Simplicity of vacuum modules and associated varieties

In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $\mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $\mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.


Introduction
Let V be a vertex algebra, and let be the state-field correspondence.The Zhu C 2 -algebra [Zhu96] of V is by definition the quotient space R V = V /C 2 (V ), where C 2 (V ) = span C {a (−2) b | a, b ∈ V }, equipped with the Poisson algebra structure given by a.b = a (−1) b, {a, b} = a (0) b, for a, b ∈ V with a := a+C 2 (V ).The associated variety X V of V is the reduced scheme X V = Specm(R V ) corresponding to R V .It is a fundamental invariant of V that captures important properties of the vertex algebra V itself (see, for example, [BFM, Zhu96, ABD04, Miy04, Ara12a, Ara15a, Ara15b, AM18a, AM17, AK18]).Moreover, the associated variety X V conjecturally [BR18] coincides with the Higgs branch of a 4D N = 2 superconformal field theory T, if V corresponds to a theory T by the 4D/2D duality discovered in [BLL + 15].Note that the Higgs branch of a 4D N = 2 superconformal field theory is a hyperkähler cone, possibly singular.
In the case where V is the universal affine vertex algebra V k (g) at level k ∈ C associated with a complex finite-dimensional simple Lie algebra g, the variety X V is just the affine space g * with Kirillov-Kostant Poisson structure.In the case where V is the unique simple graded quotient L k (g) of V k (g), the variety X V is a Poisson subscheme of g * which is G-invariant and conic, where G is the adjoint group of g.
Note that if the level k is irrational, then L k (g) = V k (g), and hence X L k (g) = g * .More generally, if L k (g) = V k (g), that is, V k (g) is simple, then obviously X L k (g) = g * .
In this article, we prove that the converse is true.
It is known by Gorelik and Kac [GK07] that V k (g) is not simple if and only if (1.1) where h ∨ is the dual Coxeter number and r ∨ is the lacing number of g.Therefore, Theorem 1.1 can be rephrased as (1.2) X L k (g) g * ⇐⇒ (1.1) holds.
Let us mention the cases when the variety X L k (g) is known for k satisfying (1.1).First, it is known [Zhu96,DM06] that X L k (g) = {0} if and only if L k (g) is integrable, that is, k is a nonnegative integer.Next, it is known that if L k (g) is admissible [KW89], or equivalently, if where h is the Coxeter number of g, then X L k (g) is the closure of some nilpotent orbit in g ( [Ara15a]).Further, it was observed in [AM18a,AM18b] that there are cases when L k (g) is non-admissible and X L k (g) is the closure of some nilpotent orbit.In fact, it was recently conjectured in physics [XY19] that, in view of the 4D/2D duality, there should be a large list of non-admissible simple affine vertex algebras whose associated varieties are the closures of some nilpotent orbits.Finally, there are also cases [AM17] where X L k (g) is neither g * nor contained in the nilpotent cone N(g) of g.
In general, the problem of determining the variety X L k (g) is wide open.Now let us explain the outline of the proof of Theorem 1.1.First, Theorem 1.1 is known for the critical level k = −h ∨ ([FF92, FG04]).Therefore, since R V k (g) is a polynomial ring C[g * ], Theorem 1.1 follows from the following fact.
The symbol σ(w) of a singular vector w in V k (g) is a singular vector in the corresponding vertex Poisson algebra grV k (g where J ∞ g * is the arc space of g * .Theorem 1.2 states that the image of σ(w) of a non-trivial singular vector w under the projection Hence, Theorem 1.2 would follow if the image of any nontrivial singular vector in 3) is nonzero.However, this is false as there are singular vectors in C[J ∞ g * ] that do not come from singular vectors of V k (g) and that belong to the kernel of (1.3) (see Section 3.4).Therefore, we do need to make use of the fact that σ(w) is the symbol of a singular vector w in V k (g).We also note that the statement of Theorem 1.2 is not true if k is critical (see Section 3.4).For this reason the proof of Theorem 1.2 is divided roughly into two parts.First, we work in the commutative setting to deduce a first important reduction (Lemma 3.1).Next, we use the Sugawara construction -which is available only at non-critical levels -in the non-commutative setting in order to complete the proof.Now, let us consider the W -algebra W k (g, f ) associated with a nilpotent element f of g at the level k defined by the generalized quantized Drinfeld-Sokolov reduction [FF90,KRW03]: Here, H • DS,f (M ) denotes the BRST cohomology of the generalized quantized Drinfeld-Sokolov reduction associated with f ∈ N(g) with coefficients in a V k (g)-module M .
By the Jacobson-Morosov theorem, f embeds into an sl 2 -triple (e, h, f ).The Slodowy slice S f at f is the affine space S f = f + g e , where g e is the centralizer of e in g.It has a natural Poisson structure induced from that of g * (see [GG02]), and we have [DSK06,Ara15a] As a consequence of Theorem 1.1, we obtain the following result.
Theorem 1.3.-Let f be any nilpotent element of g.The following assertions are equivalent: . The remainder of the paper is structured as follows.In Section 2 we set up notation in the case of affine vertex algebras that will be the framework of this note.Section 3 is devoted to the proof of Theorem 1.1.In Section 4, we have compiled some known facts on Slodowy slices, W -algebras and their associated varieties.Theorem 1.3 is proved in this section.
Acknowledgements.-T.A. and A.M. like to warmly thank Shanghai Jiao Tong University for its hospitality during their stay in September 2019.

