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Classification of transitive vertex algebroids

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Classification of transitive vertex algebroids

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Content CLASSIFICATION OF TRANSITIVE VERTEX ALGEBROIDS by Dmytro Chebotarov A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2011 Copyright 2011 Dmytro Chebotarov Table of Contents Abstract iii Introduction 1 Chapter 1: Preliminaries 9 1.1 Denitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Vertex algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 2: Classication of transitive vertex algebroids 20 2.1 Gerbes and torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 3: Twisted CDO and their deformations 35 3.1 Twisted chiral dierential operators . . . . . . . . . . . . . . . . . . . . . . 35 3.2 A deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Deformed TCDO on ag varieties . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Embedding of the ane vertex algebra . . . . . . . . . . . . . . . . . . . . . 52 Chapter 4: Half-integrable modules over transitive vertex algebroids 63 4.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Bibliography 79 ii Abstract In this dissertation we accomplish the following: We present a classication of transitive vertex algebroids on a smooth varietyX carried out in the spirit of Bressler's classication of Courant algebroids. In particular, we compute the class of the stack of transitive vertex algebroids. We dene deformations of sheaves of twisted chiral dierential operators (TCDO) introduced in [AChM] and use the classication result to describe and classify such deformations. As a particular case, we obtain a localization of Wakimoto modules at non-critical level on ag manifolds. We study representation theory of TCDO and their deformations. In particular, we show an equivalence between certain categories of modules over (deformed) TCDO and categories of twisted D-modules, thus extending a result of [AChM]. iii Introduction A vertex algebra consists of the following data: a graded vector space V = L 1 i=0 V i - the space of states a distinguished vectorj0i2V 0 called vacuum vector operations (n) :V i V j !V i+jn1 , (a;b)7!a (n) b. satisfying a number of axioms, most notably the Borcherds identity: X j0 m j (a (n+j) b) (m+kj) c = X j0 (1) j n j fa (m+nj) b (k+j) c(1) n b (n+kj) a (m+j) cg; a;b;c2V This notion, introduced by Richard Borcherds, captures the algebraic structure present in a 2-dimensional conformal quantum eld theory, and at the same time has connection to representations of nite groups, Lie algebras, modular forms and number theory. Many vertex algebras are \freely" generated by their subspace V 0 V 1 . The structure that vertex algebra axioms impose on the subspace V 0 V 1 is encapsulated in the notion of a vertex algebroid. It is much simpler than that of vertex algebra and, moreover, is of geometric nature. In fact, a vertex algebroid is an extension of the algebra of vector elds by 1-forms. More precisely, it comprises the following: 1) A =V 0 is a commutative associative algebra with respect to (1) ; 2) =A (1) @A is an A-module via a! =a (1) ! and the translation map @ :A! is a derivation; 1 3) T =V 1 = is a Lie algebra with bracket (0) and a left A-module via (1) ; 4) is a T -module with the action induced by (0) ; 5) the map (0) :TA!A denes an action of T on A by derivations. 