# Virasoro minimal string 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]]:$\begin{aligned} S_{\mathrm{L}}[\phi] & =\frac{1}{4 \pi} \int_{\Sigma} \mathrm{d}^2 x \sqrt{\tilde{g}}\left(\tilde{g}^{i j} \partial_i \phi \partial_j \phi+Q \widetilde{\mathcal{R}} \phi+4 \pi \mu_{\mathrm{SL}} \mathrm{e}^{2 b \phi}\right), \\ S_{\mathrm{tL}}[\chi] & =\frac{1}{4 \pi} \int_{\Sigma} \mathrm{d}^2 x \sqrt{\tilde{g}}\left(-\tilde{g}^{i j} \partial_i \chi \partial_j \chi-\widehat{Q} \widetilde{\mathcal{R}} \chi+4 \pi \mu_{\mathrm{tL}} \mathrm{e}^{2 \hat{b} \chi}\right).\end{aligned}$ ## Relation to (chiral) 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 $\bar{c}=0$. The partition function of this theory evaluated on $\Sigma_{g,n}\times \mathrm{S}^1$ is related to Virasoro minimal string as follows:$Z_{\Sigma_{g, n} \times \mathrm{S}^1}=\frac{1}{\left|\operatorname{Map}\left(\Sigma_{g, n}\right)\right|} \operatorname{dim} \mathcal{H}_{g, n}=\mathsf{V}_{g, n}^{(b)}\left(P_1, \ldots, P_n\right),$where $\mathsf{V}_{g,n}^{(b)}$ is the quantum volume. It is important in establishing this equivalence that the mapping class group of the 3D manifold $\Sigma_{g,n}\times \mathrm{S}^1$ is the same as the 2D mapping class group of $\Sigma_{g,n}$. ## Refs - main - [[2023#Collier, Eberhardt, Muhlmann, Rodriguez]] - with SUSY - [[2024#Johnson]]: $\mathcal{N}=1$ ## Related topics - [[0471 String-matrix duality]] - [[0658 FZZT brane]] - [[0652 ZZ brane]] - [[0197 Matrix model]]