Poisson algebras
Let g be the affine Kac-Moody algebra associated with g, that is, where the commutation relations are given by [x ⊗ t m , y ⊗ t n ] = [x, y] ⊗ t m+n + m(x|y)δ m+n,0 K, [K, g] = 0, for x, y ∈ g and m, n ∈ Z. Here, ( | ) = 1 2h ∨ × Killing form of g is the usual normalized inner product.For x ∈ g and m ∈ Z, we shall write x(m) for x ⊗ t m .

Universal affine vertex algebras
where C k is the one-dimensional representation of g[t] ⊕ CK on which K acts as multiplication by k and g ⊗ C[t] acts trivially.
By the Poincaré-Birkhoff-Witt Theorem, the direct sum decomposition, we have The space V k (g) is naturally graded, J.É.P. -M., 2021, tome 8 where the grading is defined by with 1 the image of 1 ⊗ 1 in V k (g).We have V k (g) 0 = C1, and we identify g with V k (g) 1 via the linear isomorphism defined by x → x(−1)1.
It is well-known that V k (g) has a unique vertex algebra structure such that 1 is the vacuum vector, for x ∈ g, where T is the translation operator.Here, x(n) acts on V k (g) by left multiplication, and so, one can view x(n) as an endomorphism of V k (g).The vertex algebra V k (g) is called the universal affine vertex algebra associated with g at level k [FZ92, Zhu96, LL04].The vertex algebra V k (g) is a vertex operator algebra, provided that k + h ∨ = 0, by the Sugawara construction.More specifically, set where {x i | i = 1, . . ., d} is the dual basis of a basis {x i | i = 1, . . ., dim g} of g with respect to the bilinear form ( | ), with d = dim g.Then for k = −h ∨ , the vector ω = S/(k + h ∨ ) is a conformal vector of V k (g) with central charge and L n 1 = 0 for n −1.
We have Any graded quotient of V k (g) as g-module has the structure of a quotient vertex algebra.In particular, the unique simple graded quotient L k (g) is a vertex algebra, and is called the simple affine vertex algebra associated with g at level k.

Associate graded vertex Poisson algebras of affine vertex algebras
It is known by Li [Li05] that any vertex algebra V admits a canonical filtration Here we have set Let gr F V = p F p V /F p+1 V be the associated graded vector space.The space gr F V is a vertex Poisson algebra by The filtration F • V is compatible with the grading: Then G • V defines an increasing filtration of V .We have (2.4) where In particular, we have The action of g More precisely, the element x(m), for x ∈ g J.É.P. -M., 2021, tome 8 and m ∈ Z 0 , acts on S(t −1 g[t −1 ]) as follows:

Zhu's C 2 -algebras and associated varieties of affine vertex algebras
We have [Li05, Lem.2.9] for all p 1.In particular, It is known by Zhu [Zhu96] that R V is a Poisson algebra.The Poisson algebra structure can be understood as the restriction of the vertex Poisson structure of gr F V .It is given by By definition [Ara12a], the associated variety of V is the reduced scheme It is easily seen that The following map defines an isomorphism of Poisson algebras Identifying g * with g through the bilinear form ( | ), one may view X V as a subvariety of g.
Definition 2.2.-Each element x of V k (g) is a linear combination of elements in the above PBW basis, each of them will be called a PBW monomial of x.
Definition 2.3.-For a PBW monomial v as in (2.8), we call the integer the depth of v.In other words, a PBW monomial v has depth p means that v ∈ F p V k (g) and v ∈ F p+1 V k (g).By convention, depth(1) = 0.
J.É.P. -M., 2021, tome 8 For a PBW monomial v as in (2.8), we call degree of v the integer In other words, v has degree Recall that a singular vector of a g[t]-representation M is a vector m ∈ M such that e α (0).m = 0, for all α ∈ ∆ + , and f θ (1) • m = 0, where θ is the highest positive root of g.From the identity we deduce the following easy observation, which will be useful in the proof of the main result.
Lemma 2.4.-If w is a singular vector of V k (g), then 2.5.Basis of associated graded vertex Poisson algebras.-Note that grV k (g) = S(t −1 g[t −1 ]) has a basis consisting of 1 and elements of the form (2.8).Similarly to Definition 2.2, we have the following definition.
Definition 2.5.-Each element x of S(t −1 g[t −1 ]) is a linear combination of elements in the above basis, each of them will be called a monomial of x.
As in the case of V k (g), the space S(t −1 g[t −1 ]) has two natural gradations.The first one is induced from the degree of elements as polynomials.We shall write deg(v) for the degree of a homogeneous element v ∈ S(t −1 g[t −1 ]) with respect to this gradation.
The second one is induced from the Li filtration via the isomorphism The degree of a homogeneous element v ∈ S(t −1 g[t −1 ]) with respect to the gradation induced by Li filtration will be called the depth of v, and will be denoted by depth(v).
As a consequence of (2.5), we get that (2.9) for m 0, x ∈ g, and any homogeneous element v ∈ S(t −1 g[t −1 ]) with respect to both gradations.
In the sequel, we will also use the following notation, for v of the form (2.8), viewed either as an element of V k (g) or of S(t −1 g[t −1 ]): (2.10) deg which corresponds to the degree of the element obtained from v (0) by keeping only the terms of depth 0, that is, the terms u i (−1), i = 1, . . ., .
Notice that a nonzero depth-homogeneous element of S(t −1 g[t −1 ]) has depth 0 if and only if its image in

Proof of the main result
This section is devoted to the proof of Theorem 1.1.
For k = −h ∨ , it follows from [FG04] that I k is the defining ideal of the nilpotent cone N(g) of g, and so X L k (g) = N(g) (see [Ara12b] or Section 3.4 below).Hence, there is no loss of generality in assuming that k + h ∨ = 0.
Henceforth, we suppose that k + h ∨ = 0 and that V k (g) is not simple, that is, N k = {0}.Then there exists at least one non-trivial (that is, nonzero and different from 1) singular vector w in V k (g).Theorem 1.2 states that the image of w in I k is nonzero, and this proves Theorem 1.1.The rest of this section is devoted to the proof of Theorem 1.2.
Let w be a nontrivial singular vector of V k (g).One can assume that w ∈ F p V k (g) F p+1 V k (g) for some p ∈ Z 0 .
The image Here σ : V k (g) → gr F V k (g) stands for the principal symbol map.It follows from (2.9) that one can assume that w is homogeneous with respect to both gradations on S(t −1 g[t −1 ]).In particular w has depth p.It is enough to show that p = 0, that is, w has depth zero.Write where J is a finite index set, λ j are nonzero scalar for all j ∈ J, and w j are pairwise distinct PBW monomials of the form (2.8).Let I ⊂ J be the subset of i ∈ J such that Here, w i stands for the image of w i in gr More specifically, for any j ∈ I, write (3.1) q,sq , where r 1 , . . ., r q , s 1 , . . ., s q , , t 1 , . . ., t are nonnegative integers, and a i,p , for l = 1, . . ., q, m = 1, . . ., r l , n = 1, . . ., s l , i = 1, . . ., , p = 1, . . ., t i , are nonnegative integers such that at least one of them is nonzero.
The integers r l 's, for l = 1, . . ., q, are chosen so that at least one of the a (j) l,r l 's is nonzero for j running through J if for some j ∈ J, (w j ) (+) = 1.Otherwise, we just set (w j ) (+) := 1. Similarly are defined the integers s l 's and t m 's, for l = 1, . . ., q and m = 1, . . ., .By our assumption, note that for all i ∈ I,