6) the maps (1) : T ! A and (1) : T ! A are A-bilinear pairings that satisfy (1) ! =! (1) and are determined by (1) @a = (0) a. For example one can takeA the ring of functions on a manifoldX (smooth, algebraic or complex), T =T A the Lie algebra of vector elds on X, = 1 X the module of dierential 1-forms. The result is the vertex algebroid underlying chiral dierential operators (CDO) . CDOs are sheaves of vertex algebras that resemble algebras of dierential operators in some respects. E.Witten and A.Kapustin related CDOs to the half-twisted sigma model in perturbative regime. The character of the cohomology of the super-version of CDO, known as chiral de Rham complex, coincides with Witten's elliptic genus of a manifold (Borisov-Libgober). Classication of vertex algebroids. One striking dierence between sheaves of CDO and their classical prototype is that for some manifolds X no CDO exists; or if there is one, there may be more than one isomorphism class of such sheaves. Speaking in technical language, sheaves of CDO form a stack, whose groupoid of global sections may be empty or have more than one connected component. In [GMS1] the classication of chiral dierential operators was obtained; in particular, it was established that a global sheaf of CDO exists on X if and only ifch 2 ( 1 X ) = 0 where ch 2 ( 1 X ) is the second graded piece of Chern character of 1 X . This result was re-established by Bressler [Bre] in a rather unexpected fashion. He noticed that the notion of a vertex algebroid is related to a well-known notion in dierential geometry, a Courant algebroid: the latter is a quasi-classical limit of the former. He obtained a classication of Courant algebroids extending a xed Lie algebroid and rediscovered the 2 aforementioned obstruction by connecting the existence of a CDO on X with the existence of certain Courant extensions of the Atiyah algebra of the sheaf of 1-forms. In both these classication problems the obstruction to global existence is a class in H 2 (X; 2 ! 3;cl ): This is due to a rather remarkable property of these algebroids: one can "twist" an algebroidA on U X by a closed 3-form . To be more precise, let us denote byVExt h;i L (resp.CExt h;i L ) the stack of vertex (resp. Courant) algebroid extensions of a given Lie algebroidL !T X with an invariant pairingh;i on ker (cf. section 1.2.3). Then the twisting by 3-form action onVExt h;i L andCExt h;i L extends to an action of a certain stack associated to the complex 2 X ! 3;cl X (cf. [D]) and it makes each of those a torsor over the latter. By standard abstract nonsense, to every such stackS there corresponds a class cl(S)2H 2 (X; 2 ! 3;cl ) which vanishes precisely whenS has a global object. For example, the obstruction ch 2 ( 1 X ) above is exactly the class of the stack of exact vertex algebroids on X. Our main goal in this dissertation is the classication of transitive vertex algebroids (presented in Chapter 1.2.4). Since exact vertex algebroids classied in [GMS1] are, in fact, a particular kind of transitive vertex algebroids ( those whose associated Lie algebroid is the tangent sheaf), our classication generalizes that of [GMS1]. In particular, we compute the class of the stackVExt h;i L . Bressler computes the corre- sponding class for Courant extensions ofX [Bre] and proves thatcl(CExt h;i L ) = 1 2 p 1 (L;h;i) where p 1 (L;h;i) is the Pontryagin class associated with the pair (L;h;i), a generalization of the familiar rst Pontryagin class of a vector bundle, dened in loc.cit.. Our main result is Theorem 1 below. To prove it we use the technique of Baer arithmetic for Courant algebroids developed by Bressler and use the classication of both CDO and Courant algebroids. Theorem 1 The class ofVExt h;i L in H 2 (X; 2 ! 