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

We complete the classification of conformal embeddings of a maximally reductive subalgebra $\mathfrak{k}$ into a simple Lie algebra $\mathfrak{g}$ at non-integrable non-critical levels $k$ by dealing with the case when $\mathfrak k$hasranklessthanthatof$\mathfrak g$. We describe some remarkable instances of decomposition of the vertex algebra $V_k(\mathfrak{g})$ as a module for the vertex subalgebra generated by $\mathfrak{k}$. 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 $k=-1\mathrm{/}2$, 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 $k=-1\mathrm{/}2$ we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.