Associated schemes and vertex algebras

Transcription

1 Associated schemes and vertex algebras Anne Moreau Laboratoire Paul Painlevé, CNRS UMR 8524, Villeneuve d Ascq Cedex address: anne.moreau@math.univ-lille1.fr Abstract. These notes are written in preparation for the mini-course entitled Associated schemes and vertex algebras which will take place in the summer school Current Topics in the Theory of Algebraic Groups of the GDR TLAG.

2

3 Contents Introduction 5 Part 1. Affine Kac-Moody algebras and their vacuum representations Affine Kac-Moody algebras The vacuum representations Associated varieties of vacuum representations 10 Part 2. Vertex algebras, definitions, first properties and examples Operator product expansion and definition of vertex algebras First examples of vertex algebras Modules over vertex algebras 22 Part 3. Poisson vertex algebras, arc spaces, and associated varieties Jet schemes and arc spaces Poisson vertex algebras Associated variety of a vertex algebra Lisse and quasi-lisse vertex algebras 32 Part 4. Associated varieties of affine W-algebras Slodowy slices Affine W-algebras Branching and nilpotent Slodowy slices Conclusion, open problems 46 Bibliography 49 3

4

5 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 distinguished vector, the vacuum vector, and the vertex operator map from V to the space of formal Laurent series with linear operators on V as coefficients. These data satisfy a number of axioms. Although the definition is purely algebraic, these axioms have deep geometric meaning. They reflect the fact that vertex algebras give an algebraic framework of the two-dimensional conformal field theory. The connections of this topic with other branches of mathematics and physics include algebraic geometry (moduli spaces), representation theory (modular representation theory, geometric Langlands correspondence), two dimensional conformal field theory, string theory (mirror symmetry) and four dimensional gauge theory (AGT conjecture). To each vertex algebra V one can naturally attach a certain Poisson variety X V called the associated variety of V. For an affine Poisson variety X, a vertex algebra V such that X V = X is called a chiral quantization of X. A vertex algebra V is called lisse if dim X V = 0. Lisse vertex algebras are natural generalizations of finite-dimensional algebras and possess remarkable properties. For instance, the characters of simple V -modules form vector valued modular functions. More generally, vertex algebras whose associated variety has only finitely symplectic leaves, are also of great interest for several reasons that will be addressed in the lectures. Important examples of vertex algebras are those coming from affine Kac-Moody algebra, which are called affine vertex algebras. They play a crucial role in the representation theory of affine Kac-Moody algebras, and of W-algebras. In the case that V is a simple affine vertex algebra, its associated variety is an invariant and conic subvariety of the corresponding simple Lie algebra. It plays an analog role to the associated variety of primitive ideals of the enveloping algebra of simple Lie algebras. However, associated varieties of affine vertex algebras are not necessarily contained in the nilpotent cone and it is difficult to describe them in general. Associated varieties not only capture some of the important properties of vertex algebras but also have interesting relationship with the Higgs branches of fourdimensional N = 2 superconformal field theories (SCFTs). However, their general description is fairly open, except in a few cases. It is only quite recently that the study of associated varieties of vertex algebras and their arc spaces, has been more intensively developed. In this mini-course I wish to highlight this aspect of the theory of vertex algebras which seems to be very promising. In particular, I will include open problems on associated varieties in the setting of affine vertex algebras (vertex algebras associated with Kac-Moody 5

6 6 INTRODUCTION algebras) and W-algebras (they are certain vertex algebras attached with nilpotent elements of a simple Lie algebra) raised by my recent works with Tomoyuki Arakawa.

7 PART 1 Affine Kac-Moody algebras and their vacuum representations References: [Kac1, Moody-Pianzola]. The goal of this lecture is to introduce the universal vacuum representation V k (g) attached to an affine Kac-Moody algebra ĝ and a complex number k. We will see next lecture that it has a natural vertex algebra structure. We also attach to any quotient V k (g) a certain Poisson algebra and a corresponding Poisson variety Affine Kac-Moody algebras 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 a certain linear algebraic group G, g = Lie(G). Let ( ) Kil be the Killing form of g, ( ) Kil : g g C, (x, y) tr(ad x ad y). It is a nondegenerate symmetric bilinear form of g which is G-invariant, that is, or else, (g.x g.y) Kil ) = (x y) Kil for all x, y g, g G, ([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 the Killing form. Example 1.1. Let g be the Lie algebra sl n, n 2, which is the set of traceless complex n-size square matrices, with bracket [A, B] = AB BA. The Lie algebra sl n is known to be simple and its Killing form is given by (A, B) 2n tr(ab) Dual Coxeter number. Let h be a Cartan subalgebra of g, and let g = h α g α, g α := {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 = {α 1,..., α r } be a basis of, with r the rank of g, and let α1,..., αr be the coroots of α 1,..., α r respectively. The element αi, i = 1,..., r, viewed as an element of (h ) = h, will be often denoted it by h i. Let + be the set of positive roots corresponding to Π, and let (1) g = n h n + 7

8 8 1. AFFINE KAC-MOODY ALGEBRAS AND THEIR VACUUM REPRESENTATIONS be the corresponding triangular decomposition. Thus n + = α + g α and n = α g α are both nilpotent Lie subalgebras of g. Any positive root α + can be written as r α = n i α i, n i Z 0. i=1 The height of α is ht(α) = r i=1 n i. 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 + α i {0} for i = 1,..., r. Similarly, we define the dual Coxeter number h of g by: h = ht(θ ) + 1. For example, if g = sl n 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) = The loop algebra. Consider the loop algebra of g which is the Lie algebra Lg := g[t, t 1 ] = g C[t, t 1 ], with commutation relations where xt m stands for x t m. [xt m, yt n ] = [x, y]t m+n, x, y g, m, n Z, Remark 1.2. The Lie algebra Lg is the Lie algebra of polynomial functions from 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)res t=0 ( df dt g), for x, y g and f, g C[t, t 1 ], where the linear map Res t=0 : C[t, t 1 ] C is defined by Res t=0 (t m ) = δ m, 1 for m Z. (2) (3) The bilinear ν is a 2-cocycle on Lg, that is, for any a, b, c Lg, ν(a, b) = ν(b, a), ν([a, b], c) + ν([b, c], a) + ν([c, a], b) = Affine Kac-Moody algebras. Definition 1.4. We define the affine Kac-Moody algebra ĝ as the vector space ĝ := Lg CK, with the commutation relations [K, ĝ] = 0 (so K is a central element), and (4) [x f, y g] = [x, y] Lg + ν(x f, y g)k, x, y g, f, g C[t, t 1 ],