3;cl ) equals cl(VExt h;i L ) =ch 2 ( 1 X ) 1 2 p 1 (L;h;i) 3 It is worthwhile to note that it is possible for a manifold X to have no global CDO and Courant extensions of a given Lie algebroidL, but still have a vertex extension ofL. Twisted CDO and their deformations. In Chapter 2.3.2 we use the classication result above to study certain deformations of sheaves of twisted chiral dierential operators (TCDO) dened in [AChM]. A TCDO is dened through a procedure that, starting with a CDO produces a sheaf which has features of both the original CDO and and the Bernstein- Beilinson algebra of twisted dierential operators ([BB1]). These sheaves have proved useful in representation theory of ane Lie algebras at the critical level. In particular, one has a localization procedure for certain classes of ^ g-modules. [AChM]. More explicitly, a sheaf of TCDO on X is a sheaf of vertex algebras that locally looks likeD ch H X whereD ch is a sheaf of CDO on X and H X is the algebra of dierential polynomials on the space H 1 (X; 1;cl ) classifying the twisted dierential operators on X. When X is a ag variety, X = G=B , the algebra H G=B is isomorphic to the algebra of dierential polynomials on the Cartan subalgebra h of g = LieG. Moreover, there is an embedding of ane vertex algebra V h _(g)! (G=B;D ch;tw G=B ) (0.0.1) which makes the space of sections of the TCDO over big cell (U e ;D ch;tw X )'D ch (U e ) H G=B a g-module of the critical level, called the Wakimoto module W 0;h _. [FF1] The Wakimoto module W 0;h _ is a member of the family W 0;k =D ch (U e ) H X;k+h _ where H X; is the Heisenberg vertex algebra associated with the space h with a bilinear form equal to times the normalized Killing form. One might ask whether W 0;k with non-critical k admits a localization similar to that of W 0;h _. We show that such a sheaf indeed exists on any ag manifold and is, in fact, a deformation of the TCDO mentioned above: there is one such sheaf for each choice of an 4 invariant inner producth;i on g. Moreover, we show that morphism (0.0.1) \deforms" to V kh _(g)! (G=B;D ch;tw G=B;k ) whereD ch;tw G=B;k is a deformed TCDO corresponding to the inner product equal to k times normalized Killing form (see Section 3.4). More generally, we dene a deformation of TCDO on an arbitrary manifoldX to be the vertex enveloping algebra of certain transitive vertex algebroid on X. We apply our main classication result (cf. Theorem 1 above) to classify the deformations. Representation theory The last chapter in this dissertation concerns the representa- tion theory of vertex algebroids. We describe the category of \nice" representations of a transitive vertex algebroid and its relation to the category of modules over an algebra of twisted dierential operators. Representation theory of TCDO has proved to be quite interesting. TCDOs possess families of modules parameterized by Laurent series (z)2 H 1 (X; 1 ! 2;cl )((z)). For each such (z) with "regular singularity" (i.e. z(z)2 H 1 (X; 1 ! 2;cl )[[z]]), there is a category of modules with central character(z); this category was proved equivalent to the category of modules over the TDO corresponding to 0 = res z=0 (z)[AChM]. We show that analogous statements are true for deformations of TCDO and their relatives { enveloping algebras of transitive vertex algebroids. The precise statements are given below. Let V be a graded vertex algebra and M a V -module, i.e. we are given a map V k 3a7!a M (z) = X a n z nk 2EndM[[z;z 1 ]] satisfying the usual axioms, cf. section 1.1 below. We call M half-integrable if for any a2 V and n > 0 the action of a n on M is locally nilpotent; that is,M is integrable with respect to the positive part of the graded Borcherds Lie algebra of V . 5 LetX be a smooth complex variety andV a graded sheaf of vertex algebras on X such thatV 0 =O X andV is a vertex envelope of the vertex algebroidV 1 . The previous denition immediately carries over to this situation, and we will apply it to thoseV that satisfy some extra conditions. In order to formulate these conditions, recall thatL :=V 1 =V 0(1) @V 0 is an O X -Lie algebroid [GMS1]; in particular, there is an anchor, anO X -Lie algebroid morphism: L!T X . We require: L is a locally freeO X -module of nite rank; L ts into an exact sequence 0! h!L!T X ! 