Some remarks on associated varieties of vertex operator superalgebras

Abstract

We study several families of vertex operator superalgebras from a jet (super)scheme point of view. We provide new examples of vertex algebras which are "chiralizations" of their Zhu's Poisson algebras R V . Our examples come from affine C(1) ℓ -series vertex algebras (ℓ ≥ 1), certain N = 1 superconformal vertex algebras, Feigin-Stoyanovsky principal subspaces, Feigin-Stoyanovsky type subspaces, graph vertex algebras W Γ , and extended Virasoro vertex algebra. We also give a counterexample to the chiralization property for the N = 2 superconformal vertex algebra of central charge 1.

Graph Schemes, Graph Series, and Modularity

“…Andrews-Gordon identities). More recently, Nahm sums have appeared in several areas including vertex algebras [26,33,37,40], DAHA [18], wall-crossing phenomena and quivers [17,28,34], in connection to colored HOMFLY-PT polynomials [29], and in formulas for the "tail" of colored Jones polynomials [9,22,27].…”

“…As explained in Section 2, for a given graph Γ, there is a graded commutative algebra J ∞ (R), the ring of functions of the infinite jet scheme X ∞ , where X = Spec(R), whose Hilbert series is given by (1.2). This infinitely-generated algebra is closely related to a certain principal subspace vertex algebra constructed from the adjacency matrix of Γ [26,33,37,40], in the sense that its character is the Hilbert series of J ∞ (R).…”

“…Provided that V is strongly finitely generated, we obtain a surjective map from J ∞ (R V ) to gr(V ), the associated graded algebra of V , where R V is Zhu's commutative algebra of a vertex algebra V . A vertex algebra for which this map is injective is said to be classically free [6,7,33,34,35]. Although this notion is relatively new, non-trivial examples of vertex algebras with this property appeared earlier in the framework of principal subspaces [15,16,20] and Virasoro minimal models [19].…”

“…Although this notion is relatively new, non-trivial examples of vertex algebras with this property appeared earlier in the framework of principal subspaces [15,16,20] and Virasoro minimal models [19]. Recall a result announced in [25,Section 9] motivated by [37], with full details provided in [32,33].…”

Jet schemes, Quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

“…Following the terminology of [vEH18] we say that V is classically free if ψ is injective. Another way of expressing injectivity is by saying that V is "chiralization of its Poisson algebra" [Li20]. Vertex algebras with this property are common and we expect that they include many (but not all) rational vertex algebras and of course large families of irrational vertex algebras.…”