Associated schemes and vertex algebrasschool-tlag2017.math.cnrs.fr/Docs/Moreau-LectureNotes.pdfPART...

Associated schemes and vertexalgebras

Anne Moreau

Laboratoire Paul Painleve, CNRS UMR 8524, 59655 Villeneuved’Ascq Cedex

E-mail address: [email protected]

Abstract. These notes are written in preparation for the mini-course entitled“Associated schemes and vertex algebras” which will take place in the summerschool “Current Topics in the Theory of Algebraic Groups” of the GDR TLAG.

Contents

Introduction 5

Part 1. Affine Kac-Moody algebras and their vacuum representations 71.1. Affine Kac-Moody algebras 71.2. The vacuum representations 91.3. Associated varieties of vacuum representations 10

Part 2. Vertex algebras, definitions, first properties and examples 152.1. Operator product expansion and definition of vertex algebras 152.2. First examples of vertex algebras 182.3. Modules over vertex algebras 22

Part 3. Poisson vertex algebras, arc spaces, and associated varieties 233.1. Jet schemes and arc spaces 233.2. Poisson vertex algebras 273.3. Associated variety of a vertex algebra 293.4. Lisse and quasi-lisse vertex algebras 32

Part 4. Associated varieties of affine W-algebras 394.1. Slodowy slices 394.2. Affine W-algebras 434.3. Branching and nilpotent Slodowy slices 444.4. Conclusion, open problems 46

Bibliography 49

3

Introduction

The goal of this series of lectures is to introduce the theory of vertex algebras,with emphasis on their geometrical aspects.

Roughly speaking, a vertex algebra is a vector space V , endowed with a dis-tinguished vector, the vacuum vector, and the vertex operator map from V to thespace of formal Laurent series with linear operators on V as coefficients. Thesedata satisfy a number of axioms. Although the definition is purely algebraic, theseaxioms have deep geometric meaning. They reflect the fact that vertex algebrasgive an algebraic framework of the two-dimensional conformal field theory. Theconnections of this topic with other branches of mathematics and physics includealgebraic geometry (moduli spaces), representation theory (modular representationtheory, geometric Langlands correspondence), two dimensional conformal field the-ory, string theory (mirror symmetry) and four dimensional gauge theory (AGTconjecture).

To each vertex algebra V one can naturally attach a certain Poisson varietyXV called the associated variety of V . For an affine Poisson variety X, a vertexalgebra V such that XV

∼= X is called a chiral quantization of X.A vertex algebra V is called lisse if dimXV = 0. Lisse vertex algebras are

natural generalizations of finite-dimensional algebras and possess remarkable prop-erties. For instance, the characters of simple V -modules form vector valued modularfunctions. More generally, vertex algebras whose associated variety has only finitelysymplectic leaves, are also of great interest for several reasons that will be addressedin the lectures.

Important examples of vertex algebras are those coming from affine Kac-Moodyalgebra, which are called affine vertex algebras. They play a crucial role in therepresentation theory of affine Kac-Moody algebras, and of W-algebras. In the casethat V is a simple affine vertex algebra, its associated variety is an invariant andconic subvariety of the corresponding simple Lie algebra. It plays an analog roleto the associated variety of primitive ideals of the enveloping algebra of simple Liealgebras. However, associated varieties of affine vertex algebras are not necessarilycontained in the nilpotent cone and it is difficult to describe them in general.

Associated varieties not only capture some of the important properties of vertexalgebras but also have interesting relationship with the Higgs branches of four-dimensional N = 2 superconformal field theories (SCFTs). However, their generaldescription is fairly open, except in a few cases.

It is only quite recently that the study of associated varieties of vertex algebrasand their arc spaces, has been more intensively developed. In this mini-course Iwish to highlight this aspect of the theory of vertex algebras which seems to bevery promising. In particular, I will include open problems on associated varietiesin the setting of affine vertex algebras (vertex algebras associated with Kac-Moody

5

6 INTRODUCTION

algebras) and W-algebras (they are certain vertex algebras attached with nilpo-tent elements of a simple Lie algebra) raised by my recent works with TomoyukiArakawa.

PART 1

Affine Kac-Moody algebras and their vacuumrepresentations

References: [Kac1, Moody-Pianzola].

The goal of this lecture is to introduce the universal vacuum representationV k(g) attached to an affine Kac-Moody algebra g and a complex number k. Wewill see next lecture that it has a natural vertex algebra structure. We also attach toany quotient V k(g) a certain Poisson algebra and a corresponding Poisson variety.

1.1. Affine Kac-Moody algebras

1.1.1. Simple Lie algebras. Let g be a complex simple Lie algebra, that is,the only ideals of g are 0 or g and dim g > 3. Hence g is the Lie algebra of acertain linear algebraic group G, g = Lie(G).

Let ( | )Kil be the Killing form of g,

( | )Kil : g× g→ C, (x, y) 7→ tr(adx ad y).

It is a nondegenerate symmetric bilinear form of g which is G-invariant, that is,

(g.x|g.y)Kil) = (x|y)Kil for all x, y ∈ g, g ∈ G,

or else,

([x, y]|z)Kil = (x|[y, z])Kil for all x, y, z ∈ g.

Since g is semisimple, any other such bilinear form is a nonzero multiple of theKilling form.

Example 1.1. Let g be the Lie algebra sln, n > 2, which is the set of tracelesscomplex n-size square matrices, with bracket [A,B] = AB − BA. The Lie algebrasln is known to be simple and its Killing form is given by

(A,B) 7→ 2n tr(AB).

1.1.2. Dual Coxeter number. Let h be a Cartan subalgebra of g, and let

g = h⊕⊕α∈∆

gα, gα := y ∈ g | [x, y] = α(x)y for all x ∈ h,

be the corresponding root decomposition of (g, h), where ∆ is the root system of(g, h). Let Π = α1, . . . , αr be a basis of ∆, with r the rank of g, and let α∨1 , . . . , α

∨r

be the coroots of α1, . . . , αr respectively. The element α∨i , i = 1, . . . , r, viewed asan element of (h∗)∗ ∼= h, will be often denoted it by hi. Let ∆+ be the set ofpositive roots corresponding to Π, and let

g = n− ⊕ h⊕ n+(1)

7

8 1. AFFINE KAC-MOODY ALGEBRAS AND THEIR VACUUM REPRESENTATIONS

be the corresponding triangular decomposition. Thus n+ =⊕

α∈∆+gα and n− =⊕

α∈∆−gα are both nilpotent Lie subalgebras of g.

Any positive root α ∈ ∆+ can be written as

α =

r∑i=1

niαi, ni ∈ Z≥0.

The height of α is ht(α) =∑ri=1 ni. The Coxeter number of g is

h := ht(θ) + 1,

where θ is the highest positive root of ∆, that is, the unique positive root θ ∈ ∆+

such that θ + αi 6∈ ∆ ∪ 0 for i = 1, . . . , r. Similarly, we define the dual Coxeternumber h∨ of g by:

h∨ = ht(θ∨) + 1.

For example, if g = sln then h = h∨ = n.We define the normalized bilinear form ( | ) on g by:

( | ) =1

2h∨( | )Kil.

Thus with respect to the induced bilinear form on h∗, (θ|θ) = 2.

1.1.3. The loop algebra. Consider the loop algebra of g which is the Liealgebra

Lg := g[t, t−1] = g⊗ C[t, t−1],

with commutation relations

[xtm, ytn] = [x, y]tm+n, x, y ∈ g, m, n ∈ Z,

where xtm stands for x⊗ tm.

Remark 1.2. The Lie algebra Lg is the Lie algebra of polynomial functionsfrom the unit circle to g. This is the reason why it is called the loop algebra.

Definition 1.3. We define a bilinear map ν on Lg by setting:

ν(x⊗ f, y ⊗ g) := (x|y)Rest=0(df

dtg),

for x, y ∈ g and f, g ∈ C[t, t−1], where the linear map Rest=0 : C[t, t−1] → C isdefined by Rest=0(tm) = δm,−1 for m ∈ Z.

The bilinear ν is a 2-cocycle on Lg, that is, for any a, b, c ∈ Lg,

ν(a, b) = −ν(b, a),(2)

ν([a, b], c) + ν([b, c], a) + ν([c, a], b) = 0.(3)

1.1.4. Affine Kac-Moody algebras.

Definition 1.4. We define the affine Kac-Moody algebra g as the vector spaceg := Lg⊕CK, with the commutation relations [K, g] = 0 (so K is a central element),and

[x⊗ f, y ⊗ g] = [x, y]Lg + ν(x⊗ f, y ⊗ g)K, x, y ∈ g, f, g ∈ C[t, t−1],(4)

1.2. THE VACUUM REPRESENTATIONS 9

where [ , ]Lg is the Lie bracket on Lg. In other words the commutation relationsare given by:

[xtm, ytn] = [x, y]tm+n +mδm+n,0(x|y)K,

[K, g] = 0,

for x, y ∈ g and m,n ∈ Z.

Exercise 1.5. Verify that the identifies (2) and (3) are true, and then that theabove commutation relations indeed define a Lie bracket on g.

1.1.5. Triangular decomposition. Recall the triangular decomposition (1)of g, and consider the following subspaces of g:

n+ := (n− ⊕ h)⊗ tC[t]⊕ n+ ⊗ C[t] = n+ + tg[t],

n− := (n+ ⊕ h)⊗ t−1C[t−1]⊕ n− ⊗ C[t−1] = n− + t−1g[t−1],

h := (h⊗ 1)⊕ CK = h + CK.

They are Lie subalgebras of g and we have

g = n− ⊕ h⊕ n+.(5)

1.2. The vacuum representations

Consider the Lie subalgebra g[t] ⊕ CK of g. It is a parabolic subalgebra of g

since it contains the Borel subalgebra h⊕ n+.

1.2.1. The universal vacuum representation. Fix k ∈ C, and considerthe one-dimensional representation Ck g[t] ⊕ CK on which g[t] acts by 0 and Kacts as a multiplication by the scalar k.

We define the universal vacuum representation of level k of g as the represen-tation induced from Ck:

V k(g) = Indgg[t]⊕CKCk = U(g)⊗U(g[t]⊕CK) Ck.(6)

It can be viewed as a generalized Verma module.

1.2.2. Level of the vacuum representation. The representation V k(g) isa highest weight representation of g with highest weight kΛ0, with Λ0 is the highestweight of the basic representation1, and highest weight vector vk, where vk denotesthe image of 1 ⊗ 1 in V k(g). We will often denote by |0〉 the vector vk (the no-tation will be justified next part when we will endow V k(g) with a vertex algebrastructure).

According to the well-known Schur Lemma, any central element of a Lie algebraacts as a scalar on a simple finite dimensional representation L. As the SchurLemma extends to a representation with countable dimension, the result holds forhighest weight g-modules.

Definition 1.6. A representation M is said to be of level k if K acts as kIdon M .

Then V k(g) is by construction of level k.

1that is, the dual of K in h∗ with respect to a basis of h adapted to the decomposition

h = h⊕ CK.


1.2.3. PBW basis and grading. By the Poincare-Birkhoff-Witt Theorem,the direct sum decomposition (as a vector space)

g = (g⊗ t−1C[t−1])⊕ (g[t]⊕ CK)

gives us the isomorphism of vector spaces

U(g) ∼= U(g⊗ t−1C[t−1])⊗ U(g[t]⊕ CK),

whence

V k(g) ∼= U(g⊗ t−1C[t−1]).

Let x1, . . . , xd, d = dim g, be an ordered basis of g. For any x ∈ g and n ∈ Z,set

x(n) := x⊗ tn = xtn ∈ L(g).

Then K,xi(n), i = 1, . . . , d, n ∈ Z forms a basis of g and K,xi(n), i = 1, . . . , d, n ∈Z>0 forms a basis of g[t]⊕ CK. By the PBW Theorem, V k(g) has a PBW basisof monomials of the form

xi1(n1) . . . xim(nm)|0〉,

where n1 6 n2 6 · · · 6 nm < 0, and if nj = nj+1, then ij 6 ij+1.The space V k(g) is naturally graded, V k(g) =

⊕∆∈Z>0

V k(g)∆, where the

grading is defined by

deg xi1(n1) . . . xim(nm)|0〉 = −

m∑i=1

ni, deg |0〉 = 0.

We have V k(g)0 = C|0〉, and we identify g with V k(g)1 via the linear isomorphismdefined by x 7→ xt−1|0〉.

Any graded quotient V of V k(g) (i.e., a quotient by a proper submodule ofV k(g)) is again a highest weight representation of g with highest weight kΛ0, andof level k. In particular, V k(g) has a unique maximal proper graded submodule Nkand so

Lk(g) := V k(g)/Nk

is an irreducible highest weight representation of g with highest weight kΛ0, and oflevel k. Note that, as a g-representation, we have

Lk(g) ∼= L(kΛ0),

where for λ ∈ h∗, L(λ) denotes the highest weight representation of g of highestweight λ.

Note that Lk(g) is of level k too.

1.3. Associated varieties of vacuum representations

Recall that the nilpotent cone of g is the (reduced) subscheme of g∗ associatedwith the augmentation ideal C[g∗]G+ of the ring of invariants C[g∗]G.

We will see that V k(g) plays the analogue of the enveloping algebra of g forthe representation theory. Because of this, it would be nice to have analogs of theassociated varieties of primitive ideals in this context. Unfortunately, one cannotexpect exactly the same theory. One of the main reason, is that the center of U(g)is trivial (unless for the critical level k = −h∨), and so we do not have analog ofthe nilpotent cone (for the critical level, the analog is played by the arc space of

1.3. ASSOCIATED VARIETIES OF VACUUM REPRESENTATIONS 11

the nilpotent cone). However, what is very interesting with primitive ideals is thattheir associated variety in contained in the nilpotent cone (cf. §1.3.3).

To encounter this problem, we consider the associated variety of a certain Pois-son algebra, this is ous next purpose.

1.3.1. Poisson algebras. Recall that C[g∗] has naturally a Poisson structureinduced from the Kirillov-Kostant-Souriau Poisson structure on g∗. Namely, forf, g ∈ C[g∗], x ∈ g∗,

f, g(x) = 〈x, [dxf, dxg]〉,where dxf, dxg are the differentials of f, g at x ∈ g∗. They are elements of (g∗)∗ ∼= g.In particular, for f, g ∈ g = (g∗)∗ ⊂ C[g∗], f, g = [f, g].

Set

RV k(g) = V k(g)/t−2g[t−1]V k(g).

We an algebra isomorphism

C[g∗]'−→ RV k(g) = V k(g)/t−2g[t−1]V k(g)

x1 . . . xm 7−→ (x1t−1) . . . (xmt

−1)|0〉+ t−2g[t−1]V k(g) (xi ∈ g).(7)

Thus RV k(g) inherits a Poisson structure given by

a(−1)|0〉, b(−1)|0〉 = a(0)b(−1)|0〉 = [a, b](−1)|0〉, a, b ∈ g.

Identifying V k(g)1 with g, it gives

a, b = a(0)b, a, b ∈ g = V k(g)1∼= g

since in V k(g),

a(0)b(−1)|0〉 = b(−1)a(0)|0〉+ [a, b](−1)|0〉 = 0 + [a, b](−1)|0〉.

