- I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.  

 On the few pages leading up to page 192 [in here][1] one can see see the analysis of the CFT obtained in the compactified directions of closed bosonic strings. 

  From what one sees in these notes it seems that a CFT with the above properties will exist if it is put on a torus $\mathbb{R}^n/\Lambda$ where $n=rank (G)$ and $\Lambda =$root lattice of $G$. 
  - Is the above correct? If it is correct then how does one write down the corresponding Lagrangian and the currents? 

  - But in the string context where these attached notes are based one is forced to have simply laced $G$ and hence only the $A,D,E$ series. How would one do this for say the group $G_2$? (..being concerned with just a CFT and not connected to string theory..)

  - The restriction of being on the $A$, $D$, $E$ series is related to the fact that in string context one has to tune all the compactification radius to the same self-dual point. If $i$ indexes the compact directions , $1\leq i \leq n$ then the current operators are possibly like $:\partial _{z} X_i (z):$ and $:e^{i\vec{\alpha}.\vec{X}}:$ where the first set is one for each Cartan and the send set is one for each root $\alpha$. But I wonder where would the radius of the circles go in these currents. 

- Finally where in this process can one tune the level of the affine Lie algebra? What choice fixes that? 



  [1]: http://srv2.fis.puc.cl/~mbanados/Cursos/Cuerdas/LustTheisen%20.pdf