Liouville theory is an interacting 2d CFT with a continuous spectrum of scalar states.

Let us start by assuming that we have a unitary theory with $c>1$. We must also assume that all Virasoro primaries are scalar fields: with these assumptions, the theory is entirely fixed.

Consider the torus partition function

$$\begin{align} Z(\tau, \bar \tau) = \sum_{h, \bar h} d_{h, \bar h} \chi_h(\tau) \bar{\chi}_{\bar h}(\bar \tau) \end{align}$$

Now $c>1$ implies that there are no null descendents, so the Virasoro characters in the partition function take the form

$$\begin{align} \chi_h(\tau) = q^{h - c/24} \prod_{n = 1} \frac{1}{1 - q^n}, \qquad q = e^{2 \pi i \tau} \end{align}$$

The weights $h$ and $\bar h$ define the scaling dimension $\Delta = h + \bar h$ and the spin $l = h - \bar h$. The assumption that all primaries are scalars means that we only need sum over $\Delta$. Therefore the partition function may be written as

$$\begin{align} Z(\tau, \bar \tau) = \frac{1}{|\eta(\tau)|^2} \sum_{\Delta} d(\Delta) e^{-2 \pi \tau_2 ( \Delta - \Delta_* )} \end{align}$$

where $\tau_2 = \text{Im}(\tau)$, $\eta(\tau) = q^{1/24} \prod_{n = 1} (1 - q^n)$ is the Dedekind eta function, and we've defined $\Delta_* = (c - 1)/12$ for convenience.

Now we will see how the spectrum is fixed by modular invariance. Modular invariance takes the form of the requirement that

$$\begin{align} Z(\tau, \bar \tau) = Z(\tau + 1, \bar \tau + 1) = Z\left( -\frac{1}{\tau}, -\frac{1}{\bar \tau} \right) \end{align}$$

The first equation is a periodicity requirement that implies all spins must be integers; this is already satisfied. The second one implies

$$\begin{align} \sqrt{\tau_2} \sum_\Delta d(\Delta) e^{-2 \pi \tau_2 (\Delta - \Delta_*)} = \frac{ \sqrt{\tau_2} }{|\tau|} \sum_\Delta d(\Delta) e^{-2 \pi \frac{\tau_2}{|\tau|^2} (\Delta - \Delta_*)} \end{align}$$

Now we need a trick. This equation is essentially $f(x) = f(x/y)$. Because $y$ is independent of $x$, the only solution is $f(x) = \text{constant}$. As a result, we have

$$\begin{align} \sum_{\Delta} d(\Delta) e^{-2 \pi \tau_2 (\Delta - \Delta_*)} \sim \tau_2^{-1/2} \end{align}$$

This requirement is not possible with a discrete spectrum. So we all $d(\Delta)$ to be continuous, and modular invariance will be satisfied if

$$\begin{align} d(\Delta) = \frac{1}{\sqrt{\Delta - \Delta_\star}} \end{align}$$

At this point, it is customary to change variables. We define the charge $Q$ and the momentum $P$ by¹⁾

$$\begin{align} \Delta = \frac{Q^2}{2} + 2 P^2 \, , \qquad c = 1 + 6 Q^2 \end{align}$$

With this normalization, $\sum d(\Delta) \to 2 \sqrt{2} \int d P$, so we find the distribution of states in uniform in $P$ for $P>0$.

Having fixed the kinematics using modular invariance, we would like to fix the dynamics using crossing symmetry. First, we choose to normalize the two point functions by $$\begin{align} \langle V_{P_1}(z_1) V_{P_2}(z_2) \rangle = \frac{\delta(P_1 - P_2)}{ |z_{12}|^{4 \Delta}} \end{align}$$