9 1.2. THE VACUUM REPRESENTATIONS 9 where [, ] Lg is the Lie bracket on Lg. In other words the commutation relations are given by: [xt m, yt n ] = [x, y]t m+n + mδ m+n,0 (x y)k, [K, ĝ] = 0, for x, y g and m, n Z. Exercise 1.5. Verify that the identifies (2) and (3) are true, and then that the above commutation relations indeed define a Lie bracket on ĝ Triangular decomposition. Recall the triangular decomposition (1) of g, and consider the following subspaces of ĝ: n + := (n h) tc[t] n + C[t] = n + + tg[t], n := (n + h) t 1 C[t 1 ] n C[t 1 ] = n + t 1 g[t 1 ], ĥ := (h 1) CK = h + CK. They are Lie subalgebras of ĝ and we have (5) ĝ = n ĥ n The vacuum representations Consider the Lie subalgebra g[t] CK of ĝ. It is a parabolic subalgebra of g since it contains the Borel subalgebra ĥ n The universal vacuum representation. Fix k C, and consider the one-dimensional representation C k g[t] CK on which g[t] acts by 0 and K acts as a multiplication by the scalar k. We define the universal vacuum representation of level k of ĝ as the representation induced from C k : (6) V k (ĝ) = Indĝ g[t] CK C k = U(ĝ) U(g[t] CK) C k. It can be viewed as a generalized Verma module Level of the vacuum representation. The representation V k (g) is a highest weight representation of ĝ with highest weight kλ 0, with Λ 0 is the highest weight of the basic representation 1, and highest weight vector v k, where v k denotes the image of 1 1 in V k (g). We will often denote by 0 the vector v k (the notation will be justified next part when we will endow V k (g) with a vertex algebra structure). According to the well-known Schur Lemma, any central element of a Lie algebra acts as a scalar on a simple finite dimensional representation L. As the Schur Lemma extends to a representation with countable dimension, the result holds for highest weight ĝ-modules. Definition 1.6. A representation M is said to be of level k if K acts as kid on M. Then V k (g) is by construction of level k. 1 that is, the dual of K in ĥ with respect to a basis of ĥ adapted to the decomposition ĥ = h CK.

10 10 1. AFFINE KAC-MOODY ALGEBRAS AND THEIR VACUUM REPRESENTATIONS PBW basis and grading. By the Poincaré-Birkhoff-Witt Theorem, the direct sum decomposition (as a vector space) ĝ = (g t 1 C[t 1 ]) (g[t] CK) gives us the isomorphism of vector spaces whence set U(ĝ) = U(g t 1 C[t 1 ]) U(g[t] CK), V k (ĝ) = U(g t 1 C[t 1 ]). Let {x 1,..., x d }, d = dim g, be an ordered basis of g. For any x g and n Z, x (n) := x t n = xt n L(g). Then {K, x i (n), i = 1,..., d, n Z} forms a basis of ĝ 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 basis of monomials of the form x i1 (n... 1) xim (n 0, m) where n 1 n 2 n m < 0, and if n j = n j+1, then i j i j+1. The space V k (g) is naturally graded, V k (g) = Z 0 V k (g), where the grading is defined by deg x i1 (n... 1) xim (n 0 = m m) n i, deg 0 = 0. We have V k (g) 0 = C 0, and we identify g with V k (g) 1 via the linear isomorphism defined by x xt 1 0. Any graded quotient V of V k (g) (i.e., a quotient by a proper submodule of V k (g)) is again a highest weight representation of ĝ with highest weight kλ 0, and of level k. In particular, V k (g) has a unique maximal proper graded submodule N k and so L k (g) := V k (g)/n k is an irreducible highest weight representation of ĝ with highest weight kλ 0, and of level k. Note that, as a ĝ-representation, we have i=1 L k (g) = L(kΛ 0 ), where for λ ĥ, L(λ) denotes the highest weight representation of ĝ of highest weight λ. Note that L k (g) is of level k too Associated varieties of vacuum representations Recall that the nilpotent cone of g is the (reduced) subscheme of g associated with 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 for the representation theory. Because of this, it would be nice to have analogs of the associated varieties of primitive ideals in this context. Unfortunately, one cannot expect exactly the same theory. One of the main reason, is that the center of U(ĝ) is trivial (unless for the critical level k = h ), and so we do not have analog of the nilpotent cone (for the critical level, the analog is played by the arc space of

