# Virasoro TQFT The Virasoro TQFT can be used to reformulate [[0002 3D gravity|3d quantum gravity]] in a precise way. In particular, the 3d gravity partition function on a (hyperbolic) manifold $M$ with fixed topology is related to the Virasoro TQFT partition function via $Z_{\text {grav }}(M)=\sum_{\gamma \in \operatorname{Map}(\partial M) / \operatorname{Map}(M, \partial M)}\left|Z_{\mathrm{Vir}}\left(M^\gamma\right)\right|^2,$where $M^\gamma$ denotes the image of $M$ under $\gamma$. The relation of Virasoro TQFT to Liouville CFT is just like [[0089 Chern-Simons theory|Chern-Simons]] to [[0601 Weiss-Zumino-Witten models|WZW]]: the Hilbert space of VTQFT is identified with the space of conformal blocks on a spatial surface $\Sigma$. The difference is that the Hilbert space is now infinite-dimensional. ## Refs - [[1990#Verlinde]]: quantisation of Teichmüller space - [[2005#Teschner]]: discusses two different quantisations of Teichmüller space - [[2011#Andersen, Kashaev]] and [[2013#Andersen, Kashaev]]: Teichmüller TQFT (should be equivalent to Virasoro TQFT but not obviously) - [[2011#Terashima, Yamazaki]] - [[2017#Mikhaylov]] - [[2023#Collier, Eberhardt, Zhang]] and [[2024#Collier, Eberhardt, Zhang]]: 3d gravity partition functions from Virasoro TQFT - [[2024#Bhattacharyya, Ghosh, Nandi, Pal]]: super-VTQFT - [[2024#Post, Tsiares]]: Verlinde formula ## Hilbert space To define the Hilbert space, one needs to quantise the phase space. Luckily, in this case, the gravity phase space consists of two copies of the Teichmüller space $\mathcal{T}$ of the spatial slice $\Sigma$, and people know how to quantise Teichmüller space. As a result of this quantisation, the Hilbert space is just the space of conformal blocks on $\Sigma$. ## Inner product and basis The inner product between conformal blocks $\mathcal{F}_1$ and $\mathcal{F}_2$ is given by$\left\langle\mathcal{F}_1 |\mathcal{F}_2\right\rangle=\int_{\mathcal{T}} Z_{\mathrm{bc}} Z_{\text {timelike Liouville }} \overline{\mathcal{F}}_1 \mathcal{F}_2,$where $\mathcal{T}$ is the Teichmüller space. With respect to this inner product, there is a natural orthogonal basis. The basis vectors are prepared by handlebodies with Wilson lines. The number of variables that label the Wilson line weights equals the dimension of the moduli space of the corresponding conformal block. This is necessary for it to be a basis. (Think of it as a momentum basis which is dual to the position basis. The number of momenta must equal the number of position coordinates.) ## Mapping class group Unlike in the 2d case, for 3d hyperbolic manifolds, the bulk mapping class group is contained in the boundary mapping class group. (You cannot change the bulk without changing the boundary.) The gravity partition function is a sum over images under the boundary mapping class group but quotiented by the bulk mapping class group.