# Virasoro minimal string The **Virasoro minimal string** consists of a timelike Liouville theory with central charge $\hat{c}<1$ and a spacelike Liouville theory with central charge $c>25$. ## Vertex operators Physical operators are built as$\mathcal{V}_P=\mathrm{N}(P) \mathfrak{c} \tilde{\mathfrak{c}} V_P \widehat{V}_{\widehat{P}=i P},$where $P$ is real, and the fact that $\widehat{P}$ is imaginary means that an analytic continuation is needed. ## The dual matrix description The (doubled-scaled) matrix integral is a dual description of VMS. The spectral density is given by the [[0406 Cardy formula|Cardy formula]]. ## The JT limit Taking $b\to 0$ or $c\to \infty$ gives [[0050 JT gravity|JT gravity]]. In this limit, the dilaton potential becomes linear, and the matrix model spectral density also reduces to that of the matrix integral dual to JT gravity: $\sinh (\sqrt{E}) \mathrm{d} E$. ## Relation to intersection numbers The quantum volumes are related to intersection numbers via$\mathsf{V}_{g, n}^{(b)}\left(P_1, \ldots, P_n\right)=\int_{\overline{\mathcal{M}}_{g, n}} \exp \left(\frac{c-13}{24} \kappa_1+\sum_{j=1}^n P_j^2 \psi_j-\sum_{m \geq 1} \frac{B_{2 m}}{(2 m)(2 m)!} \kappa_{2 m}\right).$The main step in deriving this relation is the Hirzebruch-Riemann-Roch index theorem, which relates the dimension of the Hilbert space for $\Sigma_{g,n}$ with the [[0617 Weil-Petersson volume|Weil-Peterson volume]]. ## Relation to 3D gravity The usual 3D gravity is related to two copies of [[0596 Virasoro TQFT|Virasoro TQFT]] with $c=\bar{c}$. However, we can keep $c$ but set $\bar{c}=0$, which gives what is called chiral 3d gravity. The partition function of this theory evaluated on $\Sigma_{g,n}\times {S}^1$ is related to Virasoro minimal string as follows:$Z_{\Sigma_{g, n} \times{S}^1}=\frac{1}{\left|\operatorname{Map}\left(\Sigma_{g, n}\right)\right|} \operatorname{dim} \mathcal{H}_{g, n}=\mathsf{V}_{g, n}^{(b)}\left(P_1, \ldots, P_n\right),$where $\mathsf{V}_{g,n}^{(b)}$ is the quantum volume. To establish this equivalence, the argument uses the statement that the mapping class group of the 3D manifold $\Sigma_{g,n}\times {S}^1$ is the same as the 2D mapping class group of $\Sigma_{g,n}$, but this is incorrect. To fix this, one could consider 3D gravity on $\Sigma_{g,n}\times I$ instead, where end-of-the-world boundary conditions are placed at the ends of the interval. Now the mapping class group on $\Sigma_{g,n}\times I$ is the same as that of $\Sigma_{g,n}$ (up to a factor of two that depends on the tensions of the end-of-the-world branes). A similar argument shows the duality between VMS and 3D gravity on $\Sigma_{g,n}\times I$. ## As a model of dilaton quantum gravity The sine-dilaton gravity, which has an action just like [[0050 JT gravity|JT gravity]] but with a potential given by$W(\Phi)=\frac{\sinh \left(2 \pi b^2 \Phi\right)}{\sin \left(\pi b^2\right)},$can be redefined with$\phi=b^{-1} \rho-\pi b \Phi, \quad \chi=b^{-1} \rho+\pi b \Phi,$such that the action turns into the sum of a spacelike [[0562 Liouville theory|Liouville theory]] and a [[0622 Timelike Liouville|timelike Liouville theory]]:$\begin{aligned} S_{\mathrm{L}}[\phi] & =\frac{1}{4 \pi} \int_{\Sigma} \mathrm{d}^2 x \sqrt{\tilde{g}}\left(\tilde{g}^{i j} \partial_i \phi \partial_j \phi+Q \widetilde{\mathcal{R}} \phi+4 \pi \mu_{\mathrm{sL}} \mathrm{e}^{2 b \phi}\right), \\ S_{\mathrm{tL}}[\chi] & =\frac{1}{4 \pi} \int_{\Sigma} \mathrm{d}^2 x \sqrt{\tilde{g}}\left(-\tilde{g}^{i j} \partial_i \chi \partial_j \chi-\widehat{Q} \widetilde{\mathcal{R}} \chi+4 \pi \mu_{\mathrm{tL}} \mathrm{e}^{2 \hat{b} \chi}\right).\end{aligned}$The metric appearing here $\tilde{g}$ is related to the metric in the original sine-dilaton action via $g=e^{2\rho}\tilde{g}$. This is the Lagrangian description of VMS. In practice though, the Lagrangian description is not very useful, as the theory can be studied non-perturbatively using the bootstrap approach. ## Refs - main - [[2023#Collier, Eberhardt, Muhlmann, Rodriguez]] - with SUSY - [[2024#Johnson]]: $\mathcal{N}=1$ - relation to 3D gravity - [[2023#Collier, Eberhardt, Muhlmann, Rodriguez]]: relating chiral gravity on $\Sigma_{g,n}\times S^1$ to VMS - [[2025#Jafferis, Rozenberg, Sarkar, Wang]]: relating 3D gravity on $\Sigma_{g,n}\times I$ to VMS ## Related topics - [[0471 String-matrix duality]] - [[0658 FZZT brane]] - [[0652 ZZ brane]] - [[0197 Matrix model]]