I am currently studying affine Lie algebras and the WZW coset construction. I have a minor technical problem in calculating the (specialized) character of $\widehat{\mathfrak{su}}(2)_k$ for an affine weight $\hat{\lambda} = [k-\lambda_1,\lambda_1]$. Given the generalized theta function $$\Theta_{\lambda_1}^{(k)}(z,\tau) = \sum_{n\in\mathbb Z}e^{-2\pi i\left[knz+\frac 12\lambda_1 z-kn^2\tau-n\lambda_1\tau- \lambda_1^2\tau/4k\right]}$$ I want to evaluate $$\chi^{(k)}_{\lambda_1} = \frac{\Theta^{(k+2)}_{\lambda_1+1} - \Theta^{(k+2)}_{-\lambda_1-1}}{\Theta^{(2)}_1 - \Theta^{(2)}_{-1}}$$ at $z=0$. Putting $z=0$ directly, both the numerator and denomerator vanish (since there is no difference between $\lambda_1$ and $-\lambda_1$ due to the sum). So my question is; what is the appropriate way to take the limit $z\rightarrow 0$? [This is from Di Francesco et al, section 14.4.2, page 585]. The result should be $$\chi^{(k)}_{\lambda_1} = q^{(\lambda_1+1)^2/4(k+2)-\frac 18}\frac{\sum_{n\in\mathbb Z}\left[\lambda_1 + 1 + 2n(k+2)\right]q^{n[\lambda_1+1+2(k+2)n]}}{\sum_{n\in\mathbb Z}\left[1+4n\right]q^{n[1+2n]}}$$ where $q=e^{2\pi i\tau}$.

Since I fear the solution to my question is rather trivial, I have a bonus question. Do you know any paper which works out the details for the coset $$\frac{\widehat{\mathfrak{su}}(N)_k\oplus \widehat{\mathfrak{su}}(N)_1}{\widehat{\mathfrak{su}}(N)_{k+1}}$$ for arbitrary $N$? I am thinking about something like what Di Francesco et al. does in section 18.3 for $N=2$. It would be nice if the reference relates this to $\mathcal W$-algebras.


I found the answer to my problem, and as expected it is embarrassingly trivial. Put $z=\epsilon$ and expand everything to first order, then the result comes out directly.

With respect to references, I found that the review paper $\mathcal W$ symmetry in conformal field theory contains a discussion of these kind of coset Wess-Zumino-Witten models. Furthermore papers on the newly proposed higher spin AdS$_3$/CFT$_2$ duality contains some discussions on this (for example arXiv:1011.2986, arXiv:1108.3077 and arXiv:1106.1897). But I still welcome better references!

  • 1
    $\begingroup$ Quick note: putting $z=\epsilon$ and expanding to first order is equivalent to L'Hôpital's rule, which should probably always be the first stab at an indeterminate 0/0 limit. $\endgroup$ – Emilio Pisanty Jan 14 '13 at 12:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.