FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS

Transcription

1 FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS GIORGIA FORTUNA Our goal for today will be to introduce the concept of Chiral Algebras and relate it to the one of quasi-conformal vertex algebras. Vaguely speaking, a quasi-conformal vertex algebra will be in particular a vector space V with a structure of (DerO, AutO)-module. This structure will allow us to define a left D X -module V on every curve X, and the extra data that V carries, will endow V r := V Ω with a structure of a chiral algebra. 1. Chiral algebras Let X be a smooth curve over C. A unital chiral algebra is a right D X - module A, equipped with a D X -module homomorphism µ : j j (A A)! (A) where j : X X (X) X X X :, and an embedding satisfying the following conditions: i : Ω X A (skew-symmetry) µ = σ 12 µ σ 12. (Jacobi identity) µ 1{23} = µ {12}3 + µ 2{13}. (unit) The following diagram commutes: j j (Ω X A) j j (A A)! (A) id! (A) where the vertical map on the left comes from the sequence Ω X A j j (Ω X A)!! (Ω X A)[1]! (A) and σ 12 is the induced action on A by permuting the variables of X 2. The Jacobi identity above means the following: if we denote by j 123 the inclusion of the subset of X 3 where all the x i s are different and by ij the Date: March 12,

2 2 GIORGIA FORTUNA inclusion of the diagonal x i = x j, then µ 1{23} : j j (A A A)! (A) is defined as the composition j j 123(A A A) µ j j x 1 x 2, x 1 x 3 (A (23)! A) (23)! j j x 1 x 3 (A A) µ (123)! (A), the map µ {12}3 is the composition j j 123(A A A) µ j j x 1 x 3, x 2 x 3 ( (12)! A A) (12)! (j j x 2 x 3 (A A)) µ (123)! (A), and the map µ 2{13} is gives as j j 123(A A A) µ j j x 2 x 3, x 1 x 2 ( (13)! A A) (13)! j j x 2 x 3 (A A) µ (123)! (A). The Jacobi identity means that, as a map taking place on X 3, the alternating sum of the above maps is zero. There are many examples of chiral algebras that we already know, for instance Ω X itself is a chiral algebra (see Example below). Another important class of chiral algebras is given by commutative chiral algebras. Example 1.1. Consider the right D X -module Ω X. As we said it has a structure of a chiral algebra. In fact in this case we have Ω X Ω X Ω X 2 under the natural map that in coordinates looks like dz dw dz dw. Also! (Ω X ) consists of sections of Ω 2 X 2 with arbitrary poles on the diagonal ( denote by Ω 2 X 2 ( )) modulo sections that are regular. The chiral operation µ corresponds to the composition of the identification j j (Ω X Ω X ) Ω 2 X 2 ( ) with the projection onto! (Ω X ) µ(f(z, w) dz dw) = f(z, w)dz dw (mod Ω X 2). The Jacobi identity is a consequence of the Cousin complex for X 3 with stratification given by the various diagonals. When we consider the sheaf Ω 3 X, it says that the alternating sum of principal parts of meromorphic 3 expressions on the diagonal divisors 12, 13 and 23 corresponds to zero in the space of delta-functions 123 Ω X if and only if they are principal parts of a single meromorphic expression on X 3. When we compute the Jacobi identity for Ω X, the three maps coincide with the three different compositions j j 123Ω 3 X 3 j j x j x k (ij)! Ω 2 X 2 (123)! Ω X, for i, j, k {1, 2, 3}. Hence the alternating sum of them is indeed zero. Example 1.2. Recall that a D X -scheme is the spectrum of a D X -algebra where a D X -algebra is a commutative algebra with a structure of left D X - module. For example Sym(M) is one of them for every left D X -module M. Recall moreover that for every D X module M and inclusions i : Y X X Y : j (with i a closed embedding) we have an exact triangle i! i! (M) M j j (M).

