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

Selecta Mathematicapp 1–44


Abstract: We complete the classification of conformal embeddings of a maximally reductive subalgebra \mathfrak{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(\mathfrak{k}) 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