11 1.3. ASSOCIATED VARIETIES OF VACUUM REPRESENTATIONS 11 the nilpotent cone). However, what is very interesting with primitive ideals is that their associated variety in contained in the nilpotent cone (cf ). To encounter this problem, we consider the associated variety of a certain Poisson algebra, this is ous next purpose Poisson algebras. Recall that C[g ] has naturally a Poisson structure induced from the Kirillov-Kostant-Souriau Poisson structure on g. Namely, for f, g C[g ], x g, {f, g}(x) = x, [d x f, d x g], where d x f, d x g 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 R V k (g) = V k (g)/t 2 g[t 1 ]V k (g). (7) We an algebra isomorphism C[g ] R V k (g) = V k (g)/t 2 g[t 1 ]V k (g) x 1... x m (x 1 t 1 )... (x m t 1 ) 0 + t 2 g[t 1 ]V k (g) Thus R V k (g) inherits a Poisson structure given by (x i g). {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 {ā, 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 R V = V/t 2 g[t 1 ]V, and we get a surjective morphism of Poisson algebras, (8) C[g ] R V = V/t 2 g[t 1 ]V x 1... x r x 1,( 1)... x r,( r) 0 + t 2 g[t 1 ]V (x i g), the Poisson algebra structure on R V being defined as before. This map is surjective but not an isomorphism in general Associated varieties. Continue to assume that V is a graded quotient of V k (g). Definition 1.7. We define the associated variety X V of V to be the zero locus in g of the kernel of the above map (8). The Poisson variety X V is then a G-invariant, Poisson, and conic subvariety of g. We obviously have X V k (g) = g since the map (8) is an isomorphism for V = V k (g). For V = L k (g), X V can be viewed as an analog of the associated variety of primitive ideals. However, we will see that there are substantial differences.

12 12 1. AFFINE KAC-MOODY ALGEBRAS AND THEIR VACUUM REPRESENTATIONS 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 filtration on 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) = Spec C[g ]/gr I. The variety V (gr I) = Specm C[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 of U(g). The first result is the Duflo Theorem [Duflo77], stating that any primitive ideal in U(g) is the annihilator Ann U(g) L g (λ) of some simple highest weight module L g (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. The Irreducibility Theorem says that the associated variety V (gr I) of a primitive ideal I in U(g) is irreducible, specifically, it is the closure O of some nilpotent orbit O in 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 some primitive ideal of U(g) Integrable representations. First of all, since L k (g) = V k (g) for k Q (cf. [KK79]), we see that X V k (g) is not always contained in the nilpotent cone N. One knows that L g (λ) is finite-dimensional if and only if V (gr Ann U(g) L g (λ)) = {0}, where L g (λ) denotes the irreducible highest weight representation of g, with highest weight λ h. Contrary to irreducible highest weight representations of g, the representation L(λ), λ ĥ, is finite dimensional if and only if λ = 0, that is, L(λ) is the trivial representation. The notion of finite dimensional representations has to be replaced by the notion of integrable representations in the category O. We define the category O for ĝ in the similar way that for g, except that we do not require that the object are finitely generated by g (cf. [Moody-Pianzola]). Definition 1.8. A representation M of ĝ is said to be integrable if (1) M is ĥ-diagonalizable, (2) for λ ĥ, M λ is finite dimensional, (3) for i = 0,..., r, E i and F i act locally nilpotently on M, where E i, H i, F i are Chevalley generators 2 of ĝ. Remark 1.9. As a a i -module, i = 0,..., r, an integrable representation M decomposes into a direct sum of finite dimensional irreducible ĥ-invariant modules, 2 Namely, Ei = e i 1, F i = f i 1, H i = f i 1, for i = 1,..., r, with e i, f i, h i Chevalley generators of g, and E 0 = e 0 t, F 0 = f 0 t 1, with f 0 g θ, e 0 g θ such that (f 0 e 0 ) = 1.

13 1.3. ASSOCIATED VARIETIES OF VACUUM REPRESENTATIONS 13 where a i = sl2 is the Lie algebra generated by the Chevalley generators E i, F i, H i. Hence the action of a i on M can be integrated to the action of the group SL 2 (C). The character of the simple integrable representations in the category O satisfy remarkable combinatorial identities (related to Macdonald identities). Recall that the irreducible representation L g (λ), λ h, is finite-dimensional if and only if its associated variety V (Ann U(g) (L g (λ))) is zero. Theorem The associated variety of L k (g) is {0} if and only if L k (g) is integrable as a ĝ-module, which is true if and only if k Z 0. The last part of the statement is well-known. Lemma 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 a m I for some m Z 0. Since I is -invariant, we have m a m I. But m a m m!( a) m Hence ( a) m I, and therefore, a I. (mod I). Recall that a singular vector of g of a g-module M is a vector v M such that n +.v = 0, that is, e i.v = 0 for i = 1,..., r. A singular vector of ĝ of a ĝ-representation M is a vector v M such that n +.v = 0, that is, e i.v = 0 for i = 1,..., r, and (f θ t).v = 0. In particular, regarding V k (g) as a ĝ-representation, a vector v V k (g) is singular if and only if n +.v = 0. Proof of the if part of Theorem Suppose that L k (g) is integrable. This condition is equivalent to that k Z 0 and the maximal submodule N k (g) of V k (g) is generated by the singular vector (e θ t 1 ) k+1 0 ([Kac1]). The exact sequence 0 N k (g) V k (g) L k (g) 0 induces the exact sequence 0 I k R V k (g) R Lk (g) 0, where I k is the image of N k in R V k (g) = C[g ], and so, R Lk (g) = C[g ]/I k. The image of the singular vector in I k is given by e k+1 θ. Therefore, e θ I k. On the other hand, by Lemma 1.11, I k is preserved by the adjoint action of g. Since g is simple, g I k. This proves that X Lk (g) = {0} as required. The proof of the only if part follows from [Dong-Li-Mason06]. It can also be proved using W-algebras.