3 FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS 3 Take now the maps j : X X (X) X X X :, and the D X 2-module M M. Then we have the following exact sequence (1) M M j j (M M)!! (M M). It is not hard to check that: If M is a left D X -module, then! (M M) M M. If M is a right D X -module then! (M M) M! M, where M! M := M M Ω 1 X. Let B be a chiral algebra. B is called commutative if the composition B B j j (B B)! (B) vanishes, where the last map is given by the chiral bracket. This implies (and is in fact equivalent ) that µ factors through a map µ j j (B B)! (B) i.e. we obtain a map! (B! B) B! B B. This map yields a commutative product (because of the skew-symmetry) B l B l m B l, making B l a D X -algebra. On the other hand, if we are given a D X -algebra B l we can consider B := (B l ) r and the composition j j (B B) = j j (B l B l ) j j (Ω X Ω X )!! (B l B l )! (Ω X ) = =! (B l B l )! (Ω X ) m id! (B l )! (Ω X ) =! (B) where m is the product map of B l. This is a chiral operation on B. Hence we have established the following equivalence between {D X algebras B l } {Commutative chiral algebras B} Remark 1. Note that in the above discussion, if we take the D X -algebra to be O X then the resulting chiral algebra is Ω X. 2. Vertex Algebras As we said before, the reason why we want to know what a quasi-conformal vertex algebra consist of is because it is possible to construct a chiral algebra out of it. However, for now, it is enough to know that a vertex algebra is just a vector space V with some additional structure. Hence the first question should be: how do we get a D X -module from a vector space? We clearly need something more, and a quasi-conformal vertex algebra includes in the additional structure exactly what we need for our purpose, i.e. an action of DerO = C[[z]] z such that the action of Der 0 O = zc[[z]] z can

4 4 GIORGIA FORTUNA be exponentiated to an action of the group of automorphisms of the disc D x = SpecC[[z]] (DerO, AutO)-modules and D X -modules. Let O denote the complete topological C-algebra C[[z]], and let AutO be the group of automorphism of O fixing the closed point (z). Clearly any automorphism can be represented by ρ(z) = a 1 z +, with a 1 0. Consider the following Lie groups and Lie algebras: Aut + O = {z + a 2 z 2 + } AutO = {a 1 z + a 2 z 2 + } Der + O = z 2 C[[z]] z Lie(AutO) = Der 0 O = zc[[z]] z DerO = C[[z]] z and denote by G m the multiplicative group. Then AutO is a semi-direct product of G m and Aut + O. A quasi-conformal vertex algebra V carries an action of DerO satisfying the following: The action of Lie(G m ) C z z is diagonal with integral eigenvalues. The action of Der + (O) is locally nilpotent. These two conditions exactly mean that the action of Der 0 O can be exponentiated to an action of AutO. Now let X be a smooth curve, and x a point of X. Denote by O x the completion of the local ring at x and by m x its maximal ideal. Let K x be the field of fractions of O x. Denote by D x (resp. D x ) the disc (resp. the punctured disc). The choice of a formal coordinate (i.e. a generator for m x ) is the same as an isomorphism between O x and C[[z]]. Denote by Coord x the set of all formal coordinates at x. The group AutO acts naturally on Coord x, making it into an AutO-torsor. Given our quasi-conformal vertex algebra V, denote by V x the Coord x -twist of V : V x = Coord x AutO V. This is the fiber of a bundle over D x. However we would like to construct an AutO-bundle over the entire curve X. To do this, consider the set of pairs (x, z x ), where z x is a formal coordinate at x. One can show that this is the set of points of a scheme Coord X over X, moreover the projection Coord X X is a principal AutO-bundle. The fiber of this bundle at x is the AutO-torsor Coord x. Denote by V X the bundle V X = Coord X AutO V. By now it is clear that we didn t use the action of DerO at all. All we needed was an action of AutO on V. The fact that in the case of a quasi-conformal vertex algebra this action comes from an action of DerO allows us to define a connection on V X. This should be very familiar to you, since we are in the situation of a Harish-Chandra module V for the pair (DerO, AutO). We already know (or at least we should) that we can get a D X -module from

