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liouville_theory [2021/04/10 19:17]
Scott Lawrence 		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====
Line 100: Line 100:
  
 ===Some Remaining questions=== ===Some Remaining questions===
-  * This theory has no ground state. So how do I normalize the partition function?+  * How is the partition function normalized in a theory with no ground state?
  
-====Lagrangian Picture====+=====Lagrangian Picture=====
  
-====External Links====+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   * 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]]   * 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 19:17
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