Conflicting definitions for Kazama-Suzuki models

Question

In their original paper, Kazama and Suzuki treated the coset models $G/H$ such that this quotient is a special kind of Kahler manifold called hermitian symmetric space. They give a bunch of examples, including Grassmanian manifolds $\frac{SU(m+n)}{SU(m)\times SU(n)\times U(1)}$.

However, in another paper (by Behr and Fredenhagen) they say that the Kazama-Suzuki models are of the form $\frac{G_k\times SO(2d)_1}{H}$, and gave the example of the Grassmanian manifold, which they wrote as $\frac{SU(m+n)_1\times SO(2nk)_1}{SU(m)_{k+1}\times SU(n)_{n+1}\times U(1)}$. The subscripts just refer to the level of the representation of the loop algebras that they want to take when constructing the coset using the GKO method.

Why is it OK to add the $SO(2d)$? It should affect the coset, and we get a different representation of the $N=2$ super-CFT if we add it. How can these two definitions be compatible?

Comments

SO(2N) is for the free fermions to get the rest as the bosonic coset expression.
More details in Supersymmetric holography on AdS3 by Constantin Candu, Matthias R. Gaberdiel

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@igael What do you mean by "the rest"? Also, is there another reference? Maybe more pedagogical? This one seems a bit over my head right now.

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@igael Ok, so it is partially explained later in the Kazama-Suzuki original paper, section 4.3.

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@Soap : all the linked doc is about the presence of the fermionic component. The goal was to use (3.16) because, as the authors state, "bosonic coset description ... contains implicitly the supersymmetry generators as long as we describe the so(2N)1 algebra in terms of 2N free Majorana fermions.". Finally, there is not a conflict.

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@igael Thanks, that helped. However, it hinges on the fact that $\mathfrak{su}(N)^1_{k+N}\cong \mathfrak{su}(N)_k\oplus \mathfrak{so}(\dim \mathfrak{su}(N))$, which they do not justify. Do you know where I can find the justification for this? Also, this is for the specific case of $\mathfrak{su}(N)$, whicle I want this for the general case $G$.

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@igael It seems that the appropriate generalization would be to say that $\hat{\mathfrak{g}}_k\oplus\widehat{\mathfrak{so}(\dim \mathfrak{g})}_1\cong\hat{\mathfrak{g}}^{N=1}_{k+\dim\mathfrak{g}}$. But I have no idea if this is correct (because, as I said, I do not know where the original result comes from).