More generally, if V is a graded quotient of V k(g), then one can set

RV = V/t−2g[t−1]V,

and we get a surjective morphism of Poisson algebras,

C[g∗] −→ RV = V/t−2g[t−1]V

x1 . . . xr 7−→ x1,(−1) . . . xr,(−r)|0〉+ t−2g[t−1]V (xi ∈ g),(8)

the Poisson algebra structure on RV being defined as before. This map is surjectivebut not an isomorphism in general.

1.3.2. Associated varieties. Continue to assume that V is a graded quotientof V k(g).

Definition 1.7. We define the associated variety XV of V to be the zero locusin g∗ of the kernel of the above map (8).

The Poisson variety XV is then a G-invariant, Poisson, and conic subvarietyof g∗. We obviously have XV k(g) = g∗ since the map (8) is an isomorphism for

V = V k(g).For V = Lk(g), XV can be viewed as an analog of the associated variety of

primitive ideals. However, we will see that there are substantial differences.


1.3.3. Digression on primitive ideal of the enveloping algebra of g.Let I be a two-sided ideal of U(g). The PBW filtration on U(g) induces a filtrationon I, so that gr I becomes a graded Poisson ideal in C[g∗]. Thus, U(g)/I is a

quantization of the Poisson G-scheme V (gr I) = SpecC[g∗]/gr I.The variety

V (gr I) = SpecmC[g∗]/gr I = (V (gr I))red ⊂ g∗

is usually referred to as the associated variety of I.A proper two-sided ideal I of U(g) is called primitive if it is the annihilator of

a simple left U(g)-module. There are two important results on primitive ideals ofU(g). The first result is the Duflo Theorem [Duflo77], stating that any primitiveideal in U(g) is the annihilator AnnU(g)Lg(λ) of some simple highest weight moduleLg(λ), λ ∈ h∗, of g.

The second result is the Irreducibility Theorem. Identifying g∗ with g through( | ), we shall often view associated varieties of ideals of U(g) as subsets of g. TheIrreducibility Theorem says that the associated variety V (gr I) of a primitive idealI in U(g) is irreducible, specifically, it is the closure O of some nilpotent orbit Oin g. The latter theorem was first partially proved (by a case-by-case argument) in[Borho-Brylinski82], and in a more conceptual way in [Kashiwara-Tanisaki84]and [Joseph85] (independently), using many earlier deep results due to Joseph,Gabber, Lusztig, Vogan and others.

It is possible that different primitive ideals share the same associated variety.In addition, not all nilpotent orbit closures appear as associated variety of someprimitive ideal of U(g).

1.3.4. Integrable representations. First of all, since Lk(g) ∼= V k(g) fork 6∈ Q (cf. [KK79]), we see that XV k(g) is not always contained in the nilpotentcone N .

One knows that Lg(λ) is finite-dimensional if and only if V (gr AnnU(g)Lg(λ)) =

0, where Lg(λ) denotes the irreducible highest weight representation of g, withhighest weight λ ∈ h∗.

Contrary to irreducible highest weight representations of g, the representation

L(λ), λ ∈ h∗, is finite dimensional if and only if λ = 0, that is, L(λ) is the trivialrepresentation.

The notion of finite dimensional representations has to be replaced by the notionof integrable representations in the category O.

We define the category O for g in the similar way that for g, except that we donot require that the object are finitely generated by g (cf. [Moody-Pianzola]).

Definition 1.8. A representation M of g is said to be integrable if

(1) M is h-diagonalizable,

(2) for λ ∈ h∗, Mλ is finite dimensional,(3) for i = 0, . . . , r, Ei and Fi act locally nilpotently on M , where Ei, Hi, Fi

are Chevalley generators2 of g.

Remark 1.9. As a ai-module, i = 0, . . . , r, an integrable representation M

decomposes into a direct sum of finite dimensional irreducible h-invariant modules,

2Namely, Ei = ei ⊗ 1, Fi = fi ⊗ 1, Hi = fi ⊗ 1, for i = 1, . . . , r, with ei, fi, hi Chevalleygenerators of g, and E0 = e0 ⊗ t, F0 = f0 ⊗ t−1, with f0 ∈ gθ, e0 ∈ g−θ such that (f0|e0) = 1.

1.3. ASSOCIATED VARIETIES OF VACUUM REPRESENTATIONS 13

where ai ∼= sl2 is the Lie algebra generated by the Chevalley generators Ei, Fi, Hi.Hence the action of ai on M can be “integrated” to the action of the group SL2(C).

The character of the simple integrable representations in the category O satisfyremarkable combinatorial identities (related to Macdonald identities).

Recall that the irreducible representation Lg(λ), λ ∈ h∗, is finite-dimensionalif and only if its associated variety V (AnnU(g)(Lg(λ))) is zero.

Theorem 1.10. The associated variety of Lk(g) is 0 if and only if Lk(g) isintegrable as a g-module, which is true if and only if k ∈ Z>0.

The last part of the statement is well-known.

Lemma 1.11. Let (R, ∂) be a differential algebra over Q, I a differential ideal

of R, i.e., I is an ideal of R such that ∂I ⊂ I. Then ∂√I ⊂√I.

Proof. Let a ∈√I, so that am ∈ I for some m ∈ Z>0. Since I is ∂-invariant,

we have ∂mam ∈ I. But

∂mam ≡ m!(∂a)m (mod√I).

Hence (∂a)m ∈√I, and therefore, ∂a ∈

√I.

Recall that a singular vector of g of a g-module M is a vector v ∈ M suchthat n+.v = 0, that is, ei.v = 0 for i = 1, . . . , r. A singular vector of g of ag-representation M is a vector v ∈ M such that n+.v = 0, that is, ei.v = 0 fori = 1, . . . , r, and (fθt).v = 0. In particular, regarding V k(g) as a g-representation,a vector v ∈ V k(g) is singular if and only if n+.v = 0.

Proof of the “if” part of Theorem 1.10. Suppose that Lk(g) is integrable.This condition is equivalent to that k ∈ Z>0 and the maximal submodule Nk(g)of V k(g) is generated by the singular vector (eθt

−1)k+1|0〉 ([Kac1]). The exactsequence 0→ Nk(g)→ V k(g)→ Lk(g)→ 0 induces the exact sequence

0→ Ik → RV k(g) → RLk(g) → 0,

where Ik is the image of Nk in RV k(g) = C[g∗], and so, RLk(g) = C[g∗]/Ik. The

image of the singular vector in Ik is given by ek+1θ . Therefore, eθ ∈

√Ik. On the

other hand, by Lemma 1.11,√Ik is preserved by the adjoint action of g. Since g is

simple, g ⊂√Ik. This proves that XLk(g) = 0 as required.

The proof of the “only if” part follows from [Dong-Li-Mason06]. It can alsobe proved using W-algebras.

PART 2

Vertex algebras, definitions, first properties andexamples

References: [Frenkel-BenZvi, Kac2]Vertex algebras were introduced by Borcherds in 1986 [Borcherds86]. They

give the mathematical formalism of two-dimensional conformal field theory (CFT).

2.1. Operator product expansion and definition of vertex algebras

Let V be a vector space over C. We denote by (EndV )[[z, z−1]] the set of allformal Laurent series in the variable z with coefficients in the space EndV . We callelements a(z) of (EndV )[[z, z−1]] a series on V . For a series a(z) on V , we set

a(n) = Resz=0a(z)zn

so that the expansion of a(z) is

a(z) =∑n∈Z

a(n)z−n−1.

The coefficient a(n) is called a Fourier mode of a(z). We write

a(z)b =∑n∈Z

a(n)bz−n−1

for b ∈ V .

Definition 2.1. A series a(z) ∈ (EndV )[[z, z−1]] is called a field on V if forany b ∈ V , a(z)b ∈ V ((z)), that is, for any b ∈ V , a(n)b = 0 for large enough n.

We denote by F (V ) the space of all fields on V .

2.1.1. Definition. A vertex algebra is a vector space V equipped with thefollowing data:

• (the vacuum vector) a vector |0〉 ∈ V ,• (the vertex operators) a linear map

V → F (V ), a 7→ a(z) =∑n∈Z

a(n)z−n−1,

• (the translation operator) a linear map T : V → V .

These data are subject to the following axioms:

• (the vacuum axiom) |0〉(z) = idV . Furthermore, for all a ∈ V ,

a(z)|0〉 ∈ V [[z]]

and limz→0

a(z)|0〉 = a. In other words, a(n)|0〉 = 0 for n > 0 and a(−1)|0〉 =a.

15

16 2. VERTEX ALGEBRAS, DEFINITIONS, FIRST PROPERTIES AND EXAMPLES

• (the translation axiom) for any a ∈ V ,

[T, a(z)] = ∂za(z), (Ta)(z) = ∂za(z)

and T |0〉 = 0.• (the locality axiom) for all a, b ∈ V , (z − w)Na,b [a(z), b(w)] = 0 for someNa,b ∈ Z>0.

When two fields a(z), b(z) on a vector space V verify the condition of the localityaxiom, we say that there are mutually local.

2.1.2. Operator product expansion (OPE).

Proposition 2.2. Let a(z), b(z) be two fields on a vector space V . The follow-ing assertions are equivalent.

(i) (z − w)Na,b [a(z), b(w)] = 0 for some N = Na,b ∈ Z>0.(ii)

[a(z), b(w)] =

Na,b−1∑n=0

(a(n)b)(w)1

n!∂nwδ(z − w),(9)

where δ(z−w) :=∑n∈Z w

nz−n−1 ∈ C[[z, w, z−1, w−1]] is the formal delta-function.

(iii)

a(z)b(w) =

Na,b−1∑n=0

(a(n)b)(w)τz,w(1

(z − w)j+1)+ : a(z)b(w) : ,

and

b(w)a(z) =

Na,b−1∑n=0

(a(n)b)(w)τw,z(1

(z − w)j+1)+ : a(z)b(w) : ,

where : a(z)b(w) : and the maps τz,w and τw,z are defined below.

For a(z), b(z) ∈ F (V ), set

: a(z)b(w) : = a(z)+b(w) + b(w)a(z)−,

where

a(z)+ =∑n<0

a(n)z−n−1, a(z)− =

∑n>0

a(n)z−n−1.

The normally ordered product on a vertex algebra V is defined as : ab := a(−1)b.Thus

: ab : (z) = : a(z)b(z) : .

The normally ordered product is neither commutative nor associative. By defini-tion, : a(z)b(z)c(z) : stands for : a(z) : b(z)c(z) : :.

The two maps τz,w and τw,z are the morphisms of algebras defined by:

τz,w : C[z, w, z−1, w−1,1

z − w]→ C((z))((w)),

1

z − w7→ 1

z

∑n>0

(wz

)n= δ(z − w)−,

τw,z : C[z, w, z−1, w−1,1

z − w]→ C((w))((z)),

1

z − w7→ −1

z

∑n>0

( zw

)n= −δ(z − w)+.

2.1. OPERATOR PRODUCT EXPANSION AND DEFINITION OF VERTEX ALGEBRAS 17

Thus the map τz,w is the expansion of1

z − win |z| > |w| and τw,z is the expansion

of1

z − win |w| > |z|.

Proof of the implication (ii)⇐(i). We verify only the converse implica-tion (see for instance [Frenkel-BenZvi, Chap. 3] for more details).

Let us write

δ(z − w) =1

z

∑n>0

(wz

)n︸︷︷︸

δ−

+1

z

∑n>0

( zw

)n︸︷︷︸

δ+

,

so that when |z| > |w|, the series δ(z−w)− converges to the meromorphic function1

z − wand when |z| < |w|, the series δ(z − w)+ converges to the meromorphic

function − 1

z − w.

We have

δ(z − w) = τz,w(1

z − w)− τw,z(

1

z − w).

Both morphisms τz,w and τw,z commutes with ∂w and ∂z. Therefore,

1

j!∂jwδ(z − w) = τz,w(

1

(z − w)j+1)− τw,z(

1

(z − w)j+1),

whence

(z − w)n+1 1

n!∂nwδ(z − w) = (z − w)n+1

(τz,w(

1

(z − w)n+1)− τw,z(

1

(z − w)n+1)

)= τz,w(1)− τw,z(1) = 0.

The implication (ii)⇒(i) is then clear.

Note that the translation axiom says that

[T, a(n)] = −na(n−1), n ∈ Z,(10)

and together with the vacuum axiom we get that

Ta = a(−2)|0〉.

Exercise 2.3. Verify the above assertion.

Also, the vacuum axiom implies that the map V → End(V ) defined by theformula a 7→ a(−1) is injective. Therefore the map a 7→ a(z) is also injective.

2.1.3. Borcherds identities. A consequence of the definition are the follow-ing relations, called Borcherds identities:

[a(m), b(n)] =∑i>0

(mi

)(a(i)b)(m+n−i),(11)

(a(m)b)(n) =∑j>0

(−1)j(mj

)(a(m−j)b(n+j) − (−1)mb(m+n−j)a(j)),(12)


for m,n ∈ Z. In the above formulas, the notation

(mi

)for i > 0 and m ∈ Z means(

mi

)=m(m− 1)× · · · × (m− i+ 1)

i(i− 1)× · · · × 1.

2.2. First examples of vertex algebras

2.2.1. Commutative vertex algebras. A vertex algebra V is called com-mutative if all vertex operators a(z), a ∈ V , commute each other (i.e., we haveNa,b = 0 in the locality axiom). This condition is equivalent to that

[a(m), b(n)] = 0, ∀a, b ∈ Z, m, n ∈ Z

by (11).Hence if V is a commutative vertex algebra, then a(z) ∈ EndV [[z]] for all a ∈ V .

Then a commutative vertex algebra has a structure of a unital commutative algebrawith the product:

a · b = : ab : = a(−1)b,

where the unit is given by the vacuum vector |0〉. The translation operator T of Vacts on V as a derivation with respect to this product:

T (a · b) = (Ta) · b+ a · (Tb).

Therefore a commutative vertex algebra has the structure of a differential algebra,that is, a unital commutative algebra equipped with a derivation.

Conversely, there is a unique vertex algebra structure on a differential algebraR with derivation ∂ such that:

a(z)b =(ez∂a

)b =

∑n>0

zn

n!(∂na)b,

for all a, b ∈ R. We take the unit as the vacuum vector. This correspondence givesthe following result.

Theorem 2.4 ([Borcherds86]). The category of commutative vertex algebrasis the same as that of differential algebras.

One important example of commutative vertex algebras are obtained by con-sidering the function sheaf over arc spaces of a scheme (see Section 3.1).

2.2.2. Universal affine vertex algebras. Let us consider a slightly moregeneral construction of the vacuum representation considered in Section 1.2. Wewill recover this special case with a = g, and κ = k( | ).

Let a be a Lie algebra endowed with a symmetric invariant bilinear form κ. Let

a = a[t, t−1]⊕C1

be the Kac-Moody affinization of a. It is a Lie algebra with commutation relations

[xtm, ytn] = [x, y]tm+n +mδm+n,0κ(x, y)1, x, y ∈ a, m, n ∈ Z, [1, a] = 0.