The three-point functions are fixed by conformal symmetry to be

$$\begin{align} \langle V_{P_1}(z_1) V_{P_2}(z_2) V_{P_3}(z_3) \rangle = \frac{C_{P_1, P_2, P_3}}{|z_{12}|^{2(\Delta_3 - \Delta_1 - \Delta_2)} |z_{23}|^{2(\Delta_1 - \Delta_2 - \Delta_3)} |z_{31}|^{2(\Delta_2 - \Delta_3 - \Delta_1)}} \, , \end{align}$$

up to their structure constants ${C_{P_1, P_2, P_3}}$. The coefficients are subject to a consistency condition in the form of the crossing equation. This takes the form

$$\begin{align} &\int_0^{\infty} dP \, C(P_1, P_2, P) \, C(P_3, P_4, P) \, |\mathcal{F}(P_1, P_2, P_3, P_4, P, z)|^2 \\ & \quad = \int_0^{\infty} dP \, C(P_1, P_4, P) \, C(P_2, P_3, P) \, |\mathcal{F}(P_1, P_4, P_3, P_2, P, z)|^2 \, . \end{align}$$

The integral is over all internal states, which have momentum $P$. The functions $\mathcal{F}$ are the Virasoro conformal blocks. The Liouville theory will only be consistent if there exists a function $C(P_1, P_2, P_3)$ which satisfies this equation for all values of the external momenta $P_i$.

It was shown by Dorn and Otto, and independently by Zamolodchikov and Zamolodchikov, that there do exist solutions. This is known as the DOZZ formula, and takes the form

$$\begin{align} C_{P_1, P_2, P_3} = \frac{\mu^{-Q/2} \Upsilon'_b(0)}{\Upsilon_b\left(\frac{Q}{2} + i (P_1 + P_2 + P_3)\right)} \prod_{j = 1}^3 \frac{\sqrt{\Upsilon_b(2 i P_j)\Upsilon_b(-2 i P_j)}}{\Upsilon_b\left(\frac{Q}{2} + i (P_1 + P_2 + P_3 - 2 P_j)\right)} \end{align}$$

where the Upsilons are defined by

$$\begin{align} \Upsilon_b(x) = \exp\left\{ \int_0^{\infty} \frac{dt}{t} \left[ \left(\frac{Q}{2} + x \right)^2 e^{-t} + \frac{\sinh^2 \left[ \left(\frac{Q}{2} + x \right) \frac{t}{2} \right]}{\sinh \frac{t b}{2} \sinh \frac{t}{2b}} \right] \right\} \, . \end{align}$$

With this, the theory is fully specified. We see that there is a two-parameter family of solutions, depending on $b$ (which defines $Q$ through $Q = b + b^{-1}$, and $\mu$, which shows up as an overall factor. These may be mapped to the same parameters which appear in the Lagrangian description given above.

• How is the partition function normalized in a theory with no ground state?

Alternatively, the theory may be defined through the path integral. This requires an action, which takes the form

$$\begin{align} S = \frac{1}{4 \pi} \int d^2 x (\partial_a \phi \partial^a \phi + 4 \pi \mu e^{2 b \phi} ) \end{align}$$

$\phi$ is called the Liouville field. The equations of motion which result from this theory are

$$\begin{align} \Box \phi = - 4 \pi \mu b e^{2 b \phi} \end{align}$$

This is equivalent to Liouville's equation, which is the source of the theory's name.

It is instructive to consider the wavefunction of states which ``scatter“ off the potential barrier $V(\phi)$. This takes the form

$$\begin{align} \Psi_P(\phi) \sim e^{2 i P \phi} + R(P) e^{-2 i P \phi} \end{align}$$

Here $R(P)$ is the reflection amplitude for incoming plane-waves with momentum $P$, and may be computed from the equation of motion. The wave functions $\Psi$ describe the states which map to the Virarsoro primaries of Liouville theory under the state-operator correspondence.

• Xi Yin's notes on 2D CFTs, section 3.5
• Sylvain Ribault's notes on 2D CFTs, chapter 3
• Yu Nakayama's more string-oriented review
• Notes by Zamolodchikov and Zamolodchikov]]
• Dorn and Otto result for the structure constants: https://arxiv.org/pdf/hep-th/9403141.pdf
• Zamolodchikov and Zamolodchikov structure constants: https://arxiv.org/pdf/hep-th/9506136.pdf