The same theory is also described as a unitary three-dimensional TQFT whose Hilbert spaces on closed oriented boundary surfaces are the quantization of Teichmüller space with symplectic form $H(\Sigma)$0 times the Weil–Petersson form, and whose vectors are identified with Liouville Virasoro conformal blocks (Mikhaylov, 2017). In that description, the theory is engineered from the twisted $H(\Sigma)$1 system on $H(\Sigma)$2, localizes onto Kapustin–Witten flows, and reduces to an analytically continued Chern–Simons path integral

$H(\Sigma)$3

with integration cycle $H(\Sigma)$4 determined by flows from a Nahm pole boundary condition. On hyperbolic $H(\Sigma)$5, the relevant saddle is selected by the “invertible vielbein” condition (Mikhaylov, 2017).

2. Mapping-class actions, crossing kernels, and surgery

The basic algebraic content of Virasoro TQFT is a nonrational modular functor. The Hilbert space $H(\Sigma)$6 carries a unitary, projective representation of the Moore–Seiberg groupoid, with elementary moves given by braiding, fusion, and torus $H(\Sigma)$7 and $H(\Sigma)$8 transformations. In the nonrational, continuous-spectrum setting, these are integral transforms whose kernels are built from the quantum $H(\Sigma)$9-symbols of the modular double of $\Sigma$0, with $\Sigma$1. The special functions $\Sigma$2, $\Sigma$3, and Faddeev’s quantum dilogarithm $\Sigma$4 control analytic continuation, normalization, and the pentagon identity underlying associativity (Collier et al., 2023).

The surgery algorithm for gravity expresses the partition function on a fixed topology by cutting along surfaces with well-defined Hilbert spaces, evaluating elementary amplitudes, and gluing with the Verlinde-type inner product. For hyperbolic $\Sigma$5-manifolds,

$\Sigma$6

while for closed hyperbolic $\Sigma$7,

$\Sigma$8

The chiral amplitudes are assembled from $\Sigma$9, cylinders weighted by $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$0, and modular kernels $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$1 (Collier et al., 2023).

This calculus is explicit in concrete examples. The solid torus reproduces the modular sum over $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$2, BTZ geometries arise from Dehn filling, handlebodies are represented by Virasoro identity blocks, and Euclidean wormholes $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$3 have chiral amplitudes equal to Liouville partition functions on $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$4 (Collier et al., 2023). A detailed development of this framework shows that multi-boundary wormholes with three- or four-punctured sphere boundaries can be computed at finite $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$5 by diagrammatic rules in which trivalent vertices contribute $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$6, crossings contribute Virasoro $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$7-symbols with braiding phases, and internal faces are integrated with $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$8 (Collier et al., 2024). The same work gives strong evidence for equivalence between Virasoro TQFT and Teichmüller TQFT on the figure-eight knot complement and formulates Dehn surgery directly in Virasoro-TQFT language (Collier et al., 2024).

3. Boundary sector, annuli, and nonrational open–closed duality

In the strict three-dimensional sense, “open” refers naturally to surfaces and manifolds with boundary. The boundary Hilbert spaces $$\alpha=\frac{Q}{2}+iP,\qquad P\in \mathbb R_{\ge 0},$$9 are defined for negative Euler characteristic $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$0. External punctures or geodesic boundaries are labeled by continuous Liouville momenta $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$1, and in the semiclassical interpretation of the original gravity proposal a geodesic boundary of length $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$2 corresponds to

$$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$3

whereas a later compact-region formalism uses the convention

$$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$4

for above-threshold states; the coexistence of these formulas reflects different normalizations across related open formulations (Collier et al., 2023, Jafferis et al., 10 Apr 2026).

For manifolds with boundary, the relevant topological decomposition is into compression bodies $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$5, with outer boundary $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$6 and inner boundary $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$7. The TQFT path integral prepares a state by resolving the identity on each inner boundary Hilbert space and inducing a conformal block on the outer boundary in which only identity propagation occurs through cycles filled by disks in the bulk (Collier et al., 2023). In the Euclidean wormhole $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$8, the chiral amplitude equals the Liouville CFT observable on $$c=1+6Q^2,\qquad Q=b+b^{-1},\qquad \Delta=\alpha(Q-\alpha)=\frac{c-1}{24}+P^2.$$9, analytically continued when the moduli differ across the two boundaries. This is the continuous-spectrum analog of Cardy gluing (Collier et al., 2023).

A more microscopic description of boundary states is given by the brane construction in the Hitchin sigma model. Open string states are defined between a Lagrangian brane $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$0 on the Hitchin section and a coisotropic brane $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$1, while brane corners with an oper brane $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$2 generate a state $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$3 labeled by boundary complex structure and a state $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$4 generated by a solid handlebody with monodromy defects. Their overlap $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$5 is the Virasoro conformal block (Mikhaylov, 2017).