Let

V κ(a) = U(a)⊗U(a[t]⊕C1) C,(13)

2.2. FIRST EXAMPLES OF VERTEX ALGEBRAS 19

where C is a one-dimensional representation of a[t] ⊕ C1 on which a[t] acts triv-ially and 1 acts as the identity. By the PBW Theorem, we have the followingisomorphism of vector spaces:

V κ(a) ∼= U(a⊗ t−1C[t−1]).

The space V κ(a) is naturally graded: V κ(a) =⊕

∆∈Z>0V κ(a)∆, where the

grading is defined by setting deg(xtn) = −n, deg |0〉 = 0. Here |0〉 is the image of1⊗ 1 in V κ(a). We have V κ(a)0 = C|0〉. We identify a with V κ(a)1 via the linearisomorphism defined by x 7→ xt−1|0〉.

There is a unique vertex algebra structure on V κ(a), such that |0〉 is the vacuumvector and

x(z) :=∑n∈Z

(xtn)z−n−1, x ∈ a.

(So x(n) = xtn for x ∈ a = V κ(a)1, n ∈ Z.) The vertex algebra V κ(a) is called theuniversal affine vertex algebra associated with (a, κ).

Let us describe the vertex algebra structure in more details. Set

x(n) = xtn, ∀x ∈ a, n ∈ Z,

and let |0〉 be the image of 1⊗ 1 ∈ U(a)⊗C in V κ(a). Let (xi ; i = 1 . . . ,dim a) bean ordered basis of a. By the PBW Theorem, V κ(a) has a basis of the form

xi1(n1) . . . xim(nm)|0〉,

where n1 6 n2 6 · · · 6 nm < 0, and if nj = nj+1, then ij 6 ij+1.Then (V κ(a), |0〉, T, Y ) is a vertex algebra where the translation operator T is

given byT |0〉 = 0, [T, xi(n)] = −nxi(n−1),

for n ∈ Z, and the vertex operators are defined inductively by:

|0〉(z) = IdV κ(a), xi(−1)|0〉(z) = xi(z) =∑n∈Z

xi(n)z−n−1,

(xi1(n1) . . . xim(nm)|0〉)(z)

=1

(−n1 − 1)! . . . (−nm − 1)!: ∂−n1−1

z xi1(z) . . . ∂−nm−1z xim(z) :

Theorem 2.5. V κ(a) is a Z>0-graded vertex algebra.

Proof. We check only the locality axiom.It is enough to check the locality on generator fields by Dong’s lemma, which

says that if a(z), b(z), c(z) are three mutually local fields on a vector space V , thenthe fields : a(z)b(z) : and c(z) are also mutually local.

Let i, j ∈ A, . . . , d. Then

[xi(z), xj(w)] =∑n,m∈Z

[xi(n), xj(m)]z

−n−1w−m−1

=∑n,m∈Z

[xi, xj ](n+m)z−n−1w−m−1 +

∑n∈Z

nκ(xi, xj)z−n−1w−n−1

=∑l∈Z

[xi, xj ](l)

(∑n∈Z

z−n−1wn

)w−l−1 + κ(xi, xj)

∑n∈Z

nz−n−1wn−1

= [xi, xj ](w)δ(z − w) + κ(xi, xj)∂wδ(z − w).


Then it follows that for any i, j,

(z − w)2[xi(z), xj(w)] = 0,

so the locality axiom holds for these fields.

Remark 2.6. (1) In fact, the equality

[xi(z), xj(w)] = [xi, xj ](w)δ(z − w) + κ(xi, xj)∂wδ(z − w)

is equivalent to the commutation relations in the Lie algebra a.(2) Using the commutation relations, one can check directly on this example

the OPE formula with N = 2:

[xi(z), xj(w)] =

N−1∑n=0

(xi(n)xj)(w)

1

n!∂nwδ(z − w).

When a = g is the simple Lie algebra as in Part 1, so that a = g is the affineKac-Moody algebra, and

κ = k( | ) =k

2h∨× ( | )Kil, for k ∈ C,

with κg the Killing form of g, then we write V k(g) for the universal affine vertexalgebra vertex algebra V κ(a). By what foregoes, V k(g) has a natural vertex algebrastructure, and it is called the universal affine vertex algebra associated with g oflevel k.

When a ∼= C is one-dimensional with κ any non-degenerate bilinear form, thenV κ(a) is the Heisenberg vertex algebra.

2.2.3. The Virasoro vertex algebra. Let V ir = C((t))∂t ⊕ CC be theVirasoro Lie algebra, with the commutation relations

[Ln, Lm] = (n−m)Ln+m +n3 − n

12δn+m,0C,(14)

[C, V ir] = 0,(15)

where Ln := −tn+1∂t for n ∈ Z.Let c ∈ C and define the induced representation

Virc = IndV irC[[t]]∂t⊕CCCc = U(V ir)⊗C[[t]]∂t⊕CC Cc,

where C acts as multiplication by c and C[[t]]∂t acts by 0 on the 1-dimensionalmodule Cc.

By the PBW Theorem, Virc has a basis of the form

Lj1 . . . Ljm |0〉, j1 6 · · · 6 jl 6 −2,

where |0〉 is the image of 1 ⊗ 1 in Virc. Then (Virc, |0〉, T, Y ) is a vertex algebra,called the universal Virasoro vertex algebra with central charge c, such that T = L−1

and:

Y (|0〉, z) = IdVircc , Y (L−2|0〉, z) =: L(z) =∑n∈Z

Lnz−n−2,

Y (Lj1 . . . Ljm |0〉, z)

=1

(−j1 − 2)! . . . (−jm − 2)!: ∂−j1−2

z L(z) . . . ∂−jm−2z L(z) :

2.2. FIRST EXAMPLES OF VERTEX ALGEBRAS 21

Moreover, Virc is Z>0-graded by deg |0〉 = 0 and degLn|0〉 = −n.

2.2.4. Conformal vertex algebras. A Hamiltonian of V is a semisimpleoperator H on V satisfying

[H, a(n)] = −(n+ 1)a(n) + (Ha)(n)

for all a ∈ V , n ∈ Z.

Definition 2.7. A vertex algebra equipped with a Hamiltonian H is calledgraded. Let V∆ = a ∈ V | Ha = ∆a for ∆ ∈ C, so that V =

⊕∆∈C V∆. For

a ∈ V∆, ∆ is called the conformal weight of a and it is denoted by ∆a. We have

a(n)b ∈ V∆a+∆b−n−1(16)

for homogeneous elements a, b ∈ V .

For example, the universal affine vertex algebra V k(g) is Z>0-graded (that is,V k(g)∆ = 0 for ∆ 6∈ Z>0) and the Hamiltonian H is defined letting H acts onV k(g)∆ as ∆IdV k(g)∆

for any ∆ ∈ Z>0.

Definition 2.8. A graded vertex algebra V =⊕

∆∈C V∆ is called conformalof central charge c ∈ C if there is a conformal vector ω ∈ V2 such that the Fouriercoefficients Ln of the corresponding vertex operators

Y (ω, z) =∑n∈Z

Lnz−n−2

satisfy the defining relations (14) of the Virasoro algebra with central charge c, andif in addition we have

ω(0) = L−1 = T,

ω(1) = L0 = H i.e., L0|V∆= ∆IdV∆

∀∆ ∈ Z.

A Z-graded conformal vertex algebra is also called a vertex operator algebra.

Example 2.9. The Virasoro vertex algebra Virc is clearly conformal with cen-tral charge c and conformal vector ω = L−2|0〉.

Example 2.10. The universal affine vertex algebra V k(g) has a natural con-formal vector, provided that k 6= −h∨. Set

S =1

2

dim g∑i=1

xi,(−1)xi(−1)|0〉,

where (xi; i = 1, . . . ,dim g) is the dual basis of (xi; i = 1, . . . ,dim g) with respectto the bilinear form ( | ), and

xi(z) =∑n∈Z

xi(n)z−n−1, xi(z) =

∑n∈Z

xi,(n)z−n−1.

Exercise 2.11. For k 6= −h∨, show that L =S

k + h∨is a conformal vector of

V k(g), called the Segal-Sugawara vector, with central charge

c(k) =k dim g

k + h∨.

(cf. [Frenkel-BenZvi, §3.4.8]).


Then we have

[Lm, x(n)] = (m− n)x(m+n) x ∈ g, m.n ∈ Z.(17)

2.3. Modules over vertex algebras

2.3.1. Definition. A module over the vertex algebra V is a vector space Mtogether with a linear map

V → F (M), a 7→ aM (z) =∑n∈Z

aM(n)z−n−1,

which satisfies the following axioms:

|0〉(z) = IdM ,(18)

(Ta)M (z) = ∂zaM (z),(19) ∑

j60

(mj

)(a(n+j)b)

M(m+k−j)(20)

=∑j>0

(−1)j(nj

)(aM(m+n−j)b

M(k+j) − (−1)nbM(n+k−j)a

M(m+j)).

Notice that (20) is equivalent to (11) and (12) for M = V .Suppose in addition V is graded (cf. Definition 2.7). A V -module M is called

graded if there is a compatible semisimple action ofH onM , that is, M =⊕

d∈CMd,

where Md = m ∈ M | Hm = dm and [H, aM(n)] = −(n + 1)aM(n) + (Ha)M(n) for all

a ∈ V . We have

aM(n)Md ⊂Md+∆a−n−1(21)

for homogeneous a ∈ V .When there is no ambiguity, we will often denote by a(n) the element aM(n) of

End(M).The axioms imply that V is a module over itself (called the adjoint module).

We have naturally the notions of submodules, quotient module and vertex ideals. Amodule whose only submodules are 0 and itself is called simple.

2.3.2. Modules of the universal affine vertex algebra. In the case thatV is the universal affine vertex algebra V k(g) associated with g at level k ∈ C,V -modules play a crucial important role in the representation theory of the affineKac-Moody algebra g.

A g-module M of level k is called smooth if x(z) is a field on M for x ∈ g, thatis, for any m ∈M there is N > 0 such that (xtn)m = 0 for all x ∈ g and n > N .

Any V k(g)-module M is naturally a smooth g-module of level k. Conversely,any smooth g-module of level k can be regarded as a V k(g)-module. It follows thata V k(g)-module is the same as a smooth g-module of level k.

Namely, we have the following.

Proposition 2.12 (See [Frenkel-BenZvi, §5.1.18] for a proof). There is anequivalence of category between the category of V k(g)-modules and the category ofsmooth g-modules of level k.

PART 3

Poisson vertex algebras, arc spaces, and associatedvarieties

3.1. Jet schemes and arc spaces

In this section, we present some general facts on jet schemes and arc spaces.Our main references on the topic are [Mustata01, Ein-Mustata09, Ishii11].

3.1.1. Definitions. Denote by Sch the category of schemes of finite type overC. Let X be an object of this category, and m ∈ Z>0.

Definition 3.1. An m-jet of X is a morphism

SpecC[t]/(tm+1) −→ X.

The set of all m-jets of X carries the structure of a scheme Jm(X), called the m-thjet scheme of X. It is a scheme of finite type over C characterized by the followingfunctorial property: for every scheme Z over C, we have

HomSch(Z, Jm(X)) = HomSch(Z ×SpecC SpecC[t]/(tm+1), X).

The C-points of Jm(X) are thus the C[t]/(tm+1)-points of X. From Defini-tion 3.1, we have for example that J0(X) ' X and that J1(X) ' TX where TXdenotes the total tangent bundle of X.

The canonical projection C[t]/(tm+1)→ C[t]/(tn+1), m > n, induces a trunca-tion morphism πX,m,n : Jm(X)→ Jn(X). The canonical injection C → C[t]/(tm+1)induces a morphism ιX,m : X → Jm(X), and we have πX,m ιX,m,0 = idX . HenceιX,m is injective and πX,m,0 is surjective.

Define the (formal) disc as

D := SpecC[[t]].

The projections πX,m,n yield a projective system Jm(X), πX,m,nm>n of schemes.

Definition 3.2. Denote by J∞(X) its projective limit in the category ofschemes,

J∞(X) = lim←− Jm(X).

It is called the arc space, or the infinite jet scheme of X.

Thus elements of J∞(X) are the morphisms

γ : D → C[[t]],

and for every scheme Z over C,

HomSch(Z, Jm(X)) = HomSch(Z×SpecCD,X),

23

24 3. POISSON VERTEX ALGEBRAS, ARC SPACES, AND ASSOCIATED VARIETIES

where Z×D is the completion of Z×D with respect to the subscheme Z×0. Inother words, the contravariant functor

Sch→ Set, Z 7→ HomSch(Z×D,X)

is represented by the scheme J∞(X). The reason why we need the completion Z×Din the definition is that, for A an algebra, A⊗C[[t]] $ A[[t]] = A⊗C[[t]] in general.

We denote by πX,∞ the canonical projection:

πX,∞ : J∞(X)→ X.

3.1.2. The affine case. In the case where X is affine, we have the followingexplicit description of J∞(X). (We describe similarly Jm(X).)

Let N ∈ Z>0 and X ⊂ CN be an affine subscheme defined by an ideal I =〈f1, . . . , fr〉 of C[x1, . . . , xN ]. Thus

X = Spec C[x1, . . . , xN ]/I.

For f ∈ C[x1, . . . , xN ], we extend f as a map from C[[t]]N to C[[t]] via base ex-tension. Then giving a morphism γ : D → X is equivalent to giving a morphismγ∗ : C[x1, . . . , xN ]/I → C[[t]], or to giving

γ∗(xi) =∑j>0

γi(−j−1)tj , i = 1, . . . , N,

such that for any k = 1, . . . , r,

fk(γ∗(x1), . . . , γ∗(xN )) = 0 in C[[t]].

For any f ∈ C[x1, . . . , xN ], there exist functions f (j), j > 0, which only depend onf , in the variables γ = (γ1

(−j−1), . . . , γN(−j−1))j>0 such that

f(γ∗(x1), . . . , γ∗(xN )

)=∑j>0

f (j)

j!(γ) tj .(22)

Regarding the coordinates xi as functions over CN , we set:

xi(−j−1) := (xi)(j), that is, xi(−j−1)(γ) = j!γi(−j−1),

for i = 1, . . . , N .The jet scheme J∞(X) is then the closed subscheme in SpecC[xi(−j−1) ; i =

1, . . . , N, j > 0] defined by the ideal generated by the polynomials f(j)k , for k =

1, . . . , r and j > 0, that is,

J∞(X) ∼= SpecC[xi(−j−1) ; i = 1, . . . , N, j > 0]/〈f (j)k ; k = 1, . . . , r, j > 0〉.

In particular, if X is an N -dimensional vector space, then

J∞(X) ∼= SpecC[xi(−j−1) ; i = 1, . . . , N, j > 0],

and for m ∈ Z>0, the projection J∞(X) → Jm(X) corresponds to the projectiononto the first (m+ 1)N coordinates.

One can also define the functions f(j)k using a derivation.

Lemma 3.3. Define the derivation T of C[xi(−j−1) ; i = 1, . . . , N, j > 0] by

Txi(−j) = jxi(−j−1), j > 0.

