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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.

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