Joseph Ideals and Lisse Minimal -Algebras

Abstract

Abstract. We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac-Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse (C 2 -cofinite) Walgebras that are not coming from admissible representations of affine KacMoody algebras.

Free field realizations from the Higgs branch

“…This family of VOAs has been singled out in [1] as the set of theories that simultaneously saturate four-dimensional unitarity bounds 8 for the central charge c 4d and the flavor central charge k 4d . Additionally, it has been proved in [14] that for this family of VOAs the associated variety is the closure of the minimal nilpotent orbit O min (g) of the corresponding Lie algebra g. 9 6 More generally, the associated variety may be a stratified symplectic space with a finite number of symplectic leaves. 7 The case of a 0 is special and is identified with the Virasoro VOA at central charge c = − 22 5 .…”

Finite vs. Infinite Decompositions in Conformal Embeddings

“…This numerological coincidence can be explained as follows. Let W min (k) be the minimal simple W-algebra of level k for g. By [8,Theorem 7…”

On the irreducibility of associated varieties of W-algebras

“…Parts (1) and (2) of Conjecture 2 are true for r = 4 by [AM15]. Also, Part (2) of the conjecture is true for k = 2 − r as mentioned just above.…”

“…Namely, w 2 = e θ e θ2 − 2 j=1 e βj+θ2 e δj +θ2 , with β 1 := α 2 , δ 1 := α 1 + α 2 + α 3 , β 2 := α 2 + α 3 , δ 2 := α 1 + α 2 . Then σ(w 2 ) is a singular vector of V k (g) if and only if k = −2 [AM15, Theorem 4.2] where σ is the natural embedding of g-modules from S 2 (g) to V k (g) 2 := {v ∈ V k (g) | Dv = −2v} (see [AM15,Lemma 4.1]).…”

“…The case where the associated variety has finitely many symplectic leaves is particularly interesting. This happens for instance when V is an admissible affine vertex algebra 1 ( [A15a]), when V is the simple affine vertex algebra associated with a simple Lie algebra that belongs to the the Deligne exceptional series [D96] at level k = −h ∨ /6 − 1 ( [AM15]) with h ∨ the dual Coxeter number, or when V is the (generalized) Drinfeld-Sokolov reduction ( [FF90,KRW03]) of the latter affine vertex algebras provided that it is nonzero ( [A15a]). This is also expected to happen for the vertex algebras obtained from four dimensional N = 2 superconformal field theories ([BLL + 15]), where the associated variety is expected to coincide with the spectrum of the chiral ring of the Higgs branch of the four dimensional theory.…”