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On the classification of non-equal rank affine conformal embeddings and applications

We complete the classification of conformal embeddings of a maximally reductive subalgebra $\k$ into a simple Lie algebra $\g$ at non- integrable non-critical levels $k$ by dealing with the case when $\k$ has rank less than that of $\g$. We describe some remarkable instances of decomposition of the vertex algebra $V_{; ; ; ; ; ; ; ; ; ; k}; ; ; ; ; ; ; ; ; ; (\g)$ as a module for the vertex subalgebra generated by $\k$. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings $A_1 \times A_1 \hookrightarrow C_3$ at level $k=-1/2$, and obtain explicit branching rules by applying certain $q$- series identity. In the analysis of conformal embedding $A_1 \times D_4 \hookrightarrow C_8$ at level $k=-1/2$ we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.

conformal embedding ; vertex operator algebra ; non-equal rank subalgebra ; Howe dual pairs ; $q$-series identity

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