* 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](http://srv2.fis.puc.cl/~mbanados/Cursos/Cuerdas/LustTheisen%20.pdf) 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?



CFTs corresponding to affine Lie algebras
You can construct Wess-Zumino-Novikov-Witten model starting with finite-dimensional subalgebra of your affine Lie algebra. The action of this model is written in terms of field $g:\mathbb{C}\cup\left\{\infty\right\}\sim S^2 \to G$, where $G$ is a Lie group, such that $\mathfrak{g}$ is finite dimensional subalgebra and $\hat{\mathfrak{g}}$ is your affine Lie aglebra. The action has following form:
$ S=-\frac{k}{8\pi}\int d^2x\; \mathcal{K} (g^{-1}\partial^{\mu}g, g^{-1} \partial\_{\mu}g) 
 \\
 - \frac{k }{24\pi^{2}} \int\_{B}\epsilon\_{ijk} \mathcal{K}\left(
 \tilde g^{-1}\frac{\partial \tilde g}{\partial y^i},\left[
 \tilde g^{-1}\frac{\partial \tilde g}{\partial y^j}
 \tilde g^{-1}\frac{\partial \tilde g}{\partial y^k}\right]\right) d^3y
$
The conserved currents are $J(z)= -k \partial\_zg g^{-1}$, $\bar J(\bar z)=k g^{-1}\partial\_{\bar z}g$. 


Modes of $J,\bar{J}$ satisfy commutation relations of affine Lie algebra $\hat{\mathfrak{g}}$ :
$ \left[J^a\_n,J^b\_m\right]=\sum\_c i f^{abc}J^c\_{n+m}+kn\delta^{ab}\delta\_{n+m,0} \; \text{where} \; J^a(z)=\sum\limits\_{n\in \mathbb Z}z^{n-1}J^a\_n $.


Some WZNW-models can be obtained by compactification of free bosons, but not all. 


See <http://arxiv.org/abs/hep-th/9911187> for very accessible introduction to the subject or part C in the book "Conformal field theory" by P. Di Francesco, P. Mathieu and D. Senechal.


