Strings on group manifolds and critical dimension

Question

In their work, Witten and Gepner in "Strings on group manifolds" have shown that the central charge of the theory is \begin{equation} c=\frac{kD}{c^{\vee}+k}+d=26, \end{equation} where $d$ is the dimension of flat spacetime, $D$ is the dimension of group manifold, $c^{\vee}$ is the dual Coxeter and $k$ is the level of representation. It seems that the dimension of spacetime can be greater than 26. For example, suppose that $G=SU(2)$, $k=1$, where the dual Coxeter of it $c^{\vee}=2$, then we have

\begin{equation} d+\frac{1×3}{2+1}=26, \end{equation} so, $d=25$, and $d+D=28$. Is this really true?

Answer 1

Yes. You've described a way for strings in 28 dimensions, 25 of which are flat, to be anomaly free. Of course neither of these numbers is 4 which is one of the many reasons why this is a toy model. But the same is true of most string backgrounds we learn about. Like the one with $(d, D) = (26, 0)$.