Characters of $\widehat{\mathfrak{su}}(2)_k$ and WZW coset construction

Question

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.

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Answer 1

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!

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