Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model

Question

According to P. Francesco et al. conformal field theory book the central charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which obtained by Sugawara construction on the generators of symmetry in the Wess-Zumino-Witten (WZW) model, is given by the following formula (Page 629 eq 15.61):

$$c=\frac{k\dim g}{k+h^\lor }$$ Where $\dim g$ is the dimension of the corresponding algebra $g$, $k$ is the level of the affine lie algebra $\hat{g}_k$ and $h^\lor$ is the dual Coxeter number of $g$.

My question is concerning the specific Lie algebra $g = sl(2,\mathbb{R})$. This Lie algebra is noncompact but has finite dimensions of $\dim sl(2,\mathbb{R})=3$, but I am concerned with finding what is value of the dual Coxeter number. According to the classification of semisimple Lie algebra visualized by Dynkin diagrams, the closest group is of the class of $A_n$ corresponding to $sl(n+1,\mathbb{C})$ with the dual Coxeter number $h^{\lor}=n+1$ according to Wikipedia but in our case, $n=1$ but the group is defined over $\mathbb{R}$ and not over $\mathbb{C}$. According to Maldecena et al. Article the result which appears in eq. 65 demanding the dual Coxeter number to have the following value: $$h^\lor=-2$$ This value is different than the complex case of $sl(2,\mathbb{C})$ what is the reason for this? moreover, what is the right class $(A_n,B_n,C_n,D_n,...)$ that is suitable for $sl(2,\mathbb{R})$ which will give eventually the right answer?

Answer 1

Not an answer, but a long comment.

I've figured it out. According to AccidentalFourierTransform here What is wrong with this proof that $h^\vee$ the dual Coxeter number is always 1?, the definition of the Killing form is: \begin{equation} \kappa(X,Y)=\frac{1}{2h^\vee}\text{tr}(\text{ad}_X\text{ad}_X) \end{equation} Thus, on can see that if the Killing form reduces to $\propto \text{Id}$, we have the same formula as the link I failed to give correctly in the comments (which is https://conf.itp.phys.ethz.ch/esi-school/Lecture_notes/WZW%20models.pdf, equation (3.21)). But there is a problem with the canonical form of the basis of $\mathfrak{sl}(2,\mathbb{R})$ in the link you gave in the comment (https://math.stackexchange.com/questions/3438595/finding-the-structure-constants-of-sl2-f), namely that the Killing form, as given in the table https://en.wikipedia.org/wiki/Killing_form#Matrix_elements, is zero for $\kappa(X_1,X_1)$ and $\kappa(X_2,X_2)$, with: \begin{equation} X_1=\left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right),\,\,\,X_2=\left( \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \right),\,\,\,X_3=\left( \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right) \end{equation} But, we can diagonalize the Killing form by posing: \begin{equation} X_x=\frac{1}{\sqrt{2}}(X_2+X_1),\,\,\,X_y=\frac{1}{\sqrt{2}}(X_2-X_1),\,\,\,X_z=X_3 \end{equation} This is OK to do so because we work in a vector space. The commutation relations are: \begin{align} &[X_x,X_y]=X_z \\ &[X_y,X_z]=2X_x \\ &[X_x,X_z]=2X_y \end{align} With this, we have the structure constants $f_{xy}{}^z=-f_{yx}{}^z=1,\,\,f_{yz}{}^x=-f_{zy}{}^x=2$ and $f_{xz}{}^y=-f_{zx}{}^y=2$. It is then easy to check that we have: \begin{equation} \text{tr}(\text{ad}_{X_i}\text{ad}_{X_j})=f_{ik}{}^l f_{jl}{}^k = 4\text{tr}(X_iX_j) =4 \kappa(X_i,X_j) \end{equation} Thus, we deduce that $h^\vee=2$, as per the Wikipedia link on the Killing form I gave. The factor $-1$ we would want to seek if we wanted to have the same claim as Maldacena et al. comes (I think) from the physicists' habit consisting of sticking an $i$ factor everywhere in the Lie algebra.