# CFTs corresponding to affine Lie algebras

• 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 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?
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What is your definition of CFT? –  S. Carnahan Mar 31 at 2:46
@S.Cranahan It is a very difficult question to really come up with a good definition of a CFT - roughly speaking its a system such that all the couplings are tuned to the zero of their respective beta-functions. But there can be a million more subtleties to this simple thinking like you can see these important recent works, arxiv.org/abs/1204.5221, arxiv.org/abs/1209.3424, arxiv.org/abs/1205.3994, arxiv.org/abs/1101.5385, arxiv.org/abs/hep-th/0006037 –  Anirbit Apr 2 at 18:16

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 $\hat{\mathfrak{g}}$ : $\left[J^a_n,J^b_m\right]=\sum_c i f^{abc}J^c_{n+m}+kn\delta^{ab}\delta_{n+m,0} \; \text{where} \; J^a(z)=\sum\limits_{n\in \mathbb Z}z^{n-1}J^a_n$.
@Anton Nazarov Aren't you needing to do a QFT of the action you have written? I mean - your currents need to be normal ordered or something? What is the fundamental Dirac bracket from which you are getting the $[J,J]$ commutation? (..also about your comment about comapactified Boson picture - thats part of my question - in closed Bosonic string theory compactified on a $(S^1)^{rank}$ isn't the affine algebra of the massless states restricted to be in the A, D, E?...And in that case what is Lagrangian?..) –  Anirbit Apr 1 at 16:56