======Liouville theory======

{{tag>2d}}

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

====Bootstrap Picture====

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.

===Modular Invariance and the Spectrum===

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 $P$ by((sometimes this is defined with $-P^2$, instead of $+P^2$, so that $P \in i \mathbb{R}$))

\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 $P>0$.

=== Crossing Invariance and the OPE Coefficients ===

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|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.

===Some Remaining questions===
  * How is the partition function normalized in a theory with no ground state?

=====Lagrangian Picture=====

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. 

=====External Links=====
  * Xi Yin's [[https://pos.sissa.it/305/003/pdf|notes]] on 2D CFTs, section 3.5
  * Sylvain Ribault's [[https://arxiv.org/pdf/1406.4290.pdf|notes]] on 2D CFTs, chapter 3
  * Yu Nakayama's more string-oriented [[https://arxiv.org/pdf/hep-th/0402009.pdf|review]]
  * [[http://qft.itp.ac.ru/ZZ.pdf|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