5 FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS 5 it. However, let s recall the construction of the connection (see [1] 16.1 for more explanations). For reasons that will be clear later let s write L n = z n+1 z. So far we only used the AutO-action on V. Now we will use the action of L 1 = z to define a connection on V. Consider a point (x, z x ) Coord X, where x X and z x is a formal coordinate around x. Note that the tangent space to Coord X at (x, z x ) is isomorphic to DerO. Thus we obtain a map α α : F un(coord X ) DerO V ect(coord X ) such that the restriction of α to Der 0 O coincides with the map corresponding to the action of AutO. Consider the trivial vector bundle Coord X V over Coord X. We can define a connection on this bundle in the following way: α(ξ) = α(ξ) ξ. Now we use the fact that the action of DerO extends the action of Der 0 O to say that this connection is AutO invariant, hence it is well defined on the quotient V X = Coord X AutO V over X Coord X /AutO (automatically flat since X is a curve). If we want to write this connection explicitly, then it is given by = d + L 1 dz Definition of the vertex algebra. In the previous paragraph we understood why it is so important to have an action of DerO on a vertex algebra. However we still don t know what a vertex algebra is. Unfortunately, as we had been told, the actual definition looks meaningless at a first glance. Hence we are going to see one example first that hopefully will motivate the actual definition of vertex algebras. The Virasoro Lie algebra Consider DerK x = C((t)) t and write L n = t n+1 t. The infinite dimensional Lie algebra V ir is defined as the central extension with bracket given by 0 CC V ir DerK x 0 [L n, L m ] = (n m)l n+m + n3 n δ n, m C, 12 where ω(l n, L m ) = δ n, m n 3 n 12 is the generator of H 2 (DerK x, C) C. Note that DerO = C[[t]] t = {L n, n 1 } is the Lie subalgebra of infinitesimal changes of variable on the disc. Sometimes it is useful to write all these operators in a compact way T (z) = n Z L n z n 2. A large class of vertex algebras comes from infinite dimensional Lie algebras as induced representations from a one dimensional vector space. Our first example of vertex algebras will be exactly of this kind. Consider the one dimensional representation C c of DerO CC, where the

6 6 GIORGIA FORTUNA action of DerO is zero and C acts by c C. Define V ir c to be the induced representation: V ir c = Ind V DerO CC ir C c = U(V ir) U(DerO CC) C c. By the PBW theorem it has a basis consisting of monomials of the form L j1 L jm v c, where v c C c and j 1 j 2 j m 2. We can define a gradation on V ir c by setting degl j1 L jm v c = m i=1 j i and degv c =0. Consider Y (L 2 v c, z) := T (z) = L n z n 2 n Z then we can think of it as an element of End(V ir c )[[z ±1 ]]. Why are we saying all this? After we define vertex algebras it will be clear that V ir c is indeed one of them (and doesn t look bad at all). Definition 2.1. A vertex algebra is a collection of data: A Z + -graded vector space V = m=0 V m with dim V m <. A vector 0 > V 0. A linear operator T : V V of degree 1. (vertex operator) A linear operation Y (., z) : V End V [[z ±1 ]] A A (n) z n 1. where, if A V m, dega (n) = n + m 1. These data are subjected to the following: For any A V Y (A, z) 0 > V [[z]] and Y (A, z) 0 > z=0 = A. For any A V [T, Y (A, z)] = z Y (A, z). All fields Y (A, z) are local with respect to each other, i.e. there exist N 0 s.t. (z w) N [Y (A, z), Y (B, w)] = 0. Now let s consider again the vacuum module V ir c. Then we can take v c = 0 > and T = L 1 but we still have to define the vertex operator T (., z). However, it can be shown that the expression given above Y (L 2 v c, z) = L n Z nz n 2 is enough to reconstruct all vertex operators on the entire V ir c, but we won t go into this (see [1] and 3.6.1). What is important to understand instead, is the locality condition. This will be important for understanding the chiral operation. Somehow we are not asking for the bracket to be zero, but we want the result to be supported on the diagonal. How can we express this in concrete terms? Consider the delta function δ(z w) = m Z z m 1 w m. It satisfies (z w) N N 1 w δ(z w) = 0.