14

15 PART 2 Vertex algebras, definitions, first properties and examples 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) Operator product expansion and definition of vertex algebras Let V be a vector space over C. We denote by (End V )[[z, z 1 ]] the set of all formal Laurent series in the variable z with coefficients in the space End V. We call elements a(z) of (End V )[[z, z 1 ]] a series on V. For a series a(z) on V, we set so that the expansion of a(z) is a (n) = Res z=0 a(z)z n a(z) = n Z a (n) z n 1. The coefficient a (n) is called a Fourier mode of a(z). We write for b V. a(z)b = n Z a (n) bz n 1 Definition 2.1. A series a(z) (End V )[[z, z 1 ]] is called a field on V if for any 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 Definition. A vertex algebra is a vector space V equipped with the following data: (the vacuum vector) a vector 0 V, (the vertex operators) a linear map V F (V ), a 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) = id V. Furthermore, for all a V, a(z) 0 V [[z]] and lim z 0 a(z) 0 = a. In other words, a (n) 0 = 0 for n 0 and a ( 1) 0 = a. 15

16 16 2. VERTEX ALGEBRAS, DEFINITIONS, FIRST PROPERTIES AND EXAMPLES (the translation axiom) for any a V, [T, a(z)] = z a(z), (T a)(z) = z a(z) and T 0 = 0. (the locality axiom) for all a, b V, (z w) N a,b [a(z), b(w)] = 0 for some N a,b Z 0. When two fields a(z), b(z) on a vector space V verify the condition of the locality axiom, we say that there are mutually local Operator product expansion (OPE). Proposition 2.2. Let a(z), b(z) be two fields on a vector space V. The following assertions are equivalent. (9) where (i) (z w) N a,b [a(z), b(w)] = 0 for some N = N a,b Z 0. (ii) (iii) [a(z), b(w)] = N a,b 1 n=0 (a (n) b)(w) 1 n! n wδ(z w), where δ(z w) := n Z wn z n 1 C[[z, w, z 1, w 1 ]] is the formal deltafunction. and a(z)b(w) = b(w)a(z) = N a,b 1 n=0 N a,b 1 n=0 1 (a (n) b)(w)τ z,w ( )+ : a(z)b(w):, (z w) j+1 1 (a (n) b)(w)τ w,z ( )+ : a(z)b(w):, (z w) j+1 where : a(z)b(w): and the maps τ z,w and τ w,z are defined below. For a(z), b(z) F (V ), set a(z) + = n<0 : a(z)b(w): = a(z) + b(w) + b(w)a(z), 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 definition, : a(z)b(z)c(z): stands for : a(z) : b(z)c(z): :. The two maps τ z,w and τ w,z are the morphisms of algebras defined by: τ z,w : C[z, w, z 1, w 1 1, z w ] C((z))((w)), 1 z w 1 ( w ) n = δ(z w), z z τ w,z : C[z, w, z 1, w 1, 1 z w ] C((w))((z)), 1 z w 1 z n 0 ( z ) n = δ(z w)+. w n>0

17 2.1. OPERATOR PRODUCT EXPANSION AND DEFINITION OF VERTEX ALGEBRAS 17 Thus the map τ z,w is the expansion of of 1 z w in w > z. 1 z w in z > w and τ w,z is the expansion Proof of the implication (ii) (i). We verify only the converse implication (see for instance [Frenkel-BenZvi, Chap. 3] for more details). Let us write δ(z w) = 1 ( w ) n + 1 ( z ) n, z z z w n 0 n>0 }{{}}{{} δ δ + so that when z > w, the series δ(z w) converges to the meromorphic function 1 z w and when z < w, the series δ(z w) + converges to the meromorphic function 1 z w. We have 1 δ(z w) = τ z,w ( z w ) τ 1 w,z( z w ). Both morphisms τ z,w and τ w,z commutes with w and z. Therefore, 1 1 j! j wδ(z w) = τ z,w ( (z w) j+1 ) τ 1 w,z( (z w) j+1 ), whence (z w) n+1 1 ) n! n wδ(z w) = (z w) (τ n+1 1 z,w ( (z w) n+1 ) τ 1 w,z( (z w) n+1 ) = τ z,w (1) τ w,z (1) = 0. The implication (ii) (i) is then clear. (10) Note that the translation axiom says that [T, a (n) ] = na (n 1), n Z, and together with the vacuum axiom we get that T a = a ( 2) 0. Exercise 2.3. Verify the above assertion. Also, the vacuum axiom implies that the map V End(V ) defined by the formula a a ( 1) is injective. Therefore the map a a(z) is also injective Borcherds identities. A consequence of the definition are the following relations, called Borcherds identities: [a (m), b (n) ] = ( ) m (11) (a i (i) b) (m+n i), i 0 (a (m) b) (n) = ( ) ( 1) j m (12) (a j (m j) b (n+j) ( 1) m b (m+n j) a (j) ), j 0

18 18 2. VERTEX ALGEBRAS, DEFINITIONS, FIRST PROPERTIES AND EXAMPLES for m, n Z. In the above formulas, the notation ( ) m = i m(m 1) (m i + 1). i(i 1) First examples of vertex algebras ( ) m for i 0 and m Z means i Commutative vertex algebras. A vertex algebra V is called commutative if all vertex operators a(z), a V, commute each other (i.e., we have N a,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) End V [[z]] for all a V. Then a commutative vertex algebra has a structure of a unital commutative algebra with the product: a b = : ab : = a ( 1) b, where the unit is given by the vacuum vector 0. The translation operator T of V acts on V as a derivation with respect to this product: T (a b) = (T a) b + a (T b). 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 algebra R with derivation such that: a(z)b = ( e z a ) b = z n n! ( n a)b, n 0 for all a, b R. We take the unit as the vacuum vector. This correspondence gives the following result. Theorem 2.4 ([Borcherds86]). The category of commutative vertex algebras is the same as that of differential algebras. One important example of commutative vertex algebras are obtained by considering the function sheaf over arc spaces of a scheme (see Section 3.1) Universal affine vertex algebras. Let us consider a slightly more general construction of the vacuum representation considered in Section 1.2. We will 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 â = a[t, t 1 ] C1 be the Kac-Moody affinization of a. It is a Lie algebra with commutation relations Let [xt m, yt n ] = [x, y]t m+n + mδ m+n,0 κ(x, y)1, x, y a, m, n Z, [1, â] = 0. (13) V κ (a) = U(â) U(a[t] C1) C,

