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

## TL;DR

This paper completes the classification of conformal embeddings for non-equal rank cases in affine Lie algebras, exploring decomposition patterns, new dual pairs, and explicit branching rules at negative levels.

## Contribution

It extends the classification of conformal embeddings to cases where the subalgebra rank is less than the ambient algebra, and constructs new affine Howe dual pairs at negative levels.

## Key findings

- Explicit branching rules for specific embeddings at negative levels.
- Identification of subsingular vectors not seen in classical dual pairs.
- Descriptions of decomposition patterns of vertex algebras under these embeddings.

## Abstract

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 $\mathfrak k$ has rank less than that of $\mathfrak g$. 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 $k=-1/2$, 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.