# A tensor model for AdS3 A tensor model for [[0002 3D gravity|3d quantum gravity]] was introduced in [[2023#Belin, de Boer, Jafferis, Nayak, Sonner]]. It is conceptually analogous to the [[0197 Matrix model|matrix model]] as a dual theory for [[0050 JT gravity|JT gravity]], but technically very different. Schematically, it is defined as the integral$\int D[\Delta] D[C](\cdot) e^{-\frac{1}{\hbar} V[\Delta, C]},$where $\{\Delta_{ij},C_{ijk}\}$ are the CFT data for operators above the black hole threshold. Data for light operators do enter the potential, but they are considered fixed. ## Refs - [[2023#Belin, de Boer, Jafferis, Nayak, Sonner]]: presents the tensor model - [[2024#Jafferis, Rozenberg, Wong]]: 3d interpretations ## Operator content - Virasoro identity block - continuum of states above the black hole threshold: $(h, \bar{h}) \geq \frac{c-1}{24}$ - optional: discrete heavy states below the threshold ## Matrix/tensor potential - total potential: $\frac{1}{2} V=V_{\mathrm{S}}+g_4 V_4$ (and $V_T$) - $V_{\mathrm{S}, \mathbb{1}}=\int \frac{d^2 P}{4}\left|\rho_0(P)\right|^2 \sum_{i, k}\left(\delta^{(2)}\left(P-P_i\right)-\left|\mathbb{S}_{P P_i}[\mathbb{1}]\right|^2\right)\left(\delta^{(2)}\left(P-P_k\right)-\left|\mathbb{S}_{P P_k}[\mathbb{1}]\right|^2\right)$ - the quartic term: $\begin{aligned} & V_4=2 \sum_{i_1 \cdots i_4}{}^{\prime} \sum_{p, q}\left(\frac{C_{i_1 i_2 p} C_{i_3 i_4 p} C_{i_1 i_2 q} C_{i_3 i_4 q}}{\left|\rho_0(p) C_0(12 p) C_0(34 p)\right|^2} \delta^{(2)}\left(P_p-P_q\right)\right. \\- &\left.\frac{C_{i_1 i_2 p} C_{i_3 i_4 p} C_{i_1 i_4 q} C_{i_2 i_3 q}}{\left|C_0(12 p) C_0(34 p) C_0(23 q) C_0(14 q)\right|^2}\left\{\begin{array}{lll}\mathcal{O}_q & \mathcal{O}_4 & \mathcal{O}_1 \\ \mathcal{O}_p & \mathcal{O}_2 & \mathcal{O}_3\end{array}\right\}\right)\end{aligned}$