A technical lemma.
-In this paragraph we remain in the commutative setting, and we only deal with w ∈ S(t −1 g[t −1 ]) and its monomials w i 's, for i ∈ I.
Recall from (2.10) that, Proof.-Suppose the assertion is false.Then for some positive roots β j1 , . . ., β jt ∈ ∆ + , one can write for any i ∈ I so that for any l ∈ {1, . . ., t}, Since w is a singular vector of S(t −1 g[t −1 ]) and s j1 − 1 ∈ Z 0 , we have On the other hand, using the action of g[t] on S(t −1 g[t −1 ]) as described by (2.5), we see that and v is a linear combination of monomials x such that −1 (I).
where y i is a linear combination of monomials y such that deg −1 (I) and, hence, e βj 1 (s j1 − 1) • w i is a linear combination of monomials z such that deg Hence by (3.3) we get a contradiction because all monomials v i , for i running through −1 .This concludes the proof of the lemma.

3.3.
Use of Sugawara operators.-Recall that w = j∈J λ j w j .Let J 1 ⊆ J be such that for i ∈ J 1 , (w i ) (−) = 1.Then by Lemma 3.1, and J (0) −1 .Our next aim is to show that for i ∈ J + , w i has depth zero, whence p = 0 since p is by definition the smallest depth of the w j 's, and so the image of This will be achieved in this paragraph through the use of the Sugawara construction.
Recall that by Lemma 2.4, since w is a singular vector of V k (g), where -Let z be a PBW monomial of the form (2.8).Then L −1 z is a linear combination of PBW monomials x satisfying all the following conditions: Proof.-Parts (a)-(c) are easy to see.We only prove (d).Assume that deg(x (0) ) = deg(z (0) ) + 1.Either x comes from the term i=1 u i (−1)u i (0)z, or it comes from a term e α (−1)f α (0)z for some α ∈ ∆ + .
We now consider the action of L −1 on particular PBW monomials.

Second, we have
It is clear that any PBW monomial y in α∈∆+ e α (−1)e βj 1 (−1)e βj 2 (−1) J.É.P. -M., 2021, tome 8 We now consider -If β jr = α + β for some α, β ∈ ∆ + , then there is a partial sum of two terms in u r : Rewriting the above sum to a linear combination of PBW monomials, and noticing that -If α − β jr ∈ ∆ + for some α ∈ ∆ + , then there is a term in u r : (3.7) c −α,βj r e α (−1)e βj 1 (−1) It is easy to see that (3.7) is a linear combination of PBW monomials y such that y satisfies one of the following: Together with (3.4), we see that Lemma 3.4.-Let z be a PBW monomial of the form (2.8) such that z (−) = 1.Then where c is a nonzero constant, γ = q j=1 rj s=1 a j,s β j , and y 1 is a linear combination of PBW monomials y such that −1 (y) deg −1 (z).
Proof.-Since the proof is similar to that of Lemma 3.3, we left the verification to the reader.
Proof.-First we have λ j w j .
Then by Lemma 3.2(b) and Lemma 3.4, we have where j,s β i , for i ∈ J 1 , and y 1 is a linear combination of PBW monomials y satisfying one of the following conditions: On the other hand, by Lemma 2.4 J.É.P. -M., 2021, tome 8 By Lemma 2.1, there is no PBW monomial y in L −1 w such that deg(y (+) ) = d + , y (−) = 1, and deg As explained at the beginning of §3.3, Theorem 1.1 will be a consequence of the following lemma.
Proof.-By definition, for i ∈ J + , (w i ) (0) = 1.Moreover, by Lemma 3.5, depth((w i ) (+) ) = 0. Hence it suffices to prove that for i ∈ J + , Suppose the contrary.Then there exists i ∈ J + such that ,m 1, with at least one of the m j 's, for j = 1, . . ., , strictly greater than 1 and c (i) j,mj = 0 for such a j.Without loss of generality, one may assume that 1 ∈ J + , that m 1 = max{m j | j = 1, . . ., } and 0 = c where for i ∈ J + , v i is the PBW monomial defined by: ,m , (3.16) and so, by definition of J + ⊂ J J.É.P. -M., 2021, tome 8 because (v 1 ) (−) = 1 by (3.14).Remember that Computing L −1 w i , we deduce that -Finally, if j ∈ J J 1 , then by Lemma 3.2(b), any PBW monomial y in L −1 w j satisfies that y (−) = 1.So v 1 cannot be a PBW monomial in L −1 w j .This concludes the proof of the lemma.
As already explained, Lemma 3.6 implies that w has zero depth and so its image in R V k (g) is nonzero, achieving the proof of Theorem 1.1.

