Schur–weyl Duality for Heisenberg Cosets

Abstract

Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com (H,V) be the coset of H in V. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational and C 2 -cofinite and CFT-type, and Com (C,V) is a rational lattice vertex operator algebra, then so is C. These results are illustrated with many examples and the C 1 -cofiniteness of certain interesting classes of modules is established. is exact, for M ′′ ∼ = (J ⊠ M)/M ′ = 0. But, fusion is right-exact [HLZ, Prop. 4.26], sois exact. However, M ′′ = 0 implies that J −1 ⊠ M ′′ is a non-zero quotient of M, by (1), so we must have J −1 ⊠ M ′′ ∼ = M, as M is simple. Fusing with J now gives J ⊠ M ∼ = M ′′ , so we conclude that M ′ = 0 and that J ⊠ M is simple. The simplicity of J ∼ = J ⊠ V now follows from that of V, completing the proof of (3).To prove (4), note that applying right-exactness to the short exact sequence 0where f is the induced map from J ⊠ M ′ to J ⊠ M that might not be an inclusion. Fusing with J −1 and applying (2.4), we arrive at 5)

Unitary and non-unitary N = 2 minimal models

“…Their subquotients (4.18) exhaust the atypical irreducible M(u, )-modules that we have constructed through branching rules. The proof that we have found all the irreducible highest-weight M(u, )-modules now follows from [45,Thm. 4.3] as in the unitary case.…”

“…The Grothendieck fusion rules of the [i ] C L p;r, 0 in fact lift to genuine fusion rules for M(u, ). This follows from the fact that the same is true for A 1 (u, ), see (A.2), and the fact that Heisenberg cosets preserve module structures [45,Thm. 3.8].…”

“…3.8]. In particular, this gives the following M(u, ) fusion rules: We remark that if we also assume a vertex tensor category structure on a category containing the [i ] C L p;r, 0 , then the results of [45,66] yield a rigorous proof of (6.14), independent of the conjectural standard Verlinde formula.…”

“…In fact, it is easy to show that there can be no more than 2u(u − 1). This relies on the result [45,Thm. 4.3] that given any irreducible M(u, 1)-module C, one can find a Fock space F p such that F p ⊗ C may be induced to an A 1 (u, 1) ⊗ bc-module using the embedding (3.2).…”

“…Here, we note that p = i + r − 1 mod 2. The Fock space fusion rules were given in (3.7), while those involving the [0] C p ′ ;1 were evaluated using [45,Prop. 3.7].…”