0 where h is an abelianO X -Lie algebra. Such sheaves of vertex algebrasV form a natural class which includes sheaves of chiral twisted dierential operatorsD ch;tw X and certain deformations of those. LetZ(V)V be the subsheaf ofV with stalksZ X;x = Z(V X;x ), x2 X, where Z(V ) denotes the center of V ; we will call it the center ofV. DeneM int + (V) to be the category of half-integrableV-modules that satisfy the fol- lowing regularity condition: z n m = 0 for z2Z(V), n> 0, m2M. Then the following result holds. (Theorem 4.1). Theorem 2 LetV be as above andM2M int + (V). Then 1) The subsheaf SingM =fm2M : v n m = 0 for all v2V; n> 0g generatesM; 2)M possesses a ltrationM =[ i0 M i such thatV i(n) M j M i+jn1 andM 0 = SingM. The result, as stated, is true in the analytic topology; when working algebraically one has to add one more requirement, cf. section 4.1.3. 6 LetX be connected andV as above. Suppose that h =O X C h r where h r is a constant sheaf such that [L;h r ] = 0. For 2 (h r ) consider the ideal I in U O X (L) generated by h(h), h2 h r . Then U O X (L)=I is an algebra of twisted dierential operators (tdo) on X. Let us denote it byD X . LetM int + (V) denote the full subcategory of modulesM2M int + (V) such that for each h2 h r ,m2M h 0 m =(h)m. Being the Zhu algebra ofV,U O X (L) acts onM 0 = SingM, and this action factors through U O X (L)=I =D X . This way we get a functor Sing :M int + (V)!M(D X ) (0.0.2) It is not an equivalence; however, it becomes one upon restriction to certain subcategories. It is not hard to verify that the centerZ(V) is a constant sheaf with values in an algebra of dierential polynomials, generated by the subspacez := (h r ) ? = h2 h r : hh;h r i = 0 , whereh;i : hh!O X is induced by (1) . Fix a series (z)2 z ((z)), (z) = P n z n1 . We say thatZ acts onM via (z) if for all h2 z, m2M h n m = n (h)m; n2Z LetM int + ;(z) (V) denote the full subcategory ofM int + (V) consisting of modules with action ofZ given by (z). For this category to be nonzero, the obvious compatibility conditions 0 =j z and (z)2 z [[z]]z 1 have to be satised. The following result holds. Theorem 3 Let (z)2 z [[z]] be such that 0 = j z . Then the restriction of (0.0.2) to M int + ;(z) (V) is an equivalence. As a particular case, this theorem can be applied whenV is the sheaf of twisted chiral dierential operatorsD ch;tw X introduced in [AChM]. In this case h r and z are both equal to the trivial local system with ber H 1 (X; 1 ! 2;cl ) and the centerZ(V) is the algebra of dierential polynomials on the ane space H 1 (X; 1 ! 2;cl ). Theorem 3 becomes an 7 equivalence between the category of half-integrableD ch;tw X -modules with central character (z) and the category of modules over the tdoD 0 . Furthermore, when h = 0 this becomes an equivalence between the category of half- integrable modules over a CDOD ch X and the category of usualD X -modules. 8 Chapter 1: Preliminaries All vector spaces will be overC. 1.1 Denitions and examples Let V be a vector space. A eld on V is a formal series a(z) = X n2Z a (n) z n1 2 (EndV )[[z;z 1 ]] such that for any v2V one has a (n) v = 0 for suciently large n. Let Fields(V ) denote the space of all elds on V . A vertex algebra is a vector space V with the following data: a linear map Y :V !Fields(V ), V 3a7!a(z) = P n2Z a (n) z n1 a vectorj0i2V , called vacuum vector a linear operator @ :V !V , called translation operator that satisfy the following axioms: 1) (Translation Covariance) (@a)(z) =@ z a(z) 2) (Vacuum) 9 j0i(z) = id; a(z)j0i2V [z] and a (1) j0i =a 