19 2.2. FIRST EXAMPLES OF VERTEX ALGEBRAS 19 where C is a one-dimensional representation of a[t] C1 on which a[t] acts trivially and 1 acts as the identity. By the PBW Theorem, we have the following isomorphism of vector spaces: V κ (a) = U(a t 1 C[t 1 ]). The space V κ (a) is naturally graded: V κ (a) = Z 0 V κ (a), where the grading is defined by setting deg(xt n ) = n, deg 0 = 0. Here 0 is the image of 1 1 in V κ (a). We have V κ (a) 0 = C 0. We identify a with V κ (a) 1 via the linear isomorphism defined by x xt 1 0. There is a unique vertex algebra structure on V κ (a), such that 0 is the vacuum vector and x(z) := n Z(xt n )z n 1, x a. (So x (n) = xt n for x a = V κ (a) 1, n Z.) The vertex algebra V κ (a) is called the universal affine vertex algebra associated with (a, κ). Let us describe the vertex algebra structure in more details. Set x (n) = xt n, x a, n Z, and let 0 be the image of 1 1 U(â) C in V κ (a). Let (x i ; i = 1..., dim a) be an ordered basis of a. By the PBW Theorem, V κ (a) has a basis of the form x i1 (n 1)... xim (n m) 0, where n 1 n 2 n m < 0, and if n j = n j+1, then i j i j+1. Then (V κ (a), 0, T, Y ) is a vertex algebra where the translation operator T is given by T 0 = 0, [T, x i (n) ] = nxi (n 1), for n Z, and the vertex operators are defined inductively by: 0 (z) = Id V κ (a), x i ( 1) 0 (z) = xi (z) = n Z x i (n) z n 1, (x i1 (n... 1) xim (n 0 )(z) m) 1 = ( n 1 1)!... ( n m 1)! : n1 1 z x i1 (z)... z nm 1 x im (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, then the fields : a(z)b(z) : and c(z) are also mutually local. Let i, j {A,..., d}. Then [x i (z), x j (w)] = [x i (n), xj (m) ]z n 1 w m 1 n,m Z = n,m Z[x i, x j ] (n+m) z n 1 w m 1 + nκ(x i, x j )z n 1 w n 1 n Z = ( ) [x i, x j ] (l) z n 1 w n w l 1 + κ(x i, x j ) nz n 1 w n 1 l Z n Z n Z = [x i, x j ](w)δ(z w) + κ(x i, x j ) w δ(z w).

20 20 2. VERTEX ALGEBRAS, DEFINITIONS, FIRST PROPERTIES AND EXAMPLES Then it follows that for any i, j, (z w) 2 [x i (z), x j (w)] = 0, so the locality axiom holds for these fields. Remark 2.6. (1) In fact, the equality [x i (z), x j (w)] = [x i, x j ](w)δ(z w) + κ(x i, x j ) w δ(z w) is equivalent to the commutation relations in the Lie algebra â. (2) Using the commutation relations, one can check directly on this example the OPE formula with N = 2: N 1 [x i (z), x j (w)] = (x i (n) xj )(w) 1 n! n wδ(z w). n=0 When a = g is the simple Lie algebra as in Part 1, so that â = ĝ is the affine Kac-Moody algebra, and κ = k( ) = k 2h ( ) Kil, for k C, with κ g the Killing form of g, then we write V k (g) for the universal affine vertex algebra vertex algebra V κ (a). By what foregoes, V k (g) has a natural vertex algebra structure, and it is called the universal affine vertex algebra associated with ĝ of level k. When a = C is one-dimensional with κ any non-degenerate bilinear form, then V κ (a) is the Heisenberg vertex algebra The Virasoro vertex algebra. Let V ir = C((t)) t CC be the Virasoro Lie algebra, with the commutation relations (14) (15) [L n, L m ] = (n m)l n+m + n3 n δ n+m,0 C, 12 [C, V ir] = 0, where L n := t n+1 t for n Z. Let c C and define the induced representation Vir c = Ind V ir C[[t]] t CCC c = U(V ir) C[[t]] t CC C c, where C acts as multiplication by c and C[[t]] t acts by 0 on the 1-dimensional module C c. By the PBW Theorem, Vir c has a basis of the form L j1... L jm 0, j 1 j l 2, where 0 is the image of 1 1 in Vir c. Then (Vir c, 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) = Id Vircc, Y (L 2 0, z) =: L(z) = n Z L n z n 2, Y (L j1... L jm 0, z) 1 = ( j 1 2)!... ( j m 2)! : j1 2 z L(z)... z jm 2 L(z) :

21 2.2. FIRST EXAMPLES OF VERTEX ALGEBRAS 21 Moreover, Vir c is Z 0 -graded by deg 0 = 0 and deg L n 0 = n Conformal vertex algebras. A Hamiltonian of V is a semisimple operator H on V satisfying for all a V, n Z. [H, a (n) ] = (n + 1)a (n) + (Ha) (n) Definition 2.7. A vertex algebra equipped with a Hamiltonian H is called graded. 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 (16) for homogeneous elements a, b V. a (n) b V a+ b n 1 For example, the universal affine vertex algebra V k (g) is Z 0 -graded (that is, V k (g) = 0 for Z 0 ) and the Hamiltonian H is defined letting H acts on V k (g) as Id V k (g) for any Z 0. Definition 2.8. A graded vertex algebra V = C V is called conformal of central charge c C if there is a conformal vector ω V 2 such that the Fourier coefficients L n of the corresponding vertex operators Y (ω, z) = n Z L n z n 2 satisfy the defining relations (14) of the Virasoro algebra with central charge c, and if in addition we have ω (0) = L 1 = T, ω (1) = L 0 = H i.e., L 0 V = Id V Z. A Z-graded conformal vertex algebra is also called a vertex operator algebra. Example 2.9. The Virasoro vertex algebra Vir c is clearly conformal with central charge c and conformal vector ω = L 2 0. Example The universal affine vertex algebra V k (g) has a natural conformal vector, provided that k h. Set S = 1 2 dim g i=1 x i,( 1) x i ( 1) 0, where (x i ; i = 1,..., dim g) is the dual basis of (x i ; i = 1,..., dim g) with respect to the bilinear form ( ), and x i (z) = n Z x i (n) z n 1, x i (z) = n Z x i,(n) z n 1. Exercise For k h, show that L = S is a conformal vector of k + h V k (g), called the Segal-Sugawara vector, with central charge (cf. [Frenkel-BenZvi, 3.4.8]). c(k) = k dim g k + h.

