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Given a coset model $(G\times SO(2d))/H$, what is the expression for its conformal weight (in terms of its central charge or, alternatively, in terms of the highest weights of irreducible representations of $G$ and $H$?

The particular case I am trying to understand is this: in a paper by Nozaki [1] it is written that the conformal weigh for the coset model $$ \frac{\widehat{S U}(n+1)_{k} \times \widehat{S O}(2 n)_{1}}{\widehat{S U}(n)_{k+1} \times \widehat{U}(1)_{n(n+1)(k+n+1)}}, \quad n, k \in \mathbb Z^+ $$ is $$ h=\frac{1}{2(k+n+1)}\left[\Lambda\left(\Lambda+2 \rho_{G}\right)-\lambda\left(\lambda+2 \rho_{H}\right)-\frac{m^{2}}{n(n+1)}\right]+g(s) \quad \bmod \mathbf{Z} $$ so that $h=h(n,k,\Lambda,\lambda,m,s)$, and where $\rho_G$ is the Weyl vector of $G$ and $$g(s)=\left(0, \frac{1}{2}, \frac{d}{8}, \frac{d}{8}\right) \quad \text { for } \quad s=(0,2,1,-1).$$

Here $G=SU(n+1)$, $H=SU(n)\times U(1)$, $\Lambda$ is a highest weight of $\mathfrak g$ at level $k$, $\lambda$ is a highest weight of $\mathfrak{su}(n)$ at level $k+1$, $m$ is a ghest weight of $\mathfrak{u}(1)$ at level $n(n+1)(k+n+1)$ and $s$ is a highest weight of $\mathfrak{su}(2n)$ at level $1$.

I do not have any idea where this expression comes from, and so I want an expression for general coset models, to apply to this case. I could not find one in the original GKO papers, nor in di Francesco's book, nor anywhere else.


[1] Nozaki, M. (2002). Comments on d-branes in kazama-suzuki models and landau-ginzburg theories. Journal of High Energy Physics, 2002(03):027.

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