3) (Borcherds identity) X j0 m j (a (n+j) b) (m+kj) (1.1.1) = X j0 (1) j n j fa (m+nj) b (k+j) (1) n b (n+kj) a (m+j) g A vertex algebra V is graded if V = n0 V n and for a2V i , b2V j we have a (k) b2V i+jk1 for all k2Z. (We put V i = 0 for i< 0.) All our vertex algebras are graded. We say that a vector v2V m has conformal weight m and write v =m. If v2V m we denote v k =v (km+1) , this is the so-called conformal weight notation for operators. One has v k V m V mk : A morphism of vertex algebras is a mapf :V !W that preserves vacuum and satises f(v (n) v 0 ) =f(v) (n) f(v 0 ). A module over a vertex algebra V is a vector space M together with a map Y M :V !Fields(M); a!Y M (a;z) = X n2Z a M (n) z n1 ; (1.1.2) that satisfy the following axioms: 1) j0i M (z) = id M 10 2) (Borcherds identity) X j0 m j (a (n+j) b) M (m+kj) (1.1.3) = X j0 (1) j n j fa M (m+nj) b M (k+j) (1) n b M (n+kj) a M (m+j) g A module M over a graded vertex algebra V is called graded if M = n0 M n with v k M l M lk (assuming M n = 0 for negative n). A morphism of modules over a vertex algebra V is a map f : M ! N that satises f(v M (n) m) =v N (n) f(m) for v2V , m2M. f is homogeneous if f(M k )N k for all k. The Borcherds Lie algebra To any vertex algebra V one can associate a Lie algebra that acts on any V -module. It is the Borcherds Lie algebra of V dened in the following way Lie (V ) =V C[t;t 1 ]= ( (@a + (n +H)a) t n ) where Ha =ka for a2V k . This is a Lie algebra with Lie bracket given by [a t n ;b t l ] = X j0 n + a 1 j (a (j) b) t n+l for a homogeneousa2V andb2V , and extended linearly. Lie (V ) acts on anyV -module M by letting a t n act as a n . Lie (V ) has a natural grading, Lie (V ) = n2Z Lie (V ) n with Lie (V ) n equal to the image of V t n . 11 1.1.1 Examples Ane vertex algebras Let g be a semisimple Lie algebra andh;i :S 2 g!C an invariant form on g. The ane Lie algebra ^ g associated with g andh;i is a central extension of g C[t;t 1 ] dened as follows. As a vector space, ^ g = g C[t;t 1 ]CK and the Lie bracket is [x t n ;y t m ] = [x;y] t m+n +n n+m;0 hx;yiK K is a central element. We denote x t n by x n and write x(z) = P x n z n1 . Let g = n hn + be a Cartan decomposition of g. Denote ^ g < = g tC[t], ^ g > = g t 1 C[t 1 ] and ^ g = g C[t]CK. Dene ^ g + = n + ^ g > , ^ g = n ^ g < . Then ^ g = ^ g + hCK ^ g . The space of invariant forms is one-dimensional, and we will leth;i be that form for which (;) = 2 where is the longest root. Introduce the following induced module V k (g) = Ind ^ g ^ g C k ; (1.1.4) where C k is a 1-dimensional ^ g -module generated by a vector v k such that ^ g < v k = 0, gv k = 0 and Kv k =v k . V k (g) carries a vertex algebra structure that is dened by assigning to x 1 v k x2 g, the eld x(z) = P x n z n1 . These elds generate V k (g). V k (g) is a graded vertex algebra with generators having conformal weight 1. For example, V k (g) 0 =C k ; V k (g) 1 = g t 1 v k : (1.1.5) 12 Commutative vertex algebras. A vertex algebra is said to be commutative if a (n) b = 0 for a, b in V and n 0. The structure of a commutative vertex algebras is equivalent to one of commutative associative algebra with a derivation. IfW is a vector space we denote byH W the algebra of dierential polynomials onW . As an associative algebra it is a polynomial algebra in variables x i ,@x i ,@ (2) x i ,::: wherefx i g is a basis of W . A commutative vertex algebra structure on H W is uniquely determined by attaching the eld x(z) =e z@ x i to x2W . H W is equipped with grading such that (H W ) 0 =C; (H W ) 1 =W : (1.1.6) Beta{gamma system. Dene the Heisenberg Lie algebra to be the algebra with generators a i n , b i n , 1iN and K that satisfy [a i m ;b j n ] = m;n i;j K, [a i n ;a j m ] = 0, [b i n ;b j m ] = 0. Its Fock representationM