Then f(j)k = T jfk for k = 1, . . . , r and j > 0. Here we identify xi with xi(−1).

3.1. JET SCHEMES AND ARC SPACES 25

With the above lemma, we conclude that for the affine scheme X = SpecR,with R = C[x1, x2, · · · , xN ]/〈f1, f2, · · · , fr〉, its arc space J∞X is the affine schemeSpec(J∞(R)), where

J∞(R) :=C[xi(−j) ; i = 1, 2, · · · , N, j > 0]

〈T jfi ; i = 1, . . . , r, j > 0〉and T is as defined in the lemma.

The derivation T acts on the above quotient ring J∞(R). Hence for an affinescheme X = SpecR, the coordinate ring J∞(R) = C[J∞(X)] of its arc space J∞(X)is a differential algebra, hence is a commutative vertex algebra by Theorem 2.4.

Remark 3.4. The differential algebra (J∞(R), T ) is universal in the followingsense. We have a C-algebra homomorphism j : R → J∞(R) such that if (A, ∂) isanother differential algebra, and if f : R → A is a C-algebra homomorphism, thenthere is a unique differential algebra homomorphism h : J∞(R) → A making thefollowing diagram commutative.

Rj

//

f!!

(J∞(R), T )

hyy

(A, ∂)

(The map h is a differential algebra homomorphism means that it is a C-algebrahomomorphism such that ∂(h(u)) = h(T (u)) for all u ∈ J∞(R).)

Lemma 3.5 ([Ein-Mustata09]). Let m ∈ Z>0 ∪ ∞. Then for every open

subset U of X, Jm(U) = π−1X,m(U).

Then for a general scheme Y of finite type with an affine open covering Uii∈I ,its arc space J∞(Y ) is obtained by glueing J∞(Ui) (see [Ein-Mustata09, Ishii11]).In particular, the structure sheaf OJ∞(Y ) is a sheaf of commutative vertex algebras.

The natural projection πX,∞ : J∞(X)→ X corresponds to the embedding R →J∞(R), xi → xi(−1) in the case where X = SpecR is affine. In terms of arcs,

πX,∞(α) = α(0) for α ∈ HomSch(D,X), where 0 is the unique closed point of theformal disc D.

3.1.3. Basic properties. The map from a scheme to its jet schemes and arcspace is functorial. If f : X → Y is a morphism of schemes, then we naturally obtaina morphism Jmf : Jm(X)→ Jm(Y ) making the following diagram commutative,

Jm(X)Jmf //

πX,m,0

Jm(Y )

πY,m,0

Xf

// Y

In terms of arcs, it means that Jmf(α) = f α for α ∈ Jm(X). This also holds form =∞.

We have also the following for m ∈ Z>0 ∪ ∞ and for every schemes X,Y ,

Jm(X × Y ) ∼= Jm(X)× Jm(Y ).(23)


Indeed, for any scheme Z in Sch,

Hom(Z, Jm(X × Y )) = Hom(Z ×SpecC C[t]/(tm+1), X × Y )

∼= Hom(Z ×SpecC C[t]/(tm+1), X)×Hom(Z ×SpecC C[t]/(tm+1), Y )

= Hom(Z, Jm(X))×Hom(Z, Jm(Y ))

∼= Hom(Z, Jm(X)× Jm(Y )).

For m = ∞, just replace C[t]/(tm+1) with C[[t]] and take the completion in theproduct Z×SpecC[[t]] = Z×D.

If A is a group scheme over C, then Jm(A) is also a group scheme over C.Moreover, by (23), if A acts on X, then Jm(A) acts on Jm(X).

Example 3.6. Consider the algebra

g∞ := g[[t]] = g⊗C C[[t]] ∼= J∞(g).

It is naturally a Lie algebra, with Lie bracket:

[xtm, ytn)] = [x, y]tm+n, x, y ∈ g, m, n ∈ Z>0.

The arc space J∞(G) of the algebraic group G is naturally a proalgebraic group.Regarding J∞(G) as the set of C[[t]]-points of G, we have J∞(G) = G[[t]]. As Liealgebras, we have

g∞ ∼= Lie(J∞(G)).

The adjoint action of G on g induces an action of J∞(G) on g∞, and the coadjointaction of G on g∗ induces an action of J∞(G) on J∞(g∗), and so on C[J∞(g∗)].

We refer to [Mustata01, Appendix] for the following result.

Lemma 3.7. For f ∈ C[g]G, the polynomials T jf = f (j), j > 0, are elementsof C[g∞]J∞(G). In particular, the arc space J∞(N ) of the nilpotent cone is thesubscheme of g∞ defined by the equations T jPi, i = 1 . . . , r and j > 0, if P1, . . . , Prare homogeneous generators of C[g]G, that is,

J∞(N ) = SpecC[g∞]/(T jPi ; i = 1 . . . , r, j > 0).

3.1.4. Geometrical results. So far, we have stated basic properties commonfor both jet schemes Jm(X) and the arc space J∞(X) . For the geometry, arc spacesbehave rather differently. The main reason is that C[[t]] is a domain, contrary toC[t]/(tm+1). Thereby the geometry of are spaces is somehow simpler.

However, although Jm(X) is of finite type if X is, this is not anymore true forJ∞(X), and its coordinate ring C[J∞(X)] is not noetherian in general.

Lemma 3.8. Denote by Xred the reduced scheme of X. The natural morphism

Xred → X induces an isomorphism J∞Xred'−→ (J∞X)red of topological spaces.

Proof. We may assume that X = SpecR. An arc α of X corresponds to a ringhomomorphism α∗ : R → C[[t]]. Since C[[t]] is an integral domain, it decomposes

as α∗ : R→ R/√

0→ C[[t]]. Thus, α is an arc of Xred.

Similarly, if X = X1 ∪ . . . ∪Xr, where all Xi are closed in X, then

J∞(X) = J∞(X1) ∪ . . . ∪ J∞(Xr).

(Note that Lemma 3.8 is false for the schemes Jm(X).)If X is a point, then J∞(X) is also a point, since Hom(D,X) = Hom(C,C[[t]])

consists of only one element. Thus, Lemma 3.8 implies the following.

3.2. POISSON VERTEX ALGEBRAS 27

Corollary 3.9. If X is zero-dimensional then J∞(X) is also zero-dimensional.

Theorem 3.10 ([Kolchin73]). The scheme J∞(X) is irreducible if X is irre-ducible.

Theorem 3.10 is false for the jet schemes Jm(X): see for instance [Moreau-Yu16]for counter-examples in the setting of nilpotent orbit closures. We refer to loc. cit.,and reference therein, for more about existing relations between the geometry ofthe jet schemes Jm(X), m ∈ Z>0, and the singularities of X.

3.2. Poisson vertex algebras

Let V be a commutative vertex algebra (cf. §2.2.1), or equivalently, a differentialalgebra. Recall that this means: a(n) = 0 in End(V ) for all n > 0.

3.2.1. Definition. A commutative vertex algebra V is called a Poisson vertexalgebra if it is also equipped with a linear operation,

V → Hom(V, z−1V [z−1]), a 7→ a−(z),

such that

(Ta)(n) = −na(n−1),(24)

a(n)b =∑j>0

(−1)n+j+1 1

j!T j(b(n+j)a),(25)

[a(m), b(n)] =∑j>0

(mj

)(a(j)b)(m+n−j),(26)

a(n)(b · c) = (a(n)b) · c+ b · (a(n)c)(27)

for a, b, c ∈ V and n,m > 0. Here, by abuse of notations, we have set

a−(z) =∑n>0

a(n)z−n−1

so that the a(n), n > 0, are “new” operators, the “old” ones given by the field a(z)being zero for n > 0 since V is commutative.

The equation (27) says that a(n), n > 0, is a derivation of the ring V . (Donot confuse a(n) ∈ Der(V ), n > 0, with the multiplication a(n) as a vertex algebra,which should be zero for a commutative vertex algebra.)

3.2.2. Poisson vertex structure on arc spaces.

Theorem 3.11 ([Arakawa12, Proposition 2.3.1]). Let X be an affine Poissonscheme, that is, X = SpecR for some Poisson algebra R. Then there is a uniquePoisson vertex algebra structure on J∞(R) = C[J∞(X)] such that

a(n)b =

a, b if n = 0

0 if n > 0,

for a, b ∈ R.

Proof. The uniqueness is clear by (10) since J∞(R) is generated by R as adifferential algebra. We leave it to the reader to check the well-definedness. SinceJ∞(R) is generated by R, the formula a(n)b = δn,0a, b for a, b ∈ R is sufficient todefine the fields on J∞(R) by formulas (24), whence the existence.


Remark 3.12. More generally, let X be a Poisson scheme which is not nec-essarily affine. Then the structure sheaf OJ∞(X) carries a unique Poisson vertexalgebra structure such that

f(n)g = δn,0f, gfor f, g ∈ OX ⊂ OJ∞(X), see [Arakawa-Kuwabara-Malikov, Lemma 2.1.3.1].

Example 3.13. Recall that the affine space g∗ is a Poisson variety by theKirillov-Kostant-Souriau Poisson structure. If x1, . . . , xN is a basis of g, then

C[g∗] = C[x1, . . . , xN ].

Thus

J∞(g∗) = SpecC[xi(−n) ; i = 1, . . . , N, n > 1].(28)

So we may identify C[J∞(g∗)] with the symmetric algebra S(g[t−1]t−1) via

x(−n) 7−→ xt−n, x ∈ g, n > 1.

For x ∈ g, identify x with x(−1)|0〉 = (xt−1)|0〉, where we denote by |0〉 the unit

element in S(g[t−1]t−1). Then (26) gives that

(29) [x(m), y(n)] = (x(0)y)m+n = x, y(m+n) = [x, y](m+n),

for x, y ∈ g ∼= (g∗)∗ ⊂ C[g∗] ⊂ C[J∞(g∗)] and m,n ∈ Z≥0. So the Lie algebraJ∞(g) = g[[t]] acts on C[J∞(g∗)]. This action coincides with that obtained bydifferentiating the action of J∞(G) = G[[t]] on J∞(g∗) induced by the coadjointaction of G (see Example 3.6). In other words, the Poisson vertex algebra structureof C[J∞(g∗)] comes from the J∞(G)-action on J∞(g∗).

3.2.3. Canonical filtration and Poisson vertex structure. Our secondbasic example of Poisson vertex algebras comes from the graded vertex algebraassociated with the canonical filtration, that is, the Li filtration.

Haisheng Li [Li05] has shown that every vertex algebra is canonically filtered:For a vertex algebra V , let F pV be the subspace of V spanned by the elements

a1(−n1−1)a

2(−n2−1) · · · a

r(−nr−1)|0〉

with a1, a2, · · · , ar ∈ V , ni > 0, n1 + n2 + · · ·+ nr > p. Then

V = F 0V ⊃ F 1V ⊃ . . . .

It is clear that TF pV ⊂ F p+1V .Set

(F pV )(n)FqV := spanCa(n)b ; a ∈ F pV, b ∈ F qV .

Note that F 1V = spanCa(−2)b | a, b ∈ V = C2(V ).

Lemma 3.14. We have

F pV =∑j>0

(F 0V )(−j−1)Fp−jV.

Proposition 3.15. (1) (F pV )(n)(FqV ) ⊂ F p+q−n−1V . Moreover, if n >

0, we have (F pV )(n)(FqV ) ⊂ F p+q−nV . Here we have set F pV = V for

p < 0.(2) The filtration F •V is separated, that is,

⋂p>0 F

pV = 0, if V is apositive energy representation, i.e., positively graded over itself.

3.3. ASSOCIATED VARIETY OF A VERTEX ALGEBRA 29

Exercise 3.16. The verifications are straightforward and are left as an exercise.

Definition 3.17. A vertex algebra V is called finitely strongly generated ifthere exist finitely many elements a1, . . . , ar in V such that V is spanned by theelements of the form

ai1(−n1) . . . ais(−ns)|0〉

with s > 0, ni > 1.

For example, the universal affine vertex algebra and the Virasoro vertex algebraare strongly finitely generated.

In this note we always assume that a vertex algebra V is finitely strongly gen-erated. We will assume that V is conformal and positively graded, V =

⊕n∈Z>0Vn

,

so that the filtration F •V is separated.

Set

grFV =⊕p>0

F pV/F p+1V.

We denote by σp : F pV 7→ F pV/F p+1V , for p > 0, the canonical quotient map.When the filtration F is obvious, we often denote simply by grV the space grF V .

Proposition 3.18 ([Li05]). The space grFV is a Poisson vertex algebra by

σp(a) · σq(b) := σp+q(a(−1)b),

Tσp(a) := σp+1(Ta),

σp(a)(n)σq(b) := σp+q−n(a(n)b),

for a ∈ F pV \ F p−nV , b ∈ F qV , n > 0.

Set

RV := F 0V/F 1V = V/C2(V ) ⊂ grV.

Definition 3.19. The algebra RV is called the Zhu’s C2-algebra of V . Thealgebra structure is given by:

a · b := a(−1)b,(30)

where a = σ0(a).

Proposition 3.20 ([Zhu96, Li05]). The restriction of the vertex Poissonstructure on grFV gives to the Zhu’s C2-algebra RV a Poisson algebra structure,that is, RV is a Poisson algebra by

a · b := a(−1)b, a, b = a(0)b,

where a = σ0(a).

Proof. It is straightforward from Proposition 3.18.

3.3. Associated variety of a vertex algebra

We now in a position to define the main object of study of the lecture.


3.3.1. Associated variety and singular support.

Definition 3.21. Define the associated scheme XV and the associated varietyXV of a vertex algebra V as

XV := SpecRV , XV := SpecmRV = (XV )red.

It was shown in [Li05, Lemma 4.2] that grFV is generated by the subring RVas a differential algebra. Thus, we have a surjection J∞(RV )→ grFV of differentialalgebras by Remark 3.4 since RV generates J∞(RV ) as a differential algebra either.

This is in fact a homomorphism of Poisson vertex algebras.

Theorem 3.22 ([Li05, Lemma 4.2], [Arakawa12, Proposition 2.5.1]). Theidentity map RV → RV induces a surjective Poisson vertex algebra homomorphism

J∞(RV ) = C[J∞(XV )] grFV.

Definition 3.23. Define the singular support of a vertex algebra V as

SS(V ) := Spec(grFV ) ⊂ J∞(XV ).

Theorem 3.24. We have dimSS(V ) = 0 if and only if dimXV = 0.

Proof. The “only if” part is obvious since πXV ,∞(SS(V )) = XV . The “if”part follows from Corollary 3.9.

3.3.2. The lisse condition. A vertex algebra V is called lisse (or C2-cofinite)if RV = V/C2(V ) is finite dimensional.

Thus by Theorem 3.24 we get:

Lemma 3.25. The vertex algebra V is lisse if and only if dimXV = 0, that is,if and only if dimSS(V ) = 0.