22 22 2. VERTEX ALGEBRAS, DEFINITIONS, FIRST PROPERTIES AND EXAMPLES (17) Then we have [L m, x (n) ] = (m n)x (m+n) x g, m.n Z Modules over vertex algebras Definition. A module over the vertex algebra V is a vector space M together with a linear map which satisfies the following axioms: (18) (19) (20) 0 (z) = Id M, V F (M), a a M (z) = n Z a M (n) z n 1, (T a) M (z) = z a M (z), ( ) m (a j (n+j) b) M (m+k j) j 0 = ( ) ( 1) j n (a M j (m+n j) bm (k+j) ( 1)n b M (n+k j) am (m+j) ). j 0 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 of H on M, that is, M = d C M d, where M d = {m M Hm = dm} and [H, a M (n) ] = (n + 1)aM (n) + (Ha)M (n) for all a V. We have (21) a M (n) M d M d+ a n 1 for homogeneous a V. When there is no ambiguity, we will often denote by a (n) the element a M (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. A module whose only submodules are 0 and itself is called simple Modules of the universal affine vertex algebra. In the case that V is the universal affine vertex algebra V k (g) associated with ĝ at level k C, V -modules play a crucial important role in the representation theory of the affine Kac-Moody algebra ĝ. A ĝ-module M of level k is called smooth if x(z) is a field on M for x g, that is, for any m M there is N > 0 such that (xt n )m = 0 for all x g and n N. Any V k (g)-module M is naturally a smooth ĝ-module of level k. Conversely, any smooth ĝ-module of level k can be regarded as a V k (g)-module. It follows that a V k (g)-module is the same as a smooth ĝ-module of level k. Namely, we have the following. Proposition 2.12 (See [Frenkel-BenZvi, ] for a proof). There is an equivalence of category between the category of V k (g)-modules and the category of smooth ĝ-modules of level k.

23 PART 3 Poisson vertex algebras, arc spaces, and associated varieties 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] Definitions. Denote by Sch the category of schemes of finite type over C. Let X be an object of this category, and m Z 0. Definition 3.1. An m-jet of X is a morphism Spec C[t]/(t m+1 ) X. The set of all m-jets of X carries the structure of a scheme J m (X), called the m-th jet scheme of X. It is a scheme of finite type over C characterized by the following functorial property: for every scheme Z over C, we have Hom Sch (Z, J m (X)) = Hom Sch (Z Spec C Spec C[t]/(t m+1 ), X). The C-points of J m (X) are thus the C[t]/(t m+1 )-points of X. From Definition 3.1, we have for example that J 0 (X) X and that J 1 (X) TX where TX denotes the total tangent bundle of X. The canonical projection C[t]/(t m+1 ) C[t]/(t n+1 ), m n, induces a truncation morphism π X,m,n : J m (X) J n (X). The canonical injection C C[t]/(t m+1 ) induces a morphism ι X,m : X J m (X), and we have π X,m ι X,m,0 = id X. Hence ι X,m is injective and π X,m,0 is surjective. Define the (formal) disc as D := Spec C[[t]]. The projections π X,m,n yield a projective system {J m (X), π X,m,n } m n of schemes. Definition 3.2. Denote by J (X) its projective limit in the category of schemes, J (X) = lim J m (X). It is called the arc space, or the infinite jet scheme of X. Thus elements of J (X) are the morphisms and for every scheme Z over C, γ : D C[[t]], Hom Sch (Z, J m (X)) = Hom Sch (Z Spec C D, X), 23

24 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}. In other words, the contravariant functor Sch Set, Z Hom Sch (Z D, X) is represented by the scheme J (X). The reason why we need the completion Z D in the definition is that, for A an algebra, A C[[t]] A[[t]] = A C[[t]] in general. We denote by π X, the canonical projection: π X, : J (X) X The affine case. In the case where X is affine, we have the following explicit description of J (X). (We describe similarly J m (X).) Let N Z >0 and X C N be an affine subscheme defined by an ideal I = f 1,..., f r of C[x 1,..., x N ]. Thus X = Spec C[x 1,..., x N ]/I. For f C[x 1,..., x N ], we extend f as a map from C[[t]] N to C[[t]] via base extension. Then giving a morphism γ : D X is equivalent to giving a morphism γ : C[x 1,..., x N ]/I C[[t]], or to giving such that for any k = 1,..., r, γ (x i ) = j 0 γ i ( j 1) tj, i = 1,..., N, f k (γ (x 1 ),..., γ (x N )) = 0 in C[[t]]. For any f C[x 1,..., x N ], there exist functions f (j), j 0, which only depend on f, in the variables γ = (γ 1 ( j 1),..., γn ( j 1) ) j 0 such that (22) f ( γ (x 1 ),..., γ (x N ) ) = j 0 f (j) (γ) t j. j! Regarding the coordinates x i as functions over C N, we set: x i ( j 1) := (xi ) (j), that is, x i ( j 1) (γ) = j!γi ( j 1), for i = 1,..., N. The jet scheme J (X) is then the closed subscheme in Spec C[x i ( 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) = Spec C[x i (j) ( j 1) ; i = 1,..., N, j 0]/ f k ; k = 1,..., r, j 0. In particular, if X is an N-dimensional vector space, then J (X) = Spec C[x i ( j 1) ; i = 1,..., N, j 0], and for m Z 0, the projection J (X) J m (X) corresponds to the projection onto 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[x i ( j 1) ; i = 1,..., N, j 0] by T x i ( j) = jxi ( j 1), j 0. Then f (j) k = T j f k for k = 1,..., r and j 0. Here we identify x i with x i ( 1).

