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liouville_theory [2021/04/10 16:00]
Brian McPeak [External Links] 		liouville_theory [2021/06/29 22:36] (current)
Brian McPeak [Bootstrap Picture]
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-======Liouville Theory======+======Liouville theory======
  
 {{tag>2d}} {{tag>2d}}
  
-Liouville +Liouville theory is an interacting 2d CFT with a continuous spectrum of scalar states.
  
-=====Bootstrap Picture=====+====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.  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. 
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 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. 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==+===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 Now we will see how the spectrum is fixed by modular invariance. Modular invariance takes the form of the requirement that
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 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 $P>0$. 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 ==+=== 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 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
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 \begin{align} \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\}+  \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} \end{align}
  
-==Some Remaining questions== +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. 
-  * This theory has no ground state. So how do I normalize the partition function?+ 
 +===Some Remaining questions=== 
 +  * How is the partition function normalized in a theory with no ground state?
  
 =====Lagrangian Picture===== =====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===== =====External Links=====
   * Xi Yin's [[https://pos.sissa.it/305/003/pdf|notes]] on 2D CFTs, section 3.5   * 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   * 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   * 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   * Zamolodchikov and Zamolodchikov structure constants: https://arxiv.org/pdf/hep-th/9506136.pdf
  
Last modified: 2021/04/10 16:00
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