Boundary Liouville data enter in several guises. Standard boundary Liouville theory introduces a boundary cosmological constant $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$6 and a real parameter $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$7 labeling FZZT-like boundary conditions, with continuous boundary operator spectrum in $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$8. In Virasoro TQFT, geodesic boundary labels and conformal-block external weights encode the same sector from the three-dimensional side; Wilson lines threading the bulk become boundary insertions, and nonrational tangles are resolved by cutting and applying fusion kernels (Collier et al., 2023).

A major structural result is the nonrational Verlinde formula for $$\rho_0(P)=4\sqrt{2}\,\sinh(2\pi bP)\sinh(2\pi b^{-1}P),$$9,

$\rho_0$0

which expresses the Virasoro fusion kernel as an integral over a ratio of torus one-point $\rho_0$1-kernels. In boundary Liouville CFT, this identity proves open–closed duality for the annulus one-point function and shows that the one-point $\rho_0$2-kernel diagonalizes the Virasoro $\rho_0$3-symbol in Racah–Wigner normalization (Post et al., 2024). A plausible implication is that the open sector of Virasoro TQFT is not merely a formal extension of closed sewing, but an analytically controlled continuous-spectrum boundary theory.

4. Open–closed gravity, end-of-the-world branes, and hyperbolic tetrahedra

Later work promotes the open sector from a boundary-state formalism to an explicit open–closed gravitational theory with end-of-the-world branes. In the open-closed extension, the closed Hilbert space remains a direct integral of non-degenerate Virasoro representations labeled by $\rho_0$4, while for boundary conditions $\rho_0$5 one has open Hilbert spaces $\rho_0$6 spanned by boundary primaries $\rho_0$7. The canonical boundary two-point metric is

$\rho_0$8

the boundary OPE is controlled by coefficients $\rho_0$9, and bulk-to-boundary couplings are $C_0$0 (Jafferis et al., 24 Jun 2025). The corresponding classical Euclidean action includes EOW branes $C_0$1,

$C_0$2

so that $C_0$3 plays the role of a topological brane weight (Jafferis et al., 24 Jun 2025).

Within this open–closed theory, gluing along an annulus produces the open analog of handle addition: $C_0$4 and the annulus wormhole amplitude is identified with the inverse of the open Vandermonde kernel. The same framework yields a purely open tensor model enforcing the disk four-point crossing constraint $C_0$5, and interprets its Feynman rules as $C_0$6 cobordisms with EOW branes and boundary Wilson lines (Jafferis et al., 24 Jun 2025).

A more restricted “purely open” Virasoro TQFT is obtained by limiting the admissible manifolds to compact hyperbolic regions with EOW branes and, in a special class, only boundary Wilson loops. In this setting, above-threshold states are described by fixed-length boundary conditions and below-threshold states by fixed-angle boundary conditions. The open spectral density on an interval with boundary conditions $C_0$7 is

$C_0$8

the open $C_0$9-manifold is weighted by a Virasoro $\mathrm{AdS}_3$0-symbol, and the fundamental building block is a truncated hyperbolic tetrahedron whose hexagonal faces are OPE faces and whose triangular faces lie on EOW branes (Jafferis et al., 10 Apr 2026). In this normalization, above-threshold geometry uses

$\mathrm{AdS}_3$1

while below-threshold states are encoded by

$\mathrm{AdS}_3$2

This formulation makes the relation between open Virasoro TQFT and triangulated hyperbolic geometry completely explicit (Jafferis et al., 10 Apr 2026).

5. Dual formulations, minimal-model variants, and ensemble interpretations

The open Virasoro framework has several non-hyperbolic and categorical variants. One branch uses $\mathrm{AdS}_3$3–$\mathrm{AdS}_3$4 correspondence to construct $\mathrm{AdS}_3$5 bulk theories $\mathrm{AdS}_3$6 for Virasoro minimal models $\mathrm{AdS}_3$7. For unitary $\mathrm{AdS}_3$8, the bulk theory has a mass gap and flows to a unitary $\mathrm{AdS}_3$9 TQFT in the infrared; for non-unitary $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$0, it flows to a $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$1 rank-0 SCFT whose A- or B-twist is topological. Under suitable holomorphic boundary conditions, the boundary supports the chiral minimal model, and bulk Bethe vacua or irreducible $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$2 flat connections reproduce the modular data $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$3 of the boundary primaries (Gang et al., 2024). Here “Open Virasoro TQFT” denotes a rational boundary realization rather than the continuous Liouville sector.