25 3.1. JET SCHEMES AND ARC SPACES 25 With the above lemma, we conclude that for the affine scheme X = Spec R, with R = C[x 1, x 2,, x N ]/ f 1, f 2,, f r, its arc space J X is the affine scheme Spec(J (R)), where J (R) := C[xi ( j) ; i = 1, 2,, N, j 0] T j f i ; 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 affine scheme X = Spec R, 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 following sense. We have a C-algebra homomorphism j : R J (R) such that if (A, ) is another differential algebra, and if f : R A is a C-algebra homomorphism, then there is a unique differential algebra homomorphism h: J (R) A making the following diagram commutative. R j (J (R), T ) f (A, ) h (The map h is a differential algebra homomorphism means that it is a C-algebra homomorphism 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, J m (U) = π 1 X,m (U). Then for a general scheme Y of finite type with an affine open covering {U i } i I, its arc space J (Y ) is obtained by glueing J (U i ) (see [Ein-Mustata09, Ishii11]). In particular, the structure sheaf O J (Y ) is a sheaf of commutative vertex algebras. The natural projection π X, : J (X) X corresponds to the embedding R J (R), x i x i ( 1) in the case where X = Spec R is affine. In terms of arcs, π X, (α) = α(0) for α Hom Sch (D, X), where 0 is the unique closed point of the formal disc D Basic properties. The map from a scheme to its jet schemes and arc space is functorial. If f : X Y is a morphism of schemes, then we naturally obtain a morphism J m f : J m (X) J m (Y ) making the following diagram commutative, J m (X) Jmf J m (Y ) π X,m,0 X f Y π Y,m,0 In terms of arcs, it means that J m f(α) = f α for α J m (X). This also holds for m =. We have also the following for m Z 0 { } and for every schemes X, Y, (23) J m (X Y ) = J m (X) J m (Y ).

26 26 3. POISSON VERTEX ALGEBRAS, ARC SPACES, AND ASSOCIATED VARIETIES Indeed, for any scheme Z in Sch, Hom(Z, J m (X Y )) = Hom(Z SpecC C[t]/(t m+1 ), X Y ) = Hom(Z SpecC C[t]/(t m+1 ), X) Hom(Z SpecC C[t]/(t m+1 ), Y ) = Hom(Z, J m (X)) Hom(Z, J m (Y )) = Hom(Z, J m (X) J m (Y )). For m =, just replace C[t]/(t m+1 ) with C[[t]] and take the completion in the product Z Spec C[[t]] = Z D. If A is a group scheme over C, then J m (A) is also a group scheme over C. Moreover, by (23), if A acts on X, then J m (A) acts on J m (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: [xt m, yt n )] = [x, y]t m+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 Lie algebras, we have g = Lie(J (G)). The adjoint action of G on g induces an action of J (G) on g, and the coadjoint action 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 j f = f (j), j 0, are elements of C[g ] J (G). In particular, the arc space J (N ) of the nilpotent cone is the subscheme of g defined by the equations T j P i, i = 1..., r and j 0, if P 1,..., P r are homogeneous generators of C[g] G, that is, J (N ) = Spec C[g ]/(T j P i ; i = 1..., r, j 0) Geometrical results. So far, we have stated basic properties common for both jet schemes J m (X) and the arc space J (X). For the geometry, arc spaces behave rather differently. The main reason is that C[[t]] is a domain, contrary to C[t]/(t m+1 ). Thereby the geometry of are spaces is somehow simpler. However, although J m (X) is of finite type if X is, this is not anymore true for J (X), and its coordinate ring C[J (X)] is not noetherian in general. Lemma 3.8. Denote by X red the reduced scheme of X. The natural morphism X red X induces an isomorphism J X red (J X) red of topological spaces. Proof. We may assume that X = Spec R. An arc α of X corresponds to a ring homomorphism 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 X red. Similarly, if X = X 1... X r, where all X i are closed in X, then J (X) = J (X 1 )... J (X r ). (Note that Lemma 3.8 is false for the schemes J m (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.

27 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 irreducible. Theorem 3.10 is false for the jet schemes J m (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 of the jet schemes J m (X), m Z 0, and the singularities of X Poisson vertex algebras Let V be a commutative vertex algebra (cf ), or equivalently, a differential algebra. Recall that this means: a (n) = 0 in End(V ) for all n Definition. A commutative vertex algebra V is called a Poisson vertex algebra if it is also equipped with a linear operation, such that (24) V Hom(V, z 1 V [z 1 ]), (T a) (n) = na (n 1), a a (z), (25) (26) a (n) b = j 0( 1) n+j+1 1 j! T j (b (n+j) a), [a (m), b (n) ] = j 0 ( ) m (a j (j) b) (m+n j), (27) a (n) (b c) = (a (n) b) c + b (a (n) c) 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. (Do not 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.) Poisson vertex structure on arc spaces. Theorem 3.11 ([Arakawa12, Proposition 2.3.1]). Let X be an affine Poisson scheme, that is, X = Spec R for some Poisson algebra R. Then there is a unique Poisson vertex algebra structure on J (R) = C[J (X)] such that { {a, b} if n = 0 a (n) b = 0 if n > 0, for a, b R. Proof. The uniqueness is clear by (10) since J (R) is generated by R as a differential algebra. We leave it to the reader to check the well-definedness. Since J (R) is generated by R, the formula a (n) b = δ n,0 {a, b} for a, b R is sufficient to define the fields on J (R) by formulas (24), whence the existence.