7 FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS 7 Let s now compute the bracket [T (z), T (w)]. [T (z), T (w)] = [L n, L m ]z n 2 w m 2 n,m Z = (n m)l n+m z n 2 w n 2 + C n,m n n 3 n z n 2 w n 2 12 = j,l 2lL j w j 2 z l 1 w l 1 + j,l ( j 2)L j w j 3 z l 1 w l + + C l(l 1)(l 2)z l 1 w l 3 = 12 l = 2T (w) w δ(z w) + w T (w)δ(z w) + C 12 3 wδ(z w) and from this we see that (z w) 4 [T (z), T (w)] = 0. Definition 2.2. A vertex algebra V is called conformal, of central charge c, if we are given a non-zero conformal vector ω V 2 such that the Fourier coefficients L V n of the corresponding vertex operator Y (ω, z) = L V n z n 2 n z satisfy the relations of the Virsasoro algebra with central charge c C, and L V 1 = T, LV 0 V n = nid V. If V is a conformal vertex algebra, we actually have an action of the entire V ir c on it. However in order to construct a D X -module from V, we only need an action of the pair (DerO, AutO). Example 2.1. The Virasoro algebra V ir c is clearly conformal with central charge c C and conformal vector ω = L 2 v c. Let V be a conformal vertex algebra. By looking at the expression [T (z), Y (A, w)], where T (z) = n Z LV n z n 2 and A in V, we have [Tn V, Y (A, w)] = ( ) n + 1 Y (L m A, w)w n m. m + 1 m 1 Now, given a vector field v(z) z = n 1 v nz n+1 z DerO, we assign to it the operator v = n 1 v nl V n, and obtain (2) [v, Y (A, w)] = 1 (m + 1)! ( m+1 w v(w))t (L V ma, w). m 1 Definition 2.3. A vertex algebra V is called quasi-conformal if it carries an action of DerO satisfying formula (2) with T = L 1 for any A V. Ar very important example of conformal algebra is given by the vertex algebra associated to the affine Kac-Moody algebra ĝ κ, for κ κ crit.

8 8 GIORGIA FORTUNA The vertex algebra Vκ(g). 0 Consider the Kac-Moody algebra ĝ κ at level κ. Let C κ be the one dimensional representation of g[[t]] C where the action of g[[t]] is zero and C acts by 1. As we did for the Virasoro algebra, we consider the induced representation V 0 κ(g) := Ind U(ĝκ) U(g[[t]] C) C κ U(g t 1 C[t 1 ]) where the last isomorphism is understood as an isom. of vector spaces. Let {J a } a=1,...,dim g be a basis of g and write Jn a = J a t n. Then, by the PBW theorem, the monomials J a 1 n 1 Jn am m v κ, n 1 n 2 n m < 0, and if n i = n i+1 then a i a i+1 form a basis of Vκ(g). 0 Then if we set 0 >= v κ, degj a 1 n 1 Jn am m v κ = i=m i=1 n i, T v κ = 0, [T, Jn] a = njn 1 a and J a (z) := Y (J a 1, z) := n Z J a nz n 1 we obtain a vertex algebra structure on V 0 κ(g) using reconstruction process. The Sugawara construction. When κ κ crit we have a conformal structure on V 0 κ(g) coming from the Sugawara operators. In fact, for non critical values, there are elements {T n, n Z} in U(ĝ κ ) satisfying { [x m, T n ] = Cmx n+m [T m, T n ] = C((m n)t n+m + δ n, m n 3 n 12 dimg) where C was a number depending on κ that is zero iff κ = κ crit. For κ κ crit we can define L n = 1 C T n, and the above shows that {L n } n Z satisfy the Virasoro relations. These elements appear as Fourier coefficients of a vertex operator 1 C Y (S, z) associated to a conformal vector S (V0 κ(g)) 2. If you knew what normal ordering was, you could actually check that the Fourier coefficients of a=dimg 1 1 Y (S, z) = : J a (z)j a (z) : C 2C a=1 are exactly our L n satisfying [L n, J a m] = mj a n+m. Remark 2. For κ = κ crit we don t have these elements any more, but we can still define a structure of a quasi-conformal vertex algebra on V 0 κ crit since DerO acts on both g O x C, ĝ κ and C, thus it induces an action on V 0 κ(g) Remark 3. The above shows that we can construct a bundle with connection V 0 κ(g) := Coord X AutO V 0 κ(g) starting from our affine Kac-Moody algebra ĝ κ for every level κ.