Another branch is Conformal Turaev–Viro (CTV) theory, a triangulated dual presentation of Virasoro TQFT. CTV assigns continuous labels $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$4 to internal edges, weights them by $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$5, and assigns a Virasoro $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$6-symbol to each tetrahedron. For a large triangulation $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$7 of $$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$8,

$$\mathrm{SL}(2,\mathbb R)\times \mathrm{SL}(2,\mathbb R)$$9

Its main identity is a modular $H(\Sigma)$00-transform of the amplitude-squared of Virasoro TQFT,

$H(\Sigma)$01

Since the physical manifold is $H(\Sigma)$02, this state-sum naturally describes manifolds with boundary and furnishes an exact triangulation-based evaluation of the gravitational path integral with boundary data (Hartman, 15 Jul 2025).

The ensemble interpretation of open Virasoro TQFT has also become more explicit. One proposal studies a tensor-matrix integral whose Feynman rules reproduce open-closed Virasoro TQFT amplitudes with EOW branes, bulk-to-boundary couplings, and annulus constraints, and predicts GOE or GUE structure for open spectra depending on whether the boundary conditions coincide (Jafferis et al., 24 Jun 2025). Another proposal, formulated in the SymTFT language, interprets boundary theories as Lagrangian condensations of the bulk and assigns automorphism-weighted ensemble measures. In the Virasoro case, the suggested average at fixed central charge weights a boundary CFT by the inverse order of its invertible topological symmetry group, formally $H(\Sigma)$03, with open-sector branes classified by modules over the relevant Lagrangian algebra (Barbar, 6 Nov 2025). This suggests that “open Virasoro TQFT” has become not only a boundary extension of a fixed bulk theory, but also a framework for averaging over boundary theories.

The open problems emphasized in the foundational gravity proposal remain active: rigorous equivalence to Andersen–Kashaev Teichmüller TQFT, full control of inner-product normalization and convergence of modular sums, systematic treatment of FZZT/ZZ-like boundaries and defects, general noncompact mapping-class-group representations, and extensions to supergravity or higher-spin analogs obtained by quantizing higher Teichmüller components (Collier et al., 2023).

6. Open Virasoro in intersection theory and open KdV

A distinct mathematical literature uses “open Virasoro” for intersection theory on moduli spaces of Riemann surfaces with boundary. In that setting, one studies the moduli space $H(\Sigma)$04 of stable pointed bordered surfaces of doubled genus $H(\Sigma)$05, with $H(\Sigma)$06 boundary marked points and $H(\Sigma)$07 interior marked points. In genus $H(\Sigma)$08, the moduli $H(\Sigma)$09 are smooth manifolds with corners of real dimension $H(\Sigma)$10. Only interior markings carry cotangent line bundles, and open descendent integrals are defined by constructing a canonical nowhere-vanishing multisection on the boundary and using the resulting relative Euler class $H(\Sigma)$11 (Pandharipande et al., 2014).

The genus-$H(\Sigma)$12 theory proves open string, dilaton, and topological recursion relations, and defines the full partition function

$H(\Sigma)$13

The conjectural all-genus open Virasoro constraints are

$H(\Sigma)$14

where $H(\Sigma)$15 extends the closed Virasoro operators by $H(\Sigma)$16-derivative terms. The open string equation is

$H(\Sigma)$17

and the initial condition at $H(\Sigma)$18 is $H(\Sigma)$19 (Pandharipande et al., 2014). Buryak proved that the open KdV hierarchy and the open Virasoro equations for this theory are equivalent, so the two systems determine the same formal potential $H(\Sigma)$20 (Buryak, 2014). A recursive construction of the full partition function from the open Virasoro constraints and the initial disk data was then given in terms of a cut-and-join-type operator $H(\Sigma)$21 and inversion operators $H(\Sigma)$22 (Ke, 2014).

The scope of the genus-$H(\Sigma)$23 formalism was later broadened from the point target to arbitrary solutions of the open WDVV equations satisfying a homogeneity condition. In that generalization, a conformal Frobenius manifold provides the closed sector, an open WDVV solution provides the boundary extension, and a genus-$H(\Sigma)$24 open descendent potential is constructed together with open Virasoro generators $H(\Sigma)$25 satisfying

$H(\Sigma)$26

for the genus-$H(\Sigma)$27 partition function $H(\Sigma)$28 (Basalaev et al., 2019). In this literature, “open Virasoro TQFT” is therefore an open-closed descendent theory governed by Virasoro Ward identities and open WDVV/KdV recursion, rather than a three-dimensional hyperbolic TQFT.

The coexistence of these two usages is not accidental. Both are open/closed theories organized by Virasoro symmetry, both rely on exact sewing data, and both treat the partition function as determined by a constrained algebraic structure. The difference is categorical and geometric: the $H(\Sigma)$29 gravity usage quantizes Teichmüller spaces and hyperbolic manifolds, whereas the intersection-theoretic usage governs descendent integrals on moduli of bordered curves.