28 28 3. POISSON VERTEX ALGEBRAS, ARC SPACES, AND ASSOCIATED VARIETIES Remark More generally, let X be a Poisson scheme which is not necessarily affine. Then the structure sheaf O J (X) carries a unique Poisson vertex algebra structure such that f (n) g = δ n,0 {f, g} for f, g O X O J (X), see [Arakawa-Kuwabara-Malikov, Lemma ]. Example Recall that the affine space g is a Poisson variety by the Kirillov-Kostant-Souriau Poisson structure. If {x 1,..., x N } is a basis of g, then Thus (28) C[g ] = C[x 1,..., x N ]. J (g ) = Spec C[x i ( n) ; i = 1,..., N, n 1]. So we may identify C[J (g )] with the symmetric algebra S(g[t 1 ]t 1 ) via x ( n) 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 algebra J (g) = g[[t]] acts on C[J (g )]. This action coincides with that obtained by differentiating the action of J (G) = G[[t]] on J (g ) induced by the coadjoint action of G (see Example 3.6). In other words, the Poisson vertex algebra structure of C[J (g )] comes from the J (G)-action on J (g ) Canonical filtration and Poisson vertex structure. Our second basic example of Poisson vertex algebras comes from the graded vertex algebra associated 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 p V be the subspace of V spanned by the elements a 1 ( n 1 1) a2 ( n 2 1) ar ( n r 1) 0 with a 1, a 2,, a r V, n i 0, n 1 + n n r p. Then V = F 0 V F 1 V.... It is clear that T F p V F p+1 V. Set (F p V ) (n) F q V := span C {a (n) b ; a F p V, b F q V }. Note that F 1 V = span C {a ( 2) b a, b V } = C 2 (V ). Lemma We have F p V = j 0(F 0 V ) ( j 1) F p j V. Proposition (1) (F p V ) (n) (F q V ) F p+q n 1 V. Moreover, if n 0, we have (F p V ) (n) (F q V ) F p+q n V. Here we have set F p V = V for p < 0. (2) The filtration F V is separated, that is, p 0 F p V = {0}, if V is a positive energy representation, i.e., positively graded over itself.

29 3.3. ASSOCIATED VARIETY OF A VERTEX ALGEBRA 29 Exercise The verifications are straightforward and are left as an exercise. Definition A vertex algebra V is called finitely strongly generated if there exist finitely many elements a 1,..., a r in V such that V is spanned by the elements of the form a i1 ( n... 1) ais ( n 0 s) with s 0, n i 1. For example, the universal affine vertex algebra and the Virasoro vertex algebra are strongly finitely generated. In this note we always assume that a vertex algebra V is finitely strongly generated. We will assume that V is conformal and positively graded, V = n Z 0 V n, so that the filtration F V is separated. Set gr F V = p 0 F p V/F p+1 V. We denote by σ p : F p V F p V/F p+1 V, for p 0, the canonical quotient map. When the filtration F is obvious, we often denote simply by gr V the space gr F V. Proposition 3.18 ([Li05]). The space gr F V is a Poisson vertex algebra by σ p (a) σ q (b) := σ p+q (a ( 1) b), T σ p (a) := σ p+1 (T a), σ p (a) (n) σ q (b) := σ p+q n (a (n) b), for a F p V \ F p n V, b F q V, n > 0. Set R V := F 0 V/F 1 V = V/C 2 (V ) gr V. Definition The algebra R V is called the Zhu s C 2 -algebra of V. The algebra structure is given by: (30) where ā = σ 0 (a). ā b := a ( 1) b, Proposition 3.20 ([Zhu96, Li05]). The restriction of the vertex Poisson structure on gr F V gives to the Zhu s C 2 -algebra R V a Poisson algebra structure, that is, R V is a Poisson algebra by where ā = σ 0 (a). ā b := a ( 1) b, {ā, b} = a (0) b, Proof. It is straightforward from Proposition Associated variety of a vertex algebra We now in a position to define the main object of study of the lecture.

30 30 3. POISSON VERTEX ALGEBRAS, ARC SPACES, AND ASSOCIATED VARIETIES X V Associated variety and singular support. Definition Define the associated scheme X V and the associated variety of a vertex algebra V as X V := Spec R V, X V := Specm R V = ( X V ) red. It was shown in [Li05, Lemma 4.2] that gr F V is generated by the subring R V as a differential algebra. Thus, we have a surjection J (R V ) gr F V of differential algebras by Remark 3.4 since R V generates J (R V ) 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]). The identity map R V R V induces a surjective Poisson vertex algebra homomorphism J (R V ) = C[J ( X V )] gr F V. Definition Define the singular support of a vertex algebra V as SS(V ) := Spec(gr F V ) J ( X V ). Theorem We have dim SS(V ) = 0 if and only if dim X V = 0. Proof. The only if part is obvious since π XV, (SS(V )) = X V. The if part follows from Corollary The lisse condition. A vertex algebra V is called lisse (or C 2 -cofinite) if R V = V/C 2 (V ) is finite dimensional. Thus by Theorem 3.24 we get: Lemma The vertex algebra V is lisse if and only if dim X V = 0, that is, if and only if dim SS(V ) = 0. Remark Suppose that V is Z 0 -graded by some Hamiltonian H, i.e., V = i 0 V i with V i = {x V Hx = ix}, and that V 0 = C 0. Then gr F V and R V are equipped with the induced grading: gr F V = i 0(gr F V ) i, (gr F V ) 0 = C, R V = i 0(R V ) i, (R V ) 0 = C. So the following conditions are equivalent: (1) V is lisse, (2) X V = {point}, (3) the image of any vector a V i for i 1 in gr F V is nilpotent, (4) the image of any vector a V i for i 1 in R V is nilpotent. Thus, lisse vertex algebras can be regarded as a generalization of finite-dimensional algebras Comparison with weight-depending filtration. Let V be a vertex algebra 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].