Large spin expansion of affine $\mathfrak{su}(2)_k$ characters

Question

There is a problem I am trying to solve for some time now which in a few words boils down to computing the coset characters for $$ \frac{\mathfrak{su}(2)_k\oplus\mathfrak{su}(2)_\ell}{\mathfrak{su}(2)_{k+\ell}} $$ in the high spin limit. There is an expression for these characters in Sec. 18.5.4. of Di Francesco, but it is in the parafermionic formulation and it looks messy and complicated so I am not sure how to take the large spin limit there. The coset charecetrs $\Xi_{\lambda\mu}^\nu$, can be extracted from the character decomposition $$ \chi^{(k)}_{\lambda}(q)\chi^{(\ell)}_{\mu}(q)=\sum_{\nu=1}^{k+\ell}\Xi_{\lambda\mu}^\nu\chi^{(k+\ell)}_{\nu}(q), \qquad 0\leq\lambda\leq k, \ 0\leq\mu\leq\ell, $$ where $\chi^{(k)}_{\lambda}(q)$ are the affine $\mathfrak{su}(2)_k$ characters. So basically, I need to find a large spin (i.e. for large $\mu$ and $\nu$) expansion of the affine $\mathfrak{su}(2)_\ell$ characters and plug them into the above decomposition. I found some papers by Gabardiel and others where they do a similar calculation but for large level, so they use the following expansion $$ \chi_{\lambda}(q)=\frac{q^{m_{\lambda}}}{\phi(q)}\left[\mathrm{ch}_{\lambda}(g)+\mathcal{O}\left( q^{k-\lambda} \right)\right], $$ where $m_{\lambda}=\frac{C^{(2)}(\lambda)}{k+2}-\frac{c}{24}$ is the modular anomaly with $C^{(2)}(\lambda)$ the quadratic Casimir, $\phi(q)$ is the Euler function and $\mathrm{ch}_{\lambda}(g)$, for $g\in SU(2)$ is the finite character of $\mathfrak{su}(2)$. They do not explain how they get it and there is no reference to it. However, this obviously does not work in my case for lage $\lambda$. So the question I am asking is, does anyone know a similar expansion but for large $\lambda$ instead in the bibliogrphy or a way to compute it?

Answer 1

Well, I figured it out at the end. Our quest starts from the Kac character formula, which for $\mathfrak{su}(2)_\ell$ reads $$ \chi^{(\ell)}_\mu=\frac{\Theta^{(\ell+2)}_{\mu+1}-\Theta^{(\ell+2)}_{-\mu-1}}{\Theta^{(2)}_{1}-\Theta^{(2)}_{-1}}, $$ where $$ \Theta^{(\ell)}_{\mu} (q;z)= \sum_{n\in \mathbb{Z} + \frac{\mu}{2\ell}}z^{{2\ell}n}q^{\ell n^2}, \qquad q\equiv e^{2\pi i \tau} $$ is the generalized Jacobi Θ-function (at $t=0$) and $\tau$ is the torus modular parameter with $\mathrm{Im}\tau>0$. Moreover, the characters are class functions and they are completely determined by their value on the maximal torus $$\mathbb{T}=\{\mathrm{diag(e^{i\theta},e^{-i\theta})}\mid \theta\in\mathbb{R}\}. $$ This means we can choose $z=e^{i\theta}$. Thus, the Kac character formula becomes $$ \chi_{\mu}^{(\ell)}(q;e^{i\theta}) = q^{m_{\mu}}\frac{\sum_{n\in \mathbb{Z}}\left( \frac{\sin[(\mu+1+2n(\ell+2))\theta]}{\sin\theta} \right)q^{n(\mu+1)+n^2(\ell+2)}}{\prod_{n>0}( 1-q^n )( 1-e^{2i\theta}q^n )( 1-e^{-2i\theta}q^n )}. $$ The sum is uniformly convergent if $|q|<1$, i.e. for $\tau$ in the upper half-plane. I am interested in the limit where one of the spins (here Dynkin label) becomes very large in a correlated way with the level $\ell$, i.e. $$ \lambda,k<\infty, \quad \mu,\ell\gg1 \quad \Longrightarrow \quad \nu,\ell\gg1. $$ In this limit the sum over $n$ in the numerator is such that all terms with $n\neq0$ are exponentially suppressed by the $n^2$ term in the exponent. Hence, only the $n=0$ term survives in the limit $$ \chi_{\mu}^{(\ell)}(q;e^{i\theta}) \approx q^{m_{\mu}} \frac{\mathrm{ch}_{\mu}(\theta)}{{\prod_{n>0}( 1-q^n )( 1-e^{2i\theta}q^n )( 1-e^{-2i\theta}q^n)}}, $$ where $\mathrm{ch}_{\mu}(\theta)=\frac{\sin((\mu+1)\theta)}{\sin\theta}$ is the finite character of $SU(2)$.