# Liouville theory **Liouville theory** is an interesting example of a solvable [[0481 Conformal field theory|CFT]]. It can have any complex [[0033 Central charge|central charge]] ($c\in\mathbb{C}$) but unitarity requires $c>1$. It has a continuous spectrum and is interacting. The action of Liouville theory is given by$S=\frac{1}{4 \pi} \int d^2 z \sqrt{g}\left(g^{a b} \partial_a \phi \partial_b \phi+Q R \phi+4 \pi \mu e^{2 b \phi}\right).$ At large $c$, i.e., the semiclassical limit, the theory encapsulates information related to the moduli space of Riemann surfaces. Moving away from the $c=\infty$ limit should tell us about the "quantum geometry" of Riemann surfaces. ## Notations - $c=1+6 Q^2$ - $Q=b+b^{-1}$ - $h=\frac{Q^2}{4}+P^2$, $P\in \mathbb{R}$ ## Refs - reviews - [[2004#Nakayama (Review)]] - [[2024#Chatterjee, Witten (Review)]] - spectrum - [[1982#Curtright, Thorn]]: gives a conjectured spectrum from canonical quantisation - 3-point function ([[0598 DOZZ formula|DOZZ]]): - [[1994#Dorn, Otto]]: studies the three-point function - [[1995#Zamolodchikov, Zamolodchikov]]: proposes an explicit formula for the three-point function - [[1995#Teschner]]: derives the three-point function from the spectrum of [[1982#Curtright, Thorn|Curtright and Thorn]] - [[1998#O'Raifeartaigh, Pawlowski, Sreedhar]]: three-point function from path integral - crosscap one point function - [[2002#Hikida]]: from crossing symmetry of 2-point function on crosscap - fusion kernel - [[1999#Ponsot, Teschner]]: bootstrap consistency of the CFT data; fusion coefficients - timelike Liouville - [[2003#Strominger, Takayanagi]] - relation to moduli space of Riemann surfaces - [[2023#Colville, Harrison, Maloney, Namjou]]: with boundaries, conical defects, and higher genus - Liouville as a bulk theory - [[2023#Blommaert, Mertens, Yao]] - boundary Liouville - see [[0642 Boundary Liouville CFT]] - resurgence - [[2024#Benjamin, Collier, Maloney, Meruliya]] ## Three-point function $C_0\left(P_i, P_j, P_k\right)=\frac{1}{\sqrt{2}} \frac{\Gamma_b(2 Q)}{\Gamma_b(Q)^3} \frac{\prod \Gamma_b\left(\frac{Q}{2} \pm i P_i \pm i P_j \pm i P_k\right)}{\prod_{a \in\{i, j, k\}} \Gamma_b\left(Q+2 i P_a\right) \Gamma_b\left(Q-2 i P_a\right)}$ ## Spectrum - the spectrum of Liouville theory does not follow the [[0406 Cardy formula|Cardy formula]] at high energies because the identity operator is not present in Liouville (n.b. the Cardy formula can be obtained by modular transformation of the statement that the identity operator has zero conformal weights) ## Notations and relations The combination$\frac{\mathbb{S}_{P_1, P_2}\left[P_0\right]}{\mathbb{S}_{\mathbb{1}, P_2}[\mathbb{1}]} C_0\left(P_1, P_1, P_0\right)$is invariant under $P_1\leftrightarrow P_2$. Cardy:$\rho_0(P) \equiv \mathbb{S}_{\mathbb{1}, P}[\mathbb{1}]$ Reality on $\mathbb{S}$:$\mathrm{e}^{-\frac{\pi i \Delta_0}{2}} \mathbb{S}_{P_1, P_2}\left[P_0\right]$is real for real $P_i$. Identity blocks:$\mathbb{F}_{\mathbb{1}, P_5}\left[\begin{array}{ll}P_4 & P_3 \\ P_4 & P_3\end{array}\right]=C_0\left(P_3, P_4, P_5\right) \rho_0\left(P_5\right)$and$\mathbb{F}_{P_3, P_{4}}\left[\begin{array}{cc}P_3 & P_2 \\ \mathbb{1} & P_1\end{array}\right]=\delta\left(P_{4}-P_1\right).$ Identity in $C_0$: $\lim _{P_1 \rightarrow \mathbb{1}} C_0\left(P_1, P_2, P_3\right)=\rho_0\left(P_2\right)^{-1} \delta\left(P_2-P_3\right)$ Double Gamma function:$S_b(z)=\Gamma_b(z) / \Gamma_b(Q-z)$ Asymptotic ($b>0$):$S_b(z) \rightarrow \mathrm{e}^{ \pm \frac{\pi i}{2}\left(z^2-Q z+\frac{1}{6}\left(Q^2+1\right)\right)} \quad \text{for}\quad \operatorname{Im}(z) \rightarrow \mp \infty$ Special case of Gamma-$b$ function: $\Gamma_1(z)=\frac{(2 \pi)^{\frac{z}{2}}}{G(z)},$where $G(z)$ is the Barnes $G$ function. Main form for $\mathbb{S}$: $\begin{aligned} \mathbb{S}_{p_1, p_2}\left[p_0\right]= & S_b\left(\frac{Q}{2}-p_0\right) \rho_0\left(p_2\right) \frac{\Gamma_b\left(Q \pm 2 p_1\right) \Gamma_b\left(\frac{Q}{2}-p_0 \pm 2 p_2\right)}{\Gamma_b\left(Q \pm 2 p_2\right) \Gamma_b\left(\frac{Q}{2}+p_0 \pm 2 p_1\right)} \\ & \times \mathrm{e}^{\frac{\pi i}{2}\left(-p_0 Q-2 p_0^2\right)+2 \pi i\left(p_1^2+p_2^2\right)} \int_{i \mathbb{R}} \frac{\mathrm{d} p}{2 i} \mathrm{e}^{-2 \pi i p^2} \frac{S_b\left(\frac{Q}{4}+\frac{p_0}{2} \pm p_1 \pm p_2 \pm p\right)}{S_b( \pm 2 p)}\end{aligned}$ Liouville parameters: - $c=1+6 Q^2$ - $Q=b+b^{-1}$ - $h=\frac{Q^2}{4}+P^2$, $P\in \mathbb{R}$ Liouville three-point function:$C_0\left(P_i, P_j, P_k\right)=\frac{1}{\sqrt{2}} \frac{\Gamma_b(2 Q)}{\Gamma_b(Q)^3} \frac{\prod \Gamma_b\left(\frac{Q}{2} \pm i P_i \pm i P_j \pm i P_k\right)}{\prod_{a \in\{i, j, k\}} \Gamma_b\left(Q+2 i P_a\right) \Gamma_b\left(Q-2 i P_a\right)}$