Remark 3.26. Suppose that V is Z>0-graded by some Hamiltonian H, i.e.,V =

⊕i>0 Vi with Vi = x ∈ V | Hx = ix, and that V0 = C|0〉. Then grFV and

RV are equipped with the induced grading:

grFV =⊕i>0

(grFV )i, (grFV )0 = C,

RV =⊕i>0

(RV )i, (RV )0 = C.

So the following conditions are equivalent:

(1) V is lisse,(2) XV = point,(3) the image of any vector a ∈ Vi for i > 1 in grFV is nilpotent,(4) the image of any vector a ∈ Vi for i > 1 in RV is nilpotent.

Thus, lisse vertex algebras can be regarded as a generalization of finite-dimensionalalgebras.

3.3.3. Comparison with weight-depending filtration. Let V be a vertexalgebra that is Z-graded by some Hamiltonian H (see §2.2.4):

V =⊕∆∈Z

V∆ where V∆ := v ∈ V | Hv = ∆v.

Then there is another natural filtration of V defined as follows [Li04].

3.3. ASSOCIATED VARIETY OF A VERTEX ALGEBRA 31

Let GpV be the subspace of V spanned by the vectors

a1(−n1−1)a

2(−n2−1) · · · a

r(−nr−1)|0〉

with ai ∈ V homogeneous, ∆a1 + · · · + ∆ar 6 p. Then G•V defines an increasingfiltration of V :

0 = G−1V ⊂ G0V ⊂ . . . G1V ⊂ . . . , V =⋃p

GpV.

Moreover we have

TGpV ⊂ GpV,(Gp)(n)GqV ⊂ Gp+qV for n ∈ Z,(Gp)(n)GqV ⊂ Gp+q−1V for n ∈ Z>0,

It follows that grG V =⊕GpV/Gp−1V is naturally a Poisson vertex algebras.

It is not too difficult to see the following.

Lemma 3.27 ([Arakawa12, Proposition 2.6.1]). We have

F pV∆ = G∆−pV∆,

where F pV∆ = V∆ ∩ F pV , GpV∆ = V∆ ∩GpV . Therefore

grFV ∼= grGV

as Poisson vertex algebras.

Proposition 3.28 ([Arakawa12, Corollary 2.6.2]). A vertex algebra V isfinitely strongly generated if and only if RV is finitely generated as a ring.

If the images of some vectors a1, . . . , ar ∈ V in RV generate RV , we say thatV is strongly generated by a1, . . . , ar.

Proof. Suppose that a1, . . . , ar are strong generators of V . By Lemma 3.27,C2(V ) = F 1V is spanned by the vectors ai1(−n1−1) . . . a

is(−ns−1)|0〉 with s > 1 and

n1 + · · · + ns > 1. Thus a1, . . . , ar generates RV , where ai is the image of ai inRV .

Conversely, suppose that a1, . . . , ar generates RV . Then by Theorem 3.22,a1, . . . , ar generates grFV as a differential algebra. Since grFV ∼= V as C-vectorspaces by the assumption that F •V is separated, it follows that a1, . . . , ar stronglygenerates V .

Remark 3.29. In fact a stronger fact is known: V is spanned by the abovevectors with r > 0, n1 > n2 > n3 > . . . > 1, see [Gaberdiel-Neitzke03], [Li05,Theorem 4.7].

3.3.4. Universal affine vertex algebras. Consider the universal affine ver-tex algebra V κ(a) defined by (13) as in §2.2.2.

Recall that we have F 1V κ(a) = a[t−1]t−2V κ(a), and a Poisson algebra isomor-phism

C[a∗]'−→ RV κ(a) = V κ(a)/t−2a[t−1]V κ(a)

x1 . . . xr 7−→ x1t−1 . . . xrt

−1|0〉+ t−2a[t−1]V κ(a) (xi ∈ a).(31)


Thus

XV κ(a) = a∗.

We have the isomorphism

C[J∞(a∗)] ∼= grV κ(a).(32)

Indeed, the graded dimensions of both sides coincide. Moreover,

GpVκ(a) = Up(a[t−1]t−1)|0〉,

where Up(a[t−1]t−1)p is the PBW filtration of U(a[t−1]t−1), and we have theisomorphisms

grU(a[t−1]t−1) ∼= S(a[t−1]t−1) ∼= C[J∞(a∗)].

As a consequence of (32), we get

SS(V κ(a)) = J∞(a∗).

Example 3.30. Let Virc be the universal Virasoro vertex algebra with centralcharge c as in §2.2.3. Any V ir-module with central c (i.e., the central element Cof V ir acts as a multiplication by c) on which L(z) is a field can be considered asa Virc-module.

Exercise 3.31. Show that

RVirc∼= C[x],

with the trivial Poisson structure, where x is the image of L−2|0〉.

3.4. Lisse and quasi-lisse vertex algebras

3.4.1. Lisse vertex algebras. A vertex algebra V is called lisse if dimXV =0, or equivalently, if RV is finite-dimensional.

Example 3.32. The simple affine vertex algebra Lk(g) is lisse if and only ifLk(g) is integrable as a g-module, or equivalently, k ∈ Z>0.

Therefore, the lisse condition generalizes the integrability to an arbitrary vertexalgebra.

Example 3.33. Let Nc be the unique maximal submodule of the Virasorovertex algebra Virc, and Virc = Virc/Nc the unique quotient.

By [Arakawa12, Proposition 3.4.1], the following are equivalent:

(i) Virc is lisse,

(ii) c = 1 − 6(p− q)2

pqfor some p, q ∈ Z>2 such that (p, q) = 1. (These are

precisely the central charge of the minimal series representations of theVirasoro algebra V ir.)

It is known that lisse vertex algebras have various nice properties. As anexample, we state the following remarkable result.

Theorem 3.34 ([Abe-Buhl-Dong04, Zhu96, Miyamoto04]). Let V belisse.

(1) Any simple V -module is a positive energy representation. Therefore thenumber of isomorphic classes of simple V -modules is finite.

3.4. LISSE AND QUASI-LISSE VERTEX ALGEBRAS 33

(2) Let M1, . . . ,Ms be representatives of these classes, and let for i = 1, . . . , s,

χMi(τ) = TrMi

(qL0− c24 ) =

∑n>0

dim(Mi)nqn− c

24 , q = e2iπτ ,

be the normalized character of Mi. Then χMi(τ) converges in the domain

τ ∈ C | Im(τ) > 0, and the vector space generated by SL2(Z).χMi(τ) is

finite-dimensional.

Further, if it is also rational, it is known [Huang08] that under some mildassumptions, the category of V -modules forms a modular tensor category, whichfor instance yields an invariant of 3-manifolds, see [Bakalov-Kirillov01].

Definition 3.35. A conformal vertex algebra V is called rational if every Z>0-graded V -modules is completely reducible (i.e., isomorphic to a direct sum of simpleV -modules).

It is known ([Dong-Li-Mason98]) that this condition implies that V hasfinitely many simple Z>0-graded modules and that the graded components of eachof these Z>0-graded modules are finite dimensional.

In fact lisse vertex algebras also verify this property (see Theorem 3.34). It isactually conjectured by [Zhu96] that rational vertex algebras must be lisse (thisconjecture is still open).

However, there are significant vertex algebras that do not satisfy the lisse con-dition. For instance, an admissible affine vertex algebra Lk(g) (see below) has acomplete reducibility property ([Arakawa15b]) and the modular invariance prop-erty ([Kac-Wakimoto89]) in the category O still holds, although it is not lisseunless it is integrable.

So it is natural to try to relax the lisse condition.

3.4.2. Symplectic stratification and quasi-lisse vertex algebras. Anaffine Poisson scheme (resp. affine Poisson variety) is an affine scheme X = SpecA(resp. X = SpecmA) such that A is a Poisson algebra. Let X be a Poisson scheme,that is, a scheme such that the structure sheaf OX is a sheaf of Poisson algebras.

If X is smooth, then one may view X as a complex-analytic manifold equippedwith a holomorphic Poisson structure. For each point x ∈ X one defines thesymplectic leaf Sx through x to be the set of points that could be reached from xby going along Hamiltonian flows1.

If X is not necessarily smooth, let Sing(X) be the singular locus of X, and

for any k > 1 define inductively Singk(X) := Sing(Singk−1(X)). We get a finite

partition X =⊔kX

k, where the strata Xk := Singk−1(X) \ Singk(X) are smoothanalytic varieties (by definition we put X0 = X \ Sing(X)). It is known (cf. e.g.,[Brown-Gordon03]) that each Xk inherits a Poisson structure. So for any pointx ∈ Xk there is a well defined symplectic leaf Sx ⊂ Xk. In this way one definessymplectic leaves on an arbitrary Poisson variety. In general, each symplectic leaf isa connected smooth analytic (but not necessarily algebraic) subset in X. However,if the algebraic variety X consists of finitely many symplectic leaves only, then it

1A Hamiltonian flow in X from x to x′ is a curve γ defined on an open neighborhood of

[0, 1] in C, with γ(0) = x and γ(1) = x′, which is an integral curve of a Hamiltonian vector

field ξf , for some f ∈ O(X), defined on an open neighborhood of γ([0, 1]). See for example

[Laurent-Pichereau-Vanhaecke, Chapter 1] for more details.


was shown in [Brown-Gordon03] that each leaf is a smooth irreducible locally-closed algebraic subvariety in X, and the partition into symplectic leaves gives analgebraic stratification of X.

Example 3.36. The space g∗ is a (smooth) Poisson variety and the symplecticleaves of g∗ are the coadjoint orbits of g∗. The nilpotent cone N of g is an exampleof Poisson variety with finitely many symplectic leaves. The latter are precisely thenilpotent orbits of g∗ ∼= g.

Definition 3.37. A vertex algebra is called quasi-lisse if XV has only finitelymany symplectic leaves.

Clearly, lisse vertex algebras are quasi-lisse.

Example 3.38. Since symplectic leaves in XLk(g) are the coadjoint G-orbitscontained in XLk(g), it follows that Lk(g) is quasi-lisse if and only if XLk(g) ⊂ N .

Important examples of quasi-lisse simple affine vertex algebras are the admi-sisble representations Lk(g), as we explain next paragraph.

3.4.3. Admissible representations. Let ∆re be the set of real roots of g,and ∆re

+ the set of real positive roots with respect to the triangular decomposition(5) (cf. §1.1.5).

Definition 3.39 ([Kac-Wakimoto89, Kac-Wakimoto08]). A weight λ ∈h∗ is called admissible if

(1) λ is regular dominant, that is,

〈λ+ ρ, α∨〉 6∈ −Z>0 for all α ∈ ∆re+ ,

(2) Q∆λ = Q∆re, where ∆λ := α ∈ ∆re | 〈λ+ρ, α∨〉 ∈ Z with ρ = h∨Λ0+ρ.

The irreducible highest weight representation L(λ) of g with highest weight

λ ∈ h∗ is called admissible if λ is admissible. Note that an irreducible integrablerepresentation of g is admissible.

The simple affine vertex algebra Lk(g) is called admissible if it is admissible asa g-module. This happens if and only if k satisfies one of the following conditions:

(1) k = −h∨ +p

qwhere p, q ∈ Z>0, (p, q) = 1, and p > h∨,

(2) k = −h∨ +p

r∨qwhere p, q ∈ Z>0, (p, q) = 1, (p, r∨) = 1 and p > h.

Here r∨ is the lacety of g (i.e., r∨ = 1 for the types A,D,E, r∨ = 2 for thetypes B,C, F and r∨ = 3 for the type G2), h and h∨ are the Coxeter and dualCoxeter numbers.

Definition 3.40. If k satisfies one of the conditions of Proposition ??, we saythat k is an admissible level.

The following fact was conjectured by Feigin and Frenkel and proved for thecase that g = sl2 by Feigin and Malikov [Feigin-Malikov97].

Theorem 3.41 ([Arakawa15a]). If k is admissible then SS(Lk(g)) ⊂ J∞(N )or, equivalently, the associated variety XLk(g) is contained in N .

In fact, the following stronger result holds.


Theorem 3.42 ([Arakawa15a]). Assume that k is admissible. Then

XLk(g) = Oq,

where Oq is a nilpotent orbit which only depends on q, with q as above.

Remark 3.43. For g = sln, Theorem 3.42 gives the following. Let k be admis-sible, and let q ∈ Z>0 be the denominator of k, that is, k+h∨ = p/q, with p ∈ Z>0

and (p, q) = 1. Then

XLk(g) = x ∈ g | (adx)2q = 0 = Oq,

where Oq is the nilpotent orbit corresponding to the partition(n) if q > n,

(q, q, . . . , q, s) (0 6 s 6 q − 1) if q < n.

Remind that h∨ = n for g = sln.

3.4.4. Exceptional Deligne series. There was actually a “strong Feigin-Frenkel conjecture” stating that k is admissible if and only if XLk(g) ⊂ N (providedthat k is not critical, that is, k 6= −h∨ in which case it is known that XLk(g) =N ). Such a statement would be interesting because it would give a geometricaldescription of the admissible representations Lk(g).

This stronger conjecture is actually wrong, as shown the following.

Theorem 3.44 ([Arakawa-Moreau15]). Assume that g belongs to the Deligneexceptional series [Deligne96],

A1 ⊂ A2 ⊂ G2 ⊂ D4 ⊂ F4 ⊂ E6 ⊂ E7 ⊂ E8,

and that k = −h∨

6− 1 + n, n ∈ Z>0 such that k 6∈ Z>0. Then

XLk(g) = Omin.

Note that the level k = −h∨/6 − 1 is not admissible for the types D4, E6,E7, E8. Theorem 3.44 provides the first known examples of associated varietiescontained in the nilpotent cone corresponding to non-admissible levels.

The proof of this result is closely related to the Joseph primitive ideal, andits description by Gan and Savin [Gan-Savin04], associated with the minimalnilpotent orbit.

Note that the condition XLk(g) ⊂ N implies that Lk(g) has only finitely manysimple objects in the category O, and one can describe them thanks to Joseph’sclassification of irreducible highest weights representation Lg(λ) whose associated

variety is Omin.

3.4.5. Other examples. There are other examples of simple quasi-lisse affinevertex algebra Lk(g), with non-admissible level k, in type Dr, r > 5, and in type Br,r > 3; see [Arakawa-Moreau15, Arakawa-Moreau16, Arakawa-Moreau17].

Except for g = sl2, the classification problem of quasi-lisse vertex affine algebrasis wide open.


3.4.6. Sheets as associated variety. In all the above examples, XLk(g) is aclosure of a some nilpotent orbit O ⊂ N , or XLk(g) = g∗. The later happens if k is

generic, that is, k 6∈ Q in which case Lk(g) = V k(g). Therefore it is natural to askwhether there are cases when XLk(g) 6⊂ N and XLk(g) is a proper subvariety of g∗.