9 FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS 9 3. From Vertex algebras to chiral algebras. In this section we will see how quasi-conformal vertex algebras are related to chiral algebras. For this we ll now recall some basic properties of graded vertex algebras. Let V be a graded vertex algebra, then: For every A, B V we have Y (A, z)b = e zt Y (B, z)a. (OPE for graded vertex algebras) For every A, B and C V we have Y (A, z)y (B, z) = Y (Y (A, z w)b, w) = m Z Y (A (n) B, w) (z w) n+1, where by the right hand side we understand its expansion in the region z > w. (Cousin property) The expressions Y (A, z)y (B, w)c V ((z))((w)), Y (B, w)y (A, z)c V ((w))((z)), and Y (Y (A, z w)b, w)c V ((w))((z w)) are the expansions, in their respective domains, of the same element of V [[z, w]][z 1, w 1, (z w) 1 ] Let V be a quasi-conformal vertex algebra and V X the associated left D X -module with action of z given by z f(z)a = ( z f(z))a + f(z)l 1 A. Recall that a chiral algebra is first of all a right D X -module, hence if we wish to obtain a chiral algebra from V X, we ll have to consider the right D X -module V r X corresponding to it (i.e. Vr X = V X Ω X ). For a left D X -module M, denote by + (M) the module + (M) = j j (O X M)/O X M. Note that, for a right D X -module N, we have! (N) j j (Ω X N)/Ω X N. Remark 4. Let M be a left D X -module and M r the right D X -module corresponding to it, then In fact we have j j (M r M r ) j j (M M) r := j j (M M) Ω 2 X 2. j j (M r M r ) j j (M M) j j (Ω X Ω X ) j j (M M) Ω 2 X 2,

10 10 GIORGIA FORTUNA where the last isomorphism follows from the isomorphism j j (Ω X Ω X ) j j (Ω 2 X 2 ) explained in Example (1.1). Furthermore we also have! (M r ) + (M) r, under the map that in coordinates looks like f(z)dz f(w)m dw (mod Ω X M Ω X ) (f(z) f(w)m) dz dw (mod O X M). Therefore any map F : j j (M M) + (M) will induce a map F r : j j (M r M r )! (M r ). Using the above remark, we will construct a map Y : j j (V X V X ) + (V X ) and the chiral bracket µ on V r X will be given as µ := Yr Definition of the chiral bracket. Let V X be the vector bundle corresponding to a quasi-conformal chiral algebra V. For every point x X, choose a formal coordinate z around X, and use it to trivialize V Dx. Then the sheaf j j (V X V X ), when restricted to D 2 x = SpecC[[z, w]], may be described as the sheaf associated to the C[[z, w]]-module V V [[z, w]][(z w) 1 ]. Similarly, the restriction of the sheaf + (V X ) to D 2 x is associated to the module V [[z, w]][(z w) 1 ]/V [[z, w]] Define a map of O X -modules on D 2 x by the formula Y x : j j (V X V X ) + (V X ) (3) Y x (f(z, w)a B) = f(z, w)y (A, z w)b (mod V [[z, w]]). Since the module + (V X ) is supported on the diagonal, any morphism from a sheaf M to it must be zero away from the diagonal. Hence such a morphis is determined by its restriction to the formal completion of the diagonal. Equivalently it is sufficient to describe such morphism on Dx 2 for every x X. Thus we obtain the following. Proposition 1. The collection of morphisms Y x for x X define a sheaf morphism Y : j j (V X V X ) + (V X ) on X 2. From the discussion above, we automatically get a map µ : j j (V r X V r X)! (V r X). Remark 5. Note that the above map incorporates the chiral bracket on Ω X described in Example 1.1. This bracket was nothing but the canonical projection from j j (Ω X Ω X ) to! (Ω X ) j j (Ω X Ω X )/Ω X Ω X. Recall that it is given by µ ΩX (f(z, w)dz dw) = f(z, w)dx dw (mod Ω 2 X 2 ), and it satisfies σ 12 µ ΩX σ 12 = µ.

11 FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS 11 We finally have the following proposition. Proposition 2. Let V be a quasi-conformal vertex algebra and denote by V X the associated left D X module. Then the map Y : j j (V X V X ) + (M) given by (3) defines a chiral bracket on the right D X -module V r X. Proof. The complete proof can be found in [1]. We will only give a sketch of it. The embedding C 0 > gives rise to an embedding O X V X and hence A unit map Ω X V r X. The anti-commutativity of µ follows from the anti-commutativity of µ ΩX and the commutativity of Y. In fact the map Y satisfies Y σ 12 = σ 12 Y. This can be seen by using the property Y (A, z w)b = e (z w)t Y (B, w z)a. Although the above expression doesn t manifest any symmetry, when we write the map σ 12 Y, we are intrinsically identifying the two D X -modules + (V X ) and σ 12 ( +(V X )). This identification is given by 1 φ (z w) k e φ 1 (z w) w (z w) k (mod V X O X ) = k 1 m=0 m w φ 1 m!(z w) k m, where φ is a local section of V X. Under this identification the map Y satisfies Y σ 12 = σ 12 Y, hence we obtain the anti-commutativity of Y r1 The Jacobi identity follows from the operator product expansion we mentioned at the beginning of this section. In fact µ 1{23} (A B C) is the projection of the series Y (A, z u)y (B, w u)c onto the space 123! (V r X ). Similarly, the other two compositions µ 2{13} and µ {12}3 are projections of the compositions Y (B, w u)y (A, z u)c, Y (Y (A, z w)b, w u)c respectively. However, using the Cousin property (under the substitution z z u, w w u) we have that the three expressions Y (A, z u)y (B, w u)c V ((z u))((w u)), Y (B, w u)y (A, z u)c V ((w u))((z u)), and Y (Y (A, z w)b, w u)c V ((w u))((z w)) are the expansions, in their respective domains, of the same rational function on X 3 with poles only on the diagonals. By the same argument we used in Example (1.1) the alternating sum of the above three maps sum to zero. 1 the DX 2-module structure on j j (V X V X) is given by the following: z (f(z, w)a b) = zf(z, w)a b+f(z, w)l 1a b and w (f(z, w)a b) = wf(z, w)a b+f(z, w)a L 1b and the one on! (V X) is given by. w (f(z, w)a) = wf(z, w)a + f(z, w)l 1a and z (f(z, w)a) = zf(z, w)a

12 12 GIORGIA FORTUNA References [1] E. Frenkel, D. Ben-Zvi Vertex algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, American Mathematical Society.