3.4.
Remarks.-The statement of Theorem 1.2 is not true at the critical level.Also, it is not true that the depth of a depth-homogeneous singular vector of S(g where J ∞ X is the arc space of X, and so S(g This means that the invariant ring is a polynomial ring with infinitely many variables ∂ j p i , i = 1, . . ., , j 0, where p 1 , . . ., p is a set of homogeneous generators of S(g) g considered as elements of S(g[t −1 ]t −1 ) via the embedding S(g) → S(g[t −1 ]t −1 ), g x → x(−1).We have depth(∂ j p i ) = j although each ∂ j p i is a singular vector of S(g[t −1 ]t −1 ).
For k = −h ∨ , the maximal submodule N k of V k (g) is generated by Feigin-Frenkel center ( [FG04]).Hence [FF92,Fre05], gr N k is exactly the argumentation ideal of S(g[t −1 ]t −1 ) g [t] .Therefore, the above argument shows that the statement of Theorem 1.2 is false at the critical level.

W -algebras and proof of Theorem 1.3
Let f be a nilpotent element of g.By the Jacobson-Morosov theorem, it embeds into an sl 2 -triple (e, h, f ) of g.Recall that the Slodowy slice S f is the affine space f + g e , where g e is the centralizer of e in g.It has a natural Poisson structure induced from that of g * ([GG02]).
The embedding span C {e, h, f } ∼ = sl 2 → g exponentiates to a homomorphism SL 2 → G.By restriction to the one-dimensional torus consisting of diagonal matrices, we obtain a one-parameter subgroup ρ : C * → G.For t ∈ C * and x ∈ g, set ρ(t)x := t 2 ρ(t)(x).We have ρ(t)f = f , and the C * -action of ρ stabilizes S f .Moreover, it is contracting to f on S f , that is, for all x ∈ g e , lim t→0 ρ(t)(f + x) = f.
The following proposition is well-known.Since its proof is short, we give below the argument for the convenience of the reader.As in the introduction, let W k (g, f ) be the affine W -algebra associated with a nilpotent element f of g defined by the generalized quantized Drinfeld-Sokolov reduction: Here, H • DS,f (M ) denotes the BRST cohomology of the generalized quantized Drinfeld-Sokolov reduction associated with f ∈ N(g) with coefficients in a V k (g)-module M .Recall that we have [DSK06,Ara15a] a natural isomorphism R W k (g,f ) ∼ = C[S f ] of Poisson algebras, so that X W k (g,f ) = S f .We write W k (g, f ) for the unique simple (graded) quotient of W k (g, f ).Then X W k (g,f ) is a C * -invariant Poisson subvariety of the Slodowy slice S f .
Let O k be the category O of g at level k.We have a functor where W k (g, f ) -Mod denotes the category of W k (g, f )-modules.
The full subcategory of O k consisting of objects M on which g acts locally finitely will be denoted by KL k .Note that both V k (g) and L k (g) are objects of KL k .(1) H i DS,f (M ) = 0 for all i = 0, M ∈ KL k .In particular, the functor (l − 1) = depth(w) = p.
J.É.P. -M., 2021, tome 8 Proposition 4.1 ([Slo80,Pre02,CM16]). -The morphismθ f : G × S f −→ g, (g, x) −→ g • x is smooth onto a dense open subset of g * .Proof.-Since g = g e + [f, g], the map θ f is a submersion at (1 G , f ).Therefore, θ f is a submersion at all points of G × (f + g e )because it is G-equivariant for the left multiplication in G, and lim t→∞ ρ(t) • x = f for all x in f +g e .So, by [Har77, Ch.III, Prop.10.4], the map θ f is a smooth morphism onto a dense open subset of g, containing G • f .