Given m ∈ N, let g(m) be the set of elements x ∈ g such that dim gx = m,with gx the centralizer of x in g. A subset S ⊂ g is called a sheet of g if it isan irreducible component of one of the locally closed sets g(m). It is G-invariantand conic and it is smooth if g is classical. The description of sheets is closelyrelated to the Jordan classes. A sheet is a finite disjoint union of Jordan classes,[Borho-Kraft79] (or [Tauvel-Yu, 39.3.4]). As a consequence, the sheet closuresare the closures of certain Jordan classes and they are parameterized by the G-conjugacy classes of pairs (l,Ol) where l is a Levi subalgebra of g and Ol is a rigidnilpotent orbit of l, i.e., which cannot be properly induced in the sense of Lusztig-Spaltenstein [Borho-Kraft79, Borho81] (see also [Tauvel-Yu, §39]). The pair(l,Ol) is called the datum of the corresponding sheet. When Ol is zero, the sheetis called Dixmier, meaning that it contains a semisimple element. We denote by Slthe sheet with datum (l, 0).

It is known that sheets appear in the representation theory of finite-dimensionalLie algebras, see, e.g., [Borho-Brylinski82, Borho-Brylinski85, Borho-Brylinski89],and more recently of finite W -algebras, [Premet-Topley14, Premet14].

Next result is that sheets also appear as associated varieties of some affinevertex algebras.

Theorem 3.45 ([Arakawa-Moreau16]). (1) For n > 4,

XV−1(sln)∼= Smin

as schemes, where Smin is the unique sheet containing Omin. Moreover,as schemes,

SS(V−1(sln) = J∞(Smin).

(2) For m > 2,

XV−m(sl2m)∼= S0

as schemes, where S0 is the unique sheet containing the nilpotent orbitO(2m). Moreover, as schemes,

SS(V−m(sl2m) = J∞(S0).

(3) Let r be an odd integer. Then

XV2−r(so2r) = SIr ,where SIr is the unique sheets containing the nilpotent orbits O(2r−1,12).

By the Irreducibility Theorem, associated varieties of primitive ideals are irre-ducible and contained in the nilpotent cone. Theorem 3.45 shows that this is notanymore the case for affine vertex algebras.

3.4.7. Conjectures, open problems. In view of the above results, and otherresults, particularly, on associated varieties of simple affine W-algebras (cf. Part 4),we formulate a conjecture.

Conjecture 1 ([Arakawa-Moreau16, Conjecture 1]). Let V= ⊕d>0Vd bea simple, finitely strongly generated, positively graded conformal vertex operatoralgebra such that V0 = C.


(1) XV is equidimensional.(2) Assume that XV has finitely many symplectic leaves, that is, V is quasi-

lisse. Then XV is irreducible. In particular, XVk(g) is irreducible ifXVk(g) ⊂ N .

Part (1) of the conjecture is an analog of the equidimentionality theorems ofGabber and Kashiwara [Ka76]. Part (2) of the conjecture is a natural analog of theabove mentioned irreducibility result for the associated variety of primitive idealsof U(g). Note that this irreducibility theorem has been generalized to a large classof Noetherian algebras by Ginzburg [Gi03].

Theorem 3.46 ([Gi03]). Let A be a filtered unital C-algebra. Assume further-more that grA ∼= C[X] is the coordinate ring of a reduced irreducible affine algebraicvariety X, and assume that the Poisson variety Spec(grA) has only finitely manysymplectic leaves. Then for any primitive ideal I ⊂ A, the variety V (I) is theclosure of a single symplectic leaf. In particular, it is irreducible.

However, our algebra is not Noetherian.On the other hand, note that scheme-theoretic intersections of Slodowy slices

with associated varieties of simple affine vertex algebras appear as associated vari-eties of W -algebras [Arakawa15a]. These intersections are either empty of com-plete intersections [Ginzburg09]. This gives an evidence to Part (1) of the conjec-ture. However, these intersections are not always irreducible, and Part (2) of theconjecture is discussed in [Arakawa-Moreau17] (see also Part 4).

To conclude this section, note that there are other known examples of associatedvarieties with finitely many symplectic leaves: apart from the above examples, itis the case when V is the (generalized) Drinfeld-Sokolov reduction (see Part 4) ofthe above affine vertex algebras provided that it is nonzero ([Arakawa15a]). Thisis also expected to happen for the vertex algebras obtained from four dimensionalN = 2 superconformal field theories ([BLL+]), where the associated variety isexpected to coincide with the spectrum of the chiral ring of the Higgs branch of thefour dimensional theory. Of course, it also happens when the associated variety ofV is a point, that is, when V is lisse (see Section 4.4 for more details).

PART 4

Associated varieties of affine W-algebras

The study of affine W-algebras began with the work of Zamolodchikov in 1985.Mathematically, affine W-algebras are defined by the method of quantized Drinfeld-Sokolov reduction that was discovered by Feigin and Frenkel in the 1990s. The gen-eral definition of affine W-algebras were given by Kac, Roan and Wakimoto in 2003.Affine W-algebras are related with integrable systems, the two-dimensional confor-mal field theory and the geometric Langlands program. The most recent develop-ments in representation theory of affine W-algebras were done by Kac-Wakimotoand Arakawa.

The affine W-algebras are certain vertex algebras associated with nilpotentelements of simple Lie algebras. They can be regarded as affinizations of finiteW-algebras, and can also be considered as generalizations of affine Kac-Moodyalgebras and Virasoro algebras. They quantize the arc space of the Slodowy slicesassociated with nilpotent elements.

Since they are not finitely generated by Lie algebras, the formalism of vertexalgebras is necessary to study them. In this context, associated varieties of W-algebras, and their quotients, are important tools to understand some properties,such as the lisse condition and even the rationality condition.

The definition of W-algebras is quite technical and need a number a back-grounds. We do not give the precise definition in this lecture, and refer for instanceto [Arakawa-lectures] or [Arakawa-Moreau-lectures], and references therein,for more details.

4.1. Slodowy slices

Let f be a nilpotent element of g that we embed into an sl2-triple (e, h, f) of g.Let φ : g → g∗ be the isomorphism induced from the non-degenerate bilinear form( | ), and set

χ := φ(f) = (f | · ) ∈ g∗.

Then define the Slodowy slice associated with (e, h, f) to be,

Sf := φ(f + ge) = χ+ φ(ge) ⊂ g∗.

Denote by gi the i-eigenspace of ad(h) for i ∈ Z,

gi = x ∈ g | [h, x] = ix, i ∈ Z.

The restriction of the antisymmetric bilinear form,

ωχ : g× g→ C, (x, y) 7→ (f |[x, y]),

to g 12× g 1

2is nondegenerate. This results from the paring between g 1

2and g− 1

2,

and from the injectivity of the map ad f : g 12→ g− 1

2. It is called the Kirillov form

39

40 4. ASSOCIATED VARIETIES OF AFFINE W-ALGEBRAS

associated with f . Let L be a Lagrangian subspace of g 12, that is, L is maximal

isotropic which means ωχ(L,L) = 0 and dimL = 12 dim g 1

2. Set

m = mχ,L := L⊕⊕j> 1

2

gj .

Then m is an ad-nilpotent1, adh-graded subalgebra, of g. Moreover, the algebra mverifies the following properties:

(χ1) χ([m,m]) = (f |[m,m]) = 0,(χ2) m ∩ gf = 0;(χ3) dimm = 1

2 dim G.f .

Let M be the unipotent subgroup of G corresponding to m.

4.1.1. Contracting C∗-action. Let us introduce a C∗-action on g which sta-bilizes Sf

∼= f + ge. The embedding spanCe, h, f ∼= sl2 → g exponentiates toa homomorphism SL2 → G. By restriction to the 1-dimensional torus consist-ing of diagonal matrices, we obtain a one-parameter subgroup ρ : C∗ → G. Thusρ(t)x = t2jx for any x ∈ gj . For t ∈ C∗ and x ∈ g, set

ρ(t)x := t2ρ(t)(x).(33)

So, for any x ∈ gj , ρ(t)x = t2+2jx. In particular, ρ(t)f = f and the C∗-action of ρstabilizes Sf . Moreover, it is contracting to f on Sf , that is,

limt→0

ρ(t)(f + x) = f

for any x ∈ ge, because ge ⊂ m⊥ ⊆ g>−1. The same lines of arguments show thatthe action ρ stabilizes f + m⊥ and it is contracting to f on f + m⊥, too.

The affine space Sf is a “slice” according to the following result.

Theorem 4.1. The affine space Sf is transversal to the coadjoint orbits of g∗.More precisely, for any ξ ∈ Sf , Tξ(G.ξ) + Tξ(Sf ) = g∗. An analogue statementholds for the affine variety χ+ m⊥.

Sketch of proof. We have to prove that [g, x] + ge = g for any x ∈ f + ge

since Tx(G.x) = [g, x] and Tx(f + ge) = ge.It suffices to verify that the map

η : G× (f + ge)→ g

is a submersion at any point (g, x) of G× (f + ge), that is, the differential dη(g,x)

of η at (g, x) is surjective for any point (g, x) of G× (f + ge). The differential of ηis the linear map g× ge → g, (v, w) 7→ g([v, x]) + g(w).

Thus dη(Id,f)(v, w) = [v, f ]+w. Hence dη(Id,f) is surjective since [g, f ]+ge = g.Thus dη(Id,x) is surjective for any x in an open neighborhood Ω of f in f+ge. Sincethe morphism η is G-equivariant for the action by left multiplication, we deducethat dη(g,x) is surjective for any g ∈ G and any x ∈ Ω.

In particular, for any x ∈ Ω, we get

g = [g, x] + ge

Next, we use the contracting C∗-action ρ on f + ge to show that η is actually asubmersion at any point of G× (f + ge).

1i.e., m only consists of nilpotent elements of g.

4.1. SLODOWY SLICES 41

4.1.2. An isomorphism. Consider the adjoint map

M× (f + m⊥)→ g, (g, x) 7→ g.x

It image is contained in f+m⊥. Indeed, for any x ∈ n and any y ∈ m⊥, exp(adx)(f+y) ∈ f + m⊥ since [m,m] ⊂ m and χ([m,m]) = 0, and this is enough to concludebecause, m being ad-nilpotent, M is generated by the elements exp(adx) for xrunning through m. As a result, by restriction, we get a map

α : M×Sf → f + m⊥.

Theorem 4.2 ([Gan-Ginzburg02, §2.3]). The map α is an isomorphism ofaffine varieties.

Proof. We have a contracting C∗-action on M×Sf defined by:

∀ t ∈ C∗, g ∈M, x ∈ Sf , t.(g, x) := (ρ(t−1)gρ(t), ρ(t)x).

The morphism α is C∗-equivariant with respect to this contracting C∗-action, andthe C∗-action ρ on f + m⊥.

Then we conclude thanks to the following result, formulated in [Gan-Ginzburg02,Proof of Lemma 2.1]:

“a C∗-equivariant morphism α : X1 → X2 of smooth affine C∗-varieties withcontracting C∗-actions which induces an isomorphism between the tangent spacesof the C∗-fixed points is an isomorphism.”

As a consequence of this result, we get the isomorphism:

C[Sf ] ∼= C[f + m⊥]M.

4.1.3. Hamiltonian reduction. We refer to [Vaisman] or [Laurent-Pichereau-Vanhaecke,Proposition 5.39 and Definition 5.9] for the following result.

Theorem 4.3 (Marsden-Weinstein). Let X be a Poisson variety. Assume thatA is connected and that the action of A in X is Hamiltonian. Let γ ∈ a∗. Assumethat γ is a regular value2 of µ, that µ−1(γ) is A-stable and that µ−1(γ)/A is avariety. Let ι : µ−1(γ) → X and π : µ−1(γ) µ−1(γ)/A be the natural maps: ι isthe inclusion and π is the quotient map. Then the triple

(X,µ−1(γ), µ−1(γ)/A)

is Poisson-reducible, i.e., there exists a Poisson structure , ′ on µ−1(γ)/A suchthat for all open subset U ⊂ X and for all f, g ∈ OX(π(U ∩ µ−1(γ)), on has

f, g′ π(u) = f , g ι(u)

at any point u ∈ U ∩ µ−1(γ), where f , g ∈ OX(U) are arbitrary extensions off π|U∩µ−1(γ), g π|U∩µ−1(γ) to U .

We intend to apply the theorem to the connected Lie group M acting on thePoisson variety g∗ by the coadjoint action. The action is Hamiltonian and themoment map

µ : g∗ → m∗(34)

2If f : X → Y is a smooth map between varieties, we say that a point y is a regular valueof f if for all x ∈ f−1(y), the map dxf : Tx(X) → Ty(Y ) is surjective. If so, then f−1(y) is a

subvariety of X and the codimension of this variety in X is equal to the dimension of Y .


is the restriction of functions from g to m. Recall that χ = (f |·). Since χ|m is acharacter on m, it is fixed by the coadjoint action of M. As a consequence, the set

µ−1(χ|m) = ξ ∈ g∗ | µ(ξ) = χ|m

is M-stable. Moreover, we have the following lemma:

Lemma 4.4. χ|m is a regular value for the restriction of µ to each symplecticleaf of g∗.

Proof. Note that µ−1(χ|m) = χ + m⊥. Then we have to prove that for anyξ ∈ χ+ m⊥, the map

dξµ : Tξ(G.ξ)→ Tχ|m(m∗)

is surjective. But Tξ(G.ξ) ' [g, ξ] while Tχ|m(m∗) = m∗. Since χ+m⊥ is transversalto the coadjoint orbits in g∗ (cf. Theorem 4.1), we have

g = [g, ξ] + m⊥.

Let γ ∈ m∗ and write γ = x + x′, with x ∈ [g, ξ] and x′ ∈ m⊥, according to theabove decomposition of g. Then µ(x) = γ.

Since the map

M×Sf −→ χ+ m⊥

is an isomorphism of affine varieties (cf. Theorem 4.2),

Sf∼= (χ+ m⊥)/M.

Therefore, the conditions of Theorem 4.3 are fulfilled and we get a symplecticstructure on Sf .

In fact, thanks to Lemma 4.4, we have shown that the symplectic form on eachleaf on Sf is obtained by symplectic reduction from the symplectic form of thecorresponding leaf of g∗.

The Poisson structure on Sf is described as follows. Let π : χ + m⊥ (χ +m⊥)/M ' Sf be the natural projection map, and ι : χ+ m⊥ → g∗ be the naturalinclusion. Then for any f, g ∈ C[Sf ],

f, gSf π = f , g ι

where f , g are arbitrary extensions of f π, g π to g∗.

4.1.4. BRST reduction. One can also described the Hamiltonian reductionof §4.1.3 in a more factorial way, in terms of the BRST cohomology (where BRSTrefers to the physicists Becchi, Rouet, Stora and Tyutin). We do not detail herethis construction, and refer to [Arakawa-Moreau-lectures] for more details.

Let us just say that there is a certain cohomology HiBRST,χ(m,C[g∗]) group

depending on χ, the algebra m and the Poisson algebra C[g∗]), together with acertain complex, such that Hi

BRST,χ(m,C[g∗]) = 0 for i 6= 0, and

H0BRST,χ(m,C[g∗]) ∼= C[Sf ]

as Poisson algebras.

4.2. AFFINE W-ALGEBRAS 43

4.2. Affine W-algebras

For a nilpotent element f of g, let Wk(g, f) be the W -algebra associated with(g, f) at level k:

Wk(g, f) = H0BRST,f (m, V k(g)),

where H•BRST,f (m, ?) is the BRST functor of the generalized quantized Drinfeld-

Sokolov reduction associated with (g, f) with coefficients in a g-moduleM ([Feigin-Frenkel90,Kac-Roan-Wakimoto03]). Here m = m[t, t−1].

