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

Abstract

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 A1 × A1 ֒→ C3 at level k = −1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1 × D4 ֒→ C8 at level k = −1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.

Kostant Pairs of Lie Type and Conformal Embeddings

“…One instance of this phenomenon occurs in E 7 . By using similar arguments as in [4,Section 8] one can show that the affine vertex subalgebra V (−4,…”

“…The classification of maximal conformal embeddings was studied in detail in [4]. On the other hand we detected in [4] some border cases where we can not speak of conformal embeddings since the embedded affine vertex subalgebras have critical levels. In this section we provide some results on this case.…”

“…(1) The conformality of the embedding given in (4.4) can be derived in a different way using Lemma 10.4 of [4]. More precisely, result from [4] implies that the vertex subalgebra V (−2, sl(4) × Z) of V −2 (D 4 ) must be simple. Since V (−2, sl(3) × Z) is a simple vertex subalgebra of V (−2, A 3 ) we actually have conformal chain of simple affine vertex algebras…”

“…This realization provides a nice framework for studying embeddings of affine vertex algebras in V −1/2 (C ℓ ). Some interesting cases were studied in [4]. In this section we study the embeddings in M ℓ of critical level affine vertex algebras.…”

Conformal embeddings in affine vertex superalgebras

“…When g is even, k is a subalgebra of g, and V k (g) is an extension of simple current type of the conformal subalgebra V k (k), we were able (cf. [6], [8], [9]) to get explicit decomposition rules without knowing precisely the fusion rules for V k (k)-modules. We can apply such methods here to obtain decomposition formulas when V k (g) is a simple current extension of V k (g0).…”