I am trying to understand the vertex operator algebras of the following form:
$$\frac{U(M|N)_{k_1}}{U(L)_{k_2}}$$
Where $U(M|N)$ is the unitary supergroup, $U(L)$ is the usual unitary group, and $k_1$ and $k_2$ are the levels, respectively. I want to know the followings:
For a specific embedding of $U(L)$ inside $U(M|N)$, how can I determine the highest-weight states of $\frac{U(M|N)_{k_1}}{U(L)_{k_2}}$? More specifically, how can I label these highest-weight states?
What is the conformal dimension (the eigenvalue of the $\widehat{L}_0$ operator) of these highest-weight states?
What happens if $M=L$ and the embedding is the trivial embedding of $U(M)$ inside $U(M|N)$?
How can I characterize the highest-weight states in the general case, namely $\frac{G_{k_1}}{H_{k_2}}$, where $H$ is a subgroup of $G$?
A reference is highly appreciated.