The W-algebras generalize both affine vertex algebras and Virasoro vertex al-gebras. Indeed, for f = 0, we getWk(g, 0) ∼= V k(f), and for g = sl2, and f = fprinc(that is, f nonzero), then Wk(sl2, fprinc) ∼= Virc(k), provided that k 6= −2, where

c(k) =k dim g

k + h∨− 6k + h∨ − 4.

4.2.1. The BRST functor. Let us denote simply byH0f (?) the BRST functor

H0BRST,f (m, ?).

We have [DeSole-Kac06, Arakawa15a] a natural isomorphism RWk(g,f)∼=

C[Sf ] of Poisson algebras, so that

XWk(g,f) = Sf .

Moreover,

grWk(g, f) ∼= C[J∞Sf ],

so that Wk(g, f) is a quantization of C[J∞Sf ].Let Wk(g, f) be the unique simple quotient of Wk(g, f). Then XWk(g,f) is a

C∗-invariant Poisson subvariety of Sf . Since it is C∗-invariant, Wk(g, f) is lisse ifand only if XWk(g,f) = f.

Theorem 4.5 ([Arakawa15a]). For any quotient V of V k(g) we have XH0f (V )

is isomorphic to the scheme theoretic intersection XV ×g∗ Sf . So XH0f (V ) = XV ∩

Sf .

By Theorem 4.5, when XLk(g) ⊂ N , then XH0f (Lk(g)) is contained in Sf ∩ N

and so has finitely many symplectic leaves.In fact, form the above theorem, we get that:

(1) H0f (V ) 6= 0 if and only if G.f ⊂ XV ,

(2) H0f (V ) is lisse if XV = G.f ,

(3) H0f (V ) is quasi-lisse if G.f ⊂ XV ⊂ N .

The vertex algebra H0f (Lk(g)) is a quotient vertex algebra of Wk(g, f) if it is

nonzero. Conjecturally [Kac-Roan-Wakimoto03, Kac-Wakimoto08], we have

Wk(g, f) ∼= H0f (Lk(g)) provided that H0

f (Lk(g)) 6= 0.

(This conjecture has been verified in many cases [Arakawa05, Arakawa11].)


4.2.2. Lisse and quasi-lisse W-algebras. Theorem 4.5 implies that if Lk(g)is quasi-lisse and f ∈ XLk(g), then the W-algebra H0

f (Lk(g)) is quasi-lisse as well,

and so is its simple quotient Wk(g, f). In this way we obtain a huge number ofquasi-lisse W-algebras.

We discuss next sections the problem of the irreducibility of the vertex algebrasV = H0

f (Lk(g)) with XLk(g) ⊂ N .

Moreover, if XLk(g) = G.f , then XLk(g) = f by the transversality of Sf toG-orbits, so that Wk(g, f) in fact lisse. Thus, Conjecture 1 in particular says thata quasi-lisse affine vertex algebra produces exactly one lisse simple W-algebra.

For example, if k is an admissible level, then one knows that XLk(g) = O for

some nilpotent orbit (cf. Theorem 3.41). Picking f ∈ O, we obtain that Wk(g, f)is lisse.

Note by Theorem 3.44, there are other lisse W-algebras, not coming from admis-sible level. Namely, let (g, k) as in Theorem 3.44, and let f ∈ Omin. ThenWk(g, f)

is lisse. The statement is actually true for any k = −h∨

6− 1 + n, n ∈ Z>0, for

g = D4, E6, E7, E8, that is,Wk(g, f) is lisse for such (g, k), [Arakawa-Moreau15].

4.3. Branching and nilpotent Slodowy slices

We collect in this paragraph some results about branchings and nilpotentSlodowy slices. We refer to [EGA61, Chap. III, §4.3] for the definition of unibranch-ness, and to [Kraft-Procesi82] or [Fu-et-al15] for further details on branchingsand nilpotent Slodowy slices.

Consider two varieties X,Y and two points x ∈ X, y ∈ Y . The singularity ofX at x is called smoothly equivalent to the singularity of Y at y if there is a varietyZ, a point z ∈ Z and two maps

Zϕ//

ψ

X

Y

such that ϕ(z) = x, ψ(z) = y, and ϕ and ψ are smooth in z ([Hesselink76]). Thisclearly defines an equivalence relation between pointed varieties (X,x). We denotethe equivalence class of (X,x) by Sing(X,x).

Various geometric properties of X at x only depends on the equivalence classSing(X,x), for example: smoothness, normality, seminormality (cf. [Kraft-Procesi82,§16.1]), unibranchness, Cohen-Macaulay, rational singularities.

Assume that the algebraic group G acts regularly on the variety X. ThenSing(X,x) = Sing(X,x′) if x and x′ belongs to the same G-orbit O. In this case,we denote the equivalence class also by Sing(X,O).

A cross section (or transverse slice) at the point x ∈ X is defined to be a locallyclosed subvariety S ⊂ X such that x ∈ S and the map

G× S −→ X, (g, s) 7−→ g.s,

is smooth at the point (1, x). We have Sing(S, x) = Sing(X,x).In the case where X is the closure of some nilpotent G-orbit of g, there is a

natural choice of a cross section. Let O,O′ be two nonzero nilpotent orbits of g

4.3. BRANCHING AND NILPOTENT SLODOWY SLICES 45

and pick f ∈ O′ that we embed f into an sl2-triple (e, h, f) of g. The Slodowy sliceSf∼= f + ge is a transverse slice of g at f . It means that for any ξ ∈ Sf ,

Tξ(G.ξ) + Tξ(Sf ) = g∗.

The variety

SO,f := O ∩Sf

is then a transverse slice of O at f , which we call, following the terminology of[Fu-et-al15], a nilpotent Slodowy slice.

Note that SO,f = f if and only if O = G.f . Moreover, since the C∗-action of

ρ on Sf is contracting to f and stabilizes Sf,O, SO,f = ∅ if and only if G.f 6⊆ O.

Hence we can assume that O′ ⊆ O, that is, O′ 6 O for the Chevalley order onnilpotent orbits. The variety SO,f is equidimensional, and

dim SO,f = codim(O′,O).

Since any two sl2-triples containing f are conjugate by an element of theisotropy group of f in G, the isomorphism type of SO,f is independent of thechoice of such sl2-triples. Moreover, the isomorphism type of SO,f is independent

of the choice of f ∈ O′. By focussing on SO,f , we reduce the study of Sing(O,O′)to the study of the singularity of SO,f at f .

The variety SO,f is not always irreducible. We are now interested in sufficientconditions for that SO,f is irreducible.

Let X be an irreducible algebraic variety, and x ∈ X. We say that X isunibranch at x if the normalization π : (X, x)→ (X,x) of (X,x) is locally a home-omorphism at x [Fu-et-al15, §2.4]. Otherwise, we say that X has branches at xand the number of branches of X at x is the number of connected components ofπ−1(x) [Beynon-Spaltenstein84, §5,(E)].

As it is explained in [Fu-et-al15, Section 2.4], the number of irreducible com-ponents of SO,f is equal to the number of branches of O at f .

If an irreducible algebraic variety X is normal, then it is obviously unibranchat any point x ∈ X. Hence we obtain the following result.

Lemma 4.6. Let O,O′ be nilpotent orbits of g, with O′ 6 O and f ∈ O′. If Ois normal, then SO,f is irreducible.

The converse is not true. For instance, there is no branching in type G2

but one knowns that the nilpotent orbit A1 of G2 of dimension 8 is not normal[Levasseur-Smith88].

The number of branches of O at f , and so the number of irreducible compo-nents of SO,f , can be determined from the tables of Green functions in [Shoji80,Beynon-Spaltenstein84], as discussed in [Beynon-Spaltenstein84, Section 5,(E)-(F)]; see Meinolf Geck’s lectures for more details about this.

We refer to Table 2 of [Arakawa-Moreau17] for the complete list of thenilpotent orbits O which have branchings in types F4, E6, E7 and E8 (there isno branching in type G2), and Table 3 of [Arakawa-Moreau17] for the (conjec-tural) list a non-normal nilpotent orbit closures in the exceptional types. These re-sults are extracted from [Levasseur-Smith88, Kraft89, Broer98a, Broer98b,Sommers03]. The list is known to be exhaustive for the types G2, F4 and E6. Itis only conjecturally exhaustive for the types E7 and E8.


The normality question of nilpotent orbit closures in the classical types is nowcompletely answered ([Kraft-Procesi79, Kraft-Procesi79, Sommers05]). Notethat, by [Kraft-Procesi79], if g = sln, then all nilpotent orbit closures are normal.In all the other types, there is at least one non-normal nilpotent closure.

Let O be a nilpotent orbit of g. Recall that the singular locus of O is O \ O.This was shown by Namikawa [Namikawa04] using results of Kaledin and Pa-nyushev [Kaledin06, Panyushev91]; see [Henderson15, Section 2] for a re-cent review. This result also follows from Kraft and Procesi’s work in the classi-cal types [Kraft-Procesi81, Kraft-Procesi82], and from the main theorem of[Fu-et-al15] in the exceptional types.

Theorem 4.7 ([Kraft-Procesi82, Theorem 1]). Let O be a nilpotent orbit inon or spn.

(1) O is normal if and only if it is unibranch.(2) O is normal if and only if it is normal in codimension 2.

In particular, O is normal if it does not contain a nilpotent orbit O′ 6 O ofcodimension 2. Theorem 4.7 does not hold if g = so2n and if O = O1,λ, with λ

very even. To determine the equivalence class Sing(Oε,λ,Oε,η), for ε ∈ −1, 1 andη < λ, there is a combinatorial method developed in [Kraft-Procesi82]. We referto [Arakawa-Moreau17, Section 4] for more details about this.

Kraft and Procesi method together with Theorem 4.7 allow to deal with al-most all nilpotent orbits, with exceptions for the very even nilpotent orbits intype son. For these orbits, the normality question was partially answered in[Kraft-Procesi82, Theorem 17.3], the remaining cases were dealt with in [Sommers05].

4.4. Conclusion, open problems

Using the results of the previous sections we can check the following: for allknown cases where the associated variety of the simple affine vertex algebra Lk(g)is the closure of some nilpotent orbit O of g, then for any f ∈ O, the variety SO,fis irreducible.

These known cases are summarized in the following table (cf. [Arakawa-Moreau17]).

type of g k XLk(g)

(1) any −h∨ N

(2) any admissible Oq

(3) G2 −1 Omin

(4) D4, E6, E7, E8 k ∈ Z, −h∨

6− 1 ≤ k ≤ −1 Omin

(5) Dr with r ≥ 5 −2,−1 Omin

(6) Dr with r an even integer 2− r O(2r−2,14)

(7) Br −2 O(3,12r−2)

Table 1. Known pairs (g, k) for which XLk(g) ⊂ N

In case (1) of Table 1, V−h∨(g) does not satisfy the assumption of Conjec-ture 1,(2), since it is not conformal, but the irreducibility of XV−h∨ (g) holds.

4.4. CONCLUSION, OPEN PROBLEMS 47

In case (2) of Table 1, Oq is a nilpotent orbit of g described by Tables 2–10 of[Arakawa15a] which only depends on the denominator q of the admissible levelk ∈ Q.

As a consequence, we obtain that for all known cases of simple quasi-lissevertex algebras satisfying the hypothesis of Conjecture 1, the associated variety isirreducible.

To conclude, we dress the list of all these known cases. Below is the list of allknown, or expected, simple quasi-lisse vertex algebras V satisfying the hypothesisof Conjecture 1.

This happens when:

(1) V is lisse (or C2-cofinite). Since V is assumed to be positively graded, theassociated variety of V is then a point and so the conjecture is obviouslytrue,

(2) V is a simple affine vertex algebra as in Table 1,(3) V is a simple W-algebra Wk(g, f) with Lk(g) is as in Table 1,(4) this is also expected to happen for the vertex algebras obtained from four

dimensional N = 2 superconformal field theories ([BLL+]), where theassociated variety is expected to coincide with the spectrum of the chiralring of the Higgs branch of the four dimensional theory.

More precisely, the physicists Beem, Rastelli et al [BLL+] showedthat there is a remarkable map

Φ: 4d N=2 SCFTs → vertex algebras,which enjoys “nice properties”. To such a 4d N=2 SCFT, let say T ,there is important invariant, called the Higgs branch, which we denote byHiggs(T ). The Higgs branch Higgs(T ) is an affine hyperkahler variety,and hence, in particular a symplectic variety, possibly singular.

Beem and Rastelli conjecture that for a 4d N=2 SCFT T , we have

Higgs(T ) = XΦ(T ).

The main examples of vertex algebras considered in [BLL+] are the

affine vertex algebras Lk(g) of types D4, E6, E7, E8 at level k = −h∨

4 −1.Namely, for T a 4d N=2 SCFT such that Φ(T ) = Lk(g), with (g, k) asabove, it is known that the Higgs branch of T is the closure of the minimalnilpotent orbit Omin, which gives an evidence to their conjecture.

We refer to the recent survey [Arakawa-Higgs] for more detailsabout this conjecture.

Bibliography

[Abe-Buhl-Dong04] Toshiyuki Abe, Geoffrey Buhl, and Chongying Dong. Rationality, regularity,

and C2-cofiniteness. Trans. Amer. Math. Soc., 356(8):3391–3402 (electronic), 2004.[Arakawa05] Tomoyuki Arakawa. Representation theory of superconformal algebras and the Kac-

Roan-Wakimoto conjecture. Duke Math. J., 130(3):435–478, 2005.

[Arakawa07] Tomoyuki Arakawa. Representation theory of W -algebras. Invent. Math.,169(2):219–320, 2007.

[Arakawa11] Tomoyuki Arakawa. Representation theory ofW -algebras, II. In Exploring new struc-

tures and natural constructions in mathematical physics, volume 61 of Adv. Stud. Pure Math.,pages 51–90. Math. Soc. Japan, Tokyo, 2011.

[Arakawa12] Tomoyuki Arakawa. A remark on the C2-cofiniteness condition on vertex algebras.

Math. Z., 270(1-2):559–575, 2012.[Arakawa15a] Tomoyuki Arakawa. Associated varieties of modules over Kac-Moody algebras and

C2-cofiniteness of W -algebras. Int. Math. Res. Not. 2015(22): 11605-11666, 2015.[Arakawa15b] Tomoyuki Arakawa. Rationality of W-algebras: principal nilpotent cases. Ann.

Math., 182(2):565–694, 2015.

[Arakawa16] Tomoyuki Arakawa. Rationality of admissible affine vertex algebras in the categoryO. Duke Math. J., 165(1), 67–93, 2016.

[Arakawa-lectures] Tomoyuki Arakawa. Introduction to W-algebras and their representation the-

ory. Preprint ??.[Arakawa-Higgs] Tomoyuki Arakawa. Associated varieties and Higgs branches (a survey). Preprint

http://www.kurims.kyoto-u.ac.jp/~arakawa/papers/Higgs.pdf.

[Arakawa-Kawasetsu16] Tomoyuki Arakawa and Kazuya Kawasetsu. Quasi-lisse vertex algebrasand modular linear differential equations. arXiv:1610.05865 [math.QA].

[Arakawa-Kuwabara-Malikov] Tomoyuki Arakawa, Toshiro Kuwabara, and Fyodor Malikov. Lo-

calization of Affine W-Algebras. Comm. Math. Phys., 335(1):143–182, 2015.[Arakawa-Moreau-lectures] Tomoyuki Arakawa and Anne Moreau. Lectures on W-algebras (work-

shop in Melbourne). In preparation (preliminary version is available on the webpage of theworkshop).

[Arakawa-Moreau15] Tomoyuki Arakawa and Anne Moreau. Joseph ideals and lisse minimal W -

algebras. J. Inst. Math. Jussieu, published online, doi:10.1017/S1474748016000025.[Arakawa-Moreau16] Tomoyuki Arakawa and Anne Moreau. Sheets and associated varieties of

affine vertex algebras. arXiv:1601.05906[math.RT].

[Arakawa-Moreau17] Tomoyuki Arakawa and Anne Moreau. On the irreducibility of associatedvarieties of W-algebras, to appear in the special issue of J. Algebra in Honor of Efim Zelmanov

on occasion of his 60th anniversary.[Bakalov-Kirillov01] Bojko Bakalov and Alexander Kirillov, Jr. Lectures on tensor categories and

modular functors, volume 21 of University Lecture Series. American Mathematical Society,

Providence, RI, 2001.

[BLL+] Christopher Beem, Madalena Lemos, Pedro Liendo, Wolfger Peelaers, Leonardo Rastelli,and Balt C. van Rees. Infinite chiral symmetry in four dimensions. Comm. Math. Phys.,

336(3):1359–1433, 2015.[Belavin-Polyakov-Zamolodchikov84] Aleksander A. Belavin, Alexander. M. Polyakov, and Alek-

sandr. B. Zamolodchikov. Infinite conformal symmetry in two-dimensional quantum field the-

ory. Nuclear Phys. B, 241(2):333–380, 1984.[Beynon-Spaltenstein84] W. Meurig Beynon and Nicolas Spaltenstein. Green functions of finite

Chevalley groups of type En (n = 6, 7, 8). J. Algebra 88(2):584–614, 1984.

49

http://www.kurims.kyoto-u.ac.jp/~arakawa/papers/Higgs.pdf

50 BIBLIOGRAPHY

[Borcherds86] Richard E. Borcherds. Vertex algebras, Kac-Moody algebras, and the Monster.

Proc. Nat. Acad. Sci. U.S.A., 83(10):3068–3071, 1986.

[Borho81] Walter Borho. Uber schichten halbeinfacher Lie-algebren. Invent. Math., 65:283–317,1981.

[Borho-Brylinski82] Walter Borho and Jean-Luc Brylinski. Differential operators on homogeneous

spaces. I. Irreducibility of the associated variety for annihilators of induced modules. Invent.Math. 69(3):437–476, 1982.

[Borho-Brylinski85] Walter Borho and Jean-Luc Brylinski. Differential operators on homogeneousspaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals. Invent.

Math. 80(1):1–68, 1985.

[Borho-Brylinski89] Walter Borho and Jean-Luc Brylinski. Differential operators on homogeneousspaces. II. Relative enveloping algebras. Bull. Soc. Math. France 117(2):167–210, 1989.

[Borho-Kraft79] Walter Borho and Hanspeter Kraft. Uber Bahnen und deren Deformationen bei

linear Aktionen reducktiver Gruppen. Math. Helvetici., 54:61–104, 1979.[Broer98a] Abraham Broer. Decomposition varieties in semisimple Lie algebras. Canad. J. Math.,

50(5):929–971, 1998.[Broer98b] Abraham Broer. Normal nilpotent varieties in F4. J. Algebra, 207(2):427–448, 1998.

[Brown-Gordon03] Kenneth A. Brown and Iain Gordon. Poisson orders, symplectic reflection al-

gebras and representation theory. J. Reine Angew. Math. 559:193–216, 2003.[Brundan-Goodwin05] Jonathan Brundan and Simon Goodwin, Good grading polytopes, Proc.

Lond. Math. Soc. (3) 94 (2007), no. 1, 155–180.

[Chriss-Ginzburg] Neil Chriss and Victor Ginzburg. Representation theory and complex geometry.Reprint of the 1997 edition. Modern Birkhauser Classics. Birkhauser Boston, Inc., Boston,

MA, 2010.

[Collingwood-McGovern] David H. Collingwood and William M. McGovern. Nilpotent orbits insemisimple Lie algebras. Van Nostrand Reinhold Co. New York, 65, 1993.

[Deligne96] Pierre Deligne. La serie exceptionnelle de groupes de Lie. C. R. Acad. Sci. Paris, Ser

I, 322(4), 321–326, 1996.[DeSole-Kac06] Alberto De Sole and Victor Kac. Finite vs affine W-algebras. Jpn. J. Math.

1(1):137–261, 2006.[Dong-Li-Mason98] Chongying Dong, Haisheng Li and Geoffrey Mason. Twisted representations

of vertex operator algebras. Math. Ann. 310(3):571–600, 1998.

[Dong-Li-Mason06] Chongying Dong and Geoffrey Mason. Integrability of C2-cofinite vertex op-erator algebras. Int. Math. Res. Not., pages Art. ID 80468, 15, 2006.

[Duflo77] Michel Duflo. Sur la classification des ideaux primitifs dans l’algebre enveloppante d’une

algebre de Lie semisimple. Ann. of Math. 105:107–120, 1977.[Ein-Mustata09] Lawrence Ein and Mircea Mustata. Jet schemes and singularities. Proc. Sympos.

Pure Math., 80, Part 2, Amer. Math. Soc., Providence, 2009.

[Feigin-Frenkel90] Boris Feigin and Edward Frenkel. Quantization of the Drinfel′d-Sokolov reduc-tion. Phys. Lett. B, 246(1-2):75–81, 1990.

[Feigin-Malikov97] Boris Feigin and Fyodor Malikov. Modular functor and representation theory

of sl2 at a rational level. In Operads: Proceedings of Renaissance Conferences (Hartford,

CT/Luminy, 1995), volume 202 of Contemp. Math., pages 357–405, Providence, RI, 1997.Amer. Math. Soc

[Frenkel-BenZvi] Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves. Math-

ematical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, 2001.[Frenkel-Kac-Wakimoto92] Edward Frenkel, Victor Kac, and Minoru Wakimoto. Characters and

fusion rules for W -algebras via quantized Drinfel′d-Sokolov reduction. Comm. Math. Phys.,

147(2):295–328, 1992.[Fu-et-al15] Baohua Fu, Daniel Juteau, Paul Levy and Eric Sommers. Generic singularities of

nilpotent orbit closures. arXiv:1502.05770[math.RT], to appear in Adv. Math..

[Gaberdiel-Neitzke03] Matthias R. Gaberdiel and Andrew Neitzke. Rationality, quasirationalityand finite W -algebras. Comm. Math. Phys., 238(1-2):305–331, 2003.

[Gan-Ginzburg02] Wee Liang Gan and Victor Ginzburg. Quantization of Slodowy slices. Int.

Math. Res. Not. 243–255, 2002.[Gan-Savin04] Wee Teck Gan and Gordan Savin. Uniqueness of Joseph ideal. Math. Res. Lett.,

11(5-6):589–597, 2004.[Gi03] Victor Ginzburg. On primitive ideals. Sel. math., New 9:379–407, 2003.

BIBLIOGRAPHY 51

[Ginzburg09] Victor Ginzburg. Harish-Chandra bimodules for quantized Slodowy slices. Repre-

sent. Theory 13:236–271, 2009.

[EGA61] Alexander Grothendieck. Elements de geometrie algebrique. III. Etude cohomologique

des faisceaux coherents. I. Inst. Hautes Etudes Sci. Publ. Math. 11(1):5–167, 1961.[Henderson15] Anthony Henderson. Singularities of nilpotent orbit closures. Rev. Roumaine Math.

Pures Appl. 60(4):441–469, 2015.

[Hesselink76] Wim Hesselink. Singularities in the nilpotent scheme of a classical group. Trans.Amer. Math. Soc. 222:1–32, 1976.

[Huang08] Yi-Zhi Huang. Rigidity and modularity of vertex tensor categories. Commun. Con-

temp. Math., 10(suppl. 1):871–911, 2008.[Ishii11] Shihoko Ishii. Geometric properties of jet schemes. Comm. Algebra 39(5):1872–188, 2011.

[Joseph85] Anthony Joseph. On the associated variety of a primitive ideal. J. Algebra 93:509–523,

1985.[Kac1] Victor Kac. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press,

Cambridge, 1990.

[Kac2] Victor Kac. Vertex algebras for beginners. Second edition. University Lecture Series, 10.American Mathematical Society, Providence, RI, 1998. An introduction to affine Kac-Moody

algebras.

[KK79] V. G. Kac and D. A. Kazhdan. Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. in Math., 34(1):97–108, 1979.

[Kac-Roan-Wakimoto03] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto. Quantum reductionfor affine superalgebras. Comm. Math. Phys., 241(2-3):307–342, 2003.

[Kac-Wakimoto89] Victor Kac and Minoru Wakimoto. Classification of modular invariant rep-

resentations of affine algebras. In Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), volume 7 of Adv. Ser. Math. Phys., pages 138–177. World Sci. Publ., Tea-

neck, NJ, 1989.

[Kac-Wakimoto08] Victor Kac and Minoru Wakimoto. On rationality of W -algebras. Transform.Groups, 13(3-4):671–713, 2008.

[Kac-lectures] Victor Kac. Introduction to vertex algebras, Poisson vertex algebras, and integrable

Hamiltonian PDE. Preprint http://arxiv.org/pdf/1512.00821v1.pdf.[Kaledin06] Dmitry Kaledin. Symplectic singularities from the Poisson point of view. J. Reine

Angew. Math., 600:135–156, 2006.

[Kraft89] Hanspeter Kraft. Closures of conjugacy classes in G2. J. Algebra, 126(2):454–465, 1989.[Kraft-Procesi79] Hanspeter Kraft and Claudio Procesi. Closures of conjugacy classes of matrices

are normal. Invent. Math. 53(3):227–247, 1979.[Kraft-Procesi81] Hanspeter Kraft and Claudio Procesi. Minimal singularities in GLn. Invent.

Math. 62(3):503–515, 1981.

[Kraft-Procesi82] Hanspeter Kraft and Claudio Procesi. On the geometry of conjugacy classes inclassical groups. Comment. Math. Helv. 57(4):539–602, 1982.

[Levasseur-Smith88] Thierry Levasseur and S. Paul Smith. Primitive ideals and nilpotent orbits

in type G2. J. Algebra 114(1):81–105, 1988.[Ka76] Masaki Kashiwara B-functions and holonomic systems. Rationality of roots of B-functions.

Invent. Math. 38 (1976/77), no. 1, 33–53.[Kashiwara-Tanisaki84] Masaki Kashiwara and Toshiyuki Tanisaki. The characteristic cycles of

holonomic systems on a flag manifold related to the Weyl group algebra. Invent. Math. 77:185-

198, 1984.

[Kolchin73] Ellis Kolchin. Differential algebra and algebraic groups. Academic Press, New York1973.

[Laurent-Pichereau-Vanhaecke] Camille Laurent-Gengoux, Anne Pichereau and Pol Vanhaecke.Poissonstructures. Springer,Heidelberg, 347, 2013.

[Li04] Haisheng Li. Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math.,

6(1):61–110, 2004.[Li05] Haisheng Li. Abelianizing vertex algebras. Comm. Math. Phys., 259(2):391–411, 2005.

[Losev10a] Ivan Losev. Finite W-algebras. Proceedings of the International Congress of Mathe-

maticians. Volume III, 1281–1307, Hindustan Book Agency, New Delhi, 2010, 16–02.[Lusztig-Spaltenstein79] George Lusztig and Nicolas Spaltenstein. Induced unipotent classes. J.

London Math. Soc. 19:41–52, 1979.

http://arxiv.org/pdf/1512.00821v1.pdf

52 BIBLIOGRAPHY

[Miyamoto04] Masahiko Miyamoto. Modular invariance of vertex operator algebras satisfying C2-

cofiniteness. Duke Math. J., 122(1):51–91, 2004.

[Moody-Pianzola] Robert V. Moody and Arturo Pianzola. Lie algebras with triangular decompo-sitions. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-

Interscience Publication. John Wiley & Sons, Inc., New York, 1995.

[Moreau-lectures] Anne Moreau, Nilpotent orbits and finite W-algebras. Preprint http://math.

univ-lille1.fr/~amoreau/KENT2014-Walg-v2.pdf.

[Moreau-Yu16] Anne Moreau and Rupert Wei Tze Yu. Jet schemes of the closure of nilpotent

orbits. Pacific J. Math., 281(1):137–183, 2016.[Mustata01] Mircea Mustata. Jet schemes of locally complete intersection canonical singularities.

Invent. Math., 145(3):397–424, 2001. with an appendix by D. Eisenbud and E. Frenkel.

[Namikawa04] Yoshinori Namikawa. Birational geometry of symplectic resolutions of nilpotentorbits, Moduli spaces and arithmetic geometry, 75–116. Adv. Stud. Pure Math., 45, Math.

Soc. Japan, Tokyo, 2006 preprint http://arxiv.org/pdf/math/0408274v1.pdf.[Panyushev91] Dmitri Panyushev. Rationality of singularities and the Gorenstein property for

nilpotent orbits. Functional Analysis and Its Applications, 25(3):225–226, 1991.

[Premet14] Alexander Premet. Multiplicity-free primitive ideals associated with rigid nilpotentorbits. Transform. Groups 19(2):569-641, 2014.

[Premet-Topley14] Alexander Premet and Lewis Topley. Derived subalgebras of centralisers and

finite W-algebras. Compos. Math. 150(9):1485–1548, 2014.[Shoji80] Toshiaki Shoji. On the Springer representations of Chevalley groups of type F4. Comm.

Algebra 8(5):409–440, 1980.

[Sommers03] Eric Sommers. Normality of nilpotent varieties in E6. J. Algebra 270(1):288–306,2003.

[Sommers05] Eric Sommers. Normality of very even nilpotent varieties in D2l. Bull. London Math.

Soc. 37(3):351–360, 2005.[Tauvel-Yu] Patrice Tauvel and Rupert Wei Tze Yu. Lie algebras and algebraic groups. Springer

Monographs in Mathematics. Springer-Verlag, Berlin, 2005.[Vaisman] Izu Vaisman. Lectures on the geometry of Poisson manifolds. Progress in Mathematics,

118, Birkhauser Verlag, Basel, 1994.

[Zhu96] Yongchang Zhu. Modular invariance of characters of vertex operator algebras. J. Amer.Math. Soc., 9(1):237–302, 1996.

http://math.univ-lille1.fr/~amoreau/KENT2014-Walg-v2.pdf

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