is dened to be the module induced from the one-dimensional representationC 1 of its subalgebra spanned by a i n ,n 0,b i m ,m> 0 andK withK acting as identity and all the other generators acting as zero. The beta-gamma system hasM as an underlying vector space, the vertex algebra struc- ture being determined by assigning the elds a i (z) = X a i n z n1 ; b i (z) = X b i n z n to a i 1 1 and b i 0 1 resp., where 12C 1 . This vertex algebra is given a grading so that the degree of operators a i n andb i n isn. In particular, M 0 =C[b 1 0 ;:::;b N 0 ]; M 1 = N M j=1 (b j 1 M 0 a j 1 M 0 ): (1.1.7) 13 1.2 Vertex algebroids 1.2.1 Denition Let V be a vertex algebra. Dene a 1-truncated vertex algebra to be a sextuple (V 0 V 1 ;j0i;@; (1) ; (0) ; (1) ) where the operations (1) ; (0) ; (1) satisfy all the axioms of a vertex algebra that make sense upon restricting to the subspace V 0 +V 1 . (The precise denition can be found in [GMS1]). The category of 1-truncated vertex algebras will be denotedVert 1 . The denition of vertex algebroid is a reformulation of that of a sheaf of 1-truncated vertex algebras. Let (X;O X ) be a space with a sheaf ofC-algebras andT X = Der C (O X ). A vertexO X -algebroid is a sheafA of C-vector spaces equipped with C-linear maps :A!T X and@ :O X !A satisfying@ = 0 and with operations (1) :O X A!A, (0) :AA!A, (1) :AA!O X satisfying axioms: f (1) (g (1) v) (fg) (1) v = (v)(f) (1) @(g) +(v)(g) (1) @(f) (1.2.1) x (0) (f (1) y) = (x)(f) (1) y +f (1) (x (0) y) (1.2.2) x (0) y +y (0) x = @(x (1) y) (1.2.3) (f (1) v) = f(v) (1.2.4) (f (1) x) (1) y = f(x (1) y)(x)((y)(f)) (1.2.5) (v)(x (1) y) = (v (0) x) (1) y +x (1) (v (0) y) (1.2.6) @(fg) = f (1) @(g) +g (1) @(f) (1.2.7) v (0) @(f) = @((v)(f)) (1.2.8) v (1) @(f) = (v)(f) (1.2.9) (x (0) y) = [(x);(y)] (1.2.10) for v;x;y2A, f;g2O X . The map is called the anchor ofA. 14 IfV = L n0 V n is a (graded) sheaf of vertex algebras withV 0 =O X , thenA =V 1 is a vertex algebroid with @ equal to the translation operator and sending x2V 1 to the derivation f7!x (0) f. Associated Lie algebroid Recall that a Lie algebroid is a sheaf ofO X -modulesL equipped with a Lie algebra bracket [; ] and a morphism :A!T X of Lie algebra andO X -modules called anchor that satises [x;ay] =a[x;y] +(x)(a)y, x;y2A, a2O X . IfA is a vertex algebroid, then the operation (0) descends to that onL A =A=O X (1) @O X and makes it into a Lie algebroid, with the anchor induced by that ofA. L A is called the associated Lie algebroid ofA. Transitive vertex algebroids A vertex (resp., Lie) algebroid is transitive, if its anchor map is surjective. Being a derivation, see (1.2.7), @ :O X !A lifts to 1 X !A. It follows from (1.2.9) that ifA is transitive, then 1 X 'O X (1) @O X andA ts into an exact sequence 0! 1 X !A!L! 0; L =L A being an extension 0! g(L)!L!T X ! 0 where g(L) := ker(L !T X ) is anO X -Lie algebra. Note that the pairing (1) onA induces a symmetricL A -invariantO X -bilinear pairing on g(L A ) which will be denoted byh;i. We regard the pair (L A ;h;i) as "classical data" underlying the vertex algebroidA. 15 Truncation and vertex enveloping algebra functors There is an obvious truncation functor t :Vert!Vert 1 that assigns to every vertex algebra a 1-truncated vertex algebra. This functor admits a left adjoint [GMS1] u :Vert 1 !Vert called a vertex enveloping algebra functor. These functors have evident sheaf versions. In particular, one has the functor U :VertAlg!ShVert (1.2.11) from the category of vertex algebroids to the category of sheaves of vertex algebras. 