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

Authors: Dražen Adamović, Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše

https://doi.org/10.1007/s00029-017-0386-7

Abstract: We complete the classification of conformal embeddings of a maximally reductive subalgebra $k$ a simple Lie algebra $\mathfrak{g}$ at non-integrable non-critical levels k by dealing with the case when $\mathfrak{k}$ has rank less than that of $\mathfrak{g}$ . We describe some remarkable instances of decomposition of the vertex algebra $V_{k} (g)$ 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 $k = -1/2$ , 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.

Keywords: Conformal embedding Vertex operator algebra Non-equal rank subalgebra Howe dual pairs q-series identity

Mathematics Subject Classification: Primary 17B69, Secondary 17B20 17B65

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