1.2.2 Courant algebroids We give a denition of a Courant algebroid following [Bre]; see also [LWX]. A Leibniz algebra overk is ak-vector spaceA with a bracket [; ] :A k A!A satisfying [x; [y;z]] = [[x;y];z] + [y; [x;z]]: The bracket is not assumed to be skew-commutative. A CourantO X -algebroid is anO X -moduleQ equipped with 1) a structure of a LeibnizC-algebra [ ; ] :Q C Q!Q ; 2) anO X -linear map of Leibniz algebras (the anchor map) :Q!T X ; 3) a symmetricO X -bilinear pairingh;i :Q O X Q!O X ; 16 4) a derivation @ :O X !Q which satisfy @ = 0 (1.2.12) [q 1 ;fq 2 ] =f[q 1 ;q 2 ] +(q 1 )(f)q 2 (1.2.13) h[q;q 1 ];q 2 i +hq 1 ; [q;q 2 ]i =(q)(hq 1 ;q 2 i) (1.2.14) [q;@(f)] =@((q)(f)) (1.2.15) hq;@(f)i =(q)(f) (1.2.16) [q 1 ;q 2 ] + [q 2 ;q 1 ] =@(hq 1 ;q 2 i) (1.2.17) for f2O X and q;q 1 ;q 2 2Q. A morphism of CourantO X -algebroids is anO X -linear map of Leibnitz algebras which commutes with the respective anchor maps and derivations and preserves the respective pairings. A connection on a Courant algebroidQ is anO X -linear sectionr of the anchor map such thathr();r()i = 0. IfQ is a Courant algebroid, thenL Q =Q=O X @O X is a Lie algebroid; it is called the associated Lie algebroid ofQ. The pairingh;i onQ induces aL Q -invariant pairing on g(L Q ) which will be denotedh;i. 1.2.3 The category of vertex extensions LetL be a transitive Lie algebroid. A vertex extension ofL is a vertex algebroidA with an isomorphism of Lie algebroids :L A !L. In what follows we will always identifyL A andL via . A morphism of vertex extensions ofL is a morphism of vertex algebroids f :A!A 0 17 which induces the identity map onL. Thus f ts into a diagram 0! 1 X ! A ! L ! 0 ? ? y f 0! 1 X ! A 0 ! L ! 0 Vertex extensions ofL on X form a categoryVExt L (X); clearly, it is a groupoid. One can consider the category of vertex extensions ofLj U on U for any open subset UX. These categories with the obvious restriction functors form a stack on the Zariski topology of X, to be denotedVExt L LetA be a vertex extension ofL. Denote ~ g A := ker( :A!T X ); it is an extension 0! 1 X ! ~ g A ! g ! 0 It is easy to see that the operation (1) satises ~ g A(1) 1 = 0, and, therefore, induces a (symmetric,O X -bilinear) pairing h;i : gg!O X Iff :A!A 0 is a morphism of extensions,f induces the identity map on g; it also preserves (1) . ThereforeA andA 0 must have the same pairingh;i on g. It follows that the groupoid VExt L (X) is a disjoint union VExt L (X) = a h;i VExt h;i L (X) whereVExt h;i L (X) is the full subcategory of vertex extensions ofL whose induced pairing on g ish;i. Such extensions will be called vertex extensions of (L;h;i). Similarly, we dene the notion of a Courant extension ofL onX and that of a morphism of Courant extensions, the categoriesCExt h;i L (U) andCExt h;i L (U), UX. 18 1.2.4 Chiral dierential operators Vertex extensions ofT X are called exact vertex algebroids. Their vertex enveloping algebras, sheaves of chiral dierential operators (CDO). were rst introduced in [MSV] and classied in [GMS1]. Let us recall the main classication result. Let us call a smooth ane variety U = SpecA suitable for chiralization if Der(A) is a free A-module admitting an abelian framef 1 ;:::; n g. In this case there is a CDO over U, which is uniquely determined by the condition that ( i ) (1) ( j ) = ( i ) (0) ( j ) = 0. Denote this CDO by D ch U; . Theorem 1.1 LetU = SpecA be suitable for chiralization with a xed abelian framef i g DerA. (i) For each closed 3-form 2 3;cl A there is a CDO over U that is uniquely determined by the conditions ( i ) (1) j = 0; ( i ) (0) j = i j : Denote this CDO byD U; (). (ii) Each CDO over U is isomorphic toD U; () for some . (iii)D U; ( 1 ) andD U; ( 2 ) are isomorphic if and only if there is 2 2 A such that d = 1 2 . In this case the isomorphism is determined by the assignment i 7! i + i . If A =C[x 1 ;:::;x n ], one can choose @=@x j , j = 1;:::;n, for an abelian frame and check that the beta-gamma system M of sect. 1.1.1 is a unique up to isomorphism CDO over C n . A passage from M to Theorem 1.1 is accomplished by the identications b j 0 1 = x j , a j 1 1 =@=@x j . 19 Chapter 2: Classication of transitive vertex algebroids In this chapter we present the main result of the dissertation: a classication of transitive vertex algebroids in the spirit of [Bre]. In section 2.1 we recall the denition of aGr( [2;3> X )-gerbe given in [GMS1]; one of the results of [Bre] is thatVExt h;i L is aGr( [2;3> X )-gerbe. In section 2.2 we describe the core tool of the classication method: the \addition" operation on various algebroids. It enables us to construct a vertex extension starting from a Courant extension and an exact vertex algebroid. With this tool in hands we are able to compute the class of the stackVExt h;i L (Theorem 2.14). 2.1 Gerbes and torsors 2.1.1 Twisting by a 3-form LetA = (A; (1) ; (0) ; (1) ;@;) be a vertex extension ofL on U X and let 2 3;cl (U). Dene an operation (0)+ :AA!A by x (0)+ y =x (0) y + (x) (y) (2.1.1) Lemma 2.1 Let 2 3;cl (U). Then: (1) Au := (A; (1) ; (0)+ ; (1) ;@;) is a vertex extension ofL on U. 20 (2) The assignmentA7!Au can be extended to an auto-equivalence ?u :VExt h;i L !VExt h;i L (2.1.2) Proof. The proof of (1) is the same as in the case of cdo ([MSV, GMS1]) or Courant algebroids ([Bre]). To see (2), note that every morphism f :A!A 0 is automatically a morphismAu!A 0 u; this tautological action on morphisms makes ?u a functor; the composition (?u ()) (?u) is the identity functor ofVExt h;i L .N It is clear that the functors ?u, 2 3;cl (U) dene an action of the abelian group 3;cl (U) on the categoryVExt L (U). Let us show that this action in fact extends to an action of a category. For an open subset U X dene a categoryGr( [2;3> )(U) as follows. The objects of Gr( [2;3> )(U) are elements a2 3;cl (U); the morphisms Hom(; 0 ) = 2 2 (U) : d = 0 ; the composition being the addition in 2 (U). It is clear thatGr( [2;3> )(U) is a groupoid. The groupoidsGr( [2;3> )(U) form a prestack Gr( [2;3> X ); the addition of 3-forms gives it the structure of a Picard prestack. See [D], section 1.4 for generalities on Picard stacks. For f :A!A 0 and :! 0 dene (fu)(x) =f(x) + (x) (2.1.3) Proposition 2.2 (1) fu is a morphism of vertex extensions fu :Au!A 0 u 0 21 (2) The formulas (2.1.2) and (2.1.3) dene a functor u : VExt L (U)Gr( [2;3> )(U)!VExt L (U) which gives rise to an action ofGr( [2;3> )(U) onVExt L (U) The verication is, again, straightforward and repeats the analogous discussion in [GMS1]. N 2.1.2 ( 2 ! 3;cl )-gerbes We will say a stackS overX is aGr( [2;3> X )-gerbe if there is an actionu :SGr( [2;3> X )! S and a cover U =fU i g i2I such that for any i2I and x2S(U i ) the functor xu? : Gr( [2;3> X )(U)!S(U) is an equivalence. (In other words,S is a torsor over the associated stack). Theorem 2.3 [Bre] The stacksVExt h;i L andCExt h;i L , when locally nonempty, areGr( [2;3> )- gerbes. Remark 2.4 The categoriesGr( [2;3> )(U), UX form a Picard prestack (cf.[D], section 1.4.11) whose associated stack is the stack of ( 2 ! 3;cl )-torsors. What Bressler shows in [Bre] is that this stack is equivalent to the stackECA X of exact Courant algebroids, and that the stacksVExt h;i L ,CExt h;i L are, in fact,ECA X -torsors. Observe that forVExt h;i L being aGr( [2;3> )-gerbe means that for small enough UX andA2VExt h;i L one has an equivalence Au? :Gr( [2;3> )(U)!VExt h;i L (U) In particular, there is an isomorphism Hom(; 0 )' Hom(A +;A + 0 ) 22 Under this isomorphism, an element 2 2 with d = 0 , is mapped to the morphism (cf. (2.1.3)) exp() := idu : x7!x + (x) (2.1.4) The same is true for Courant algebroids and we will use the notation exp() in both cases. 2.1.3 The class of a gerbe LetS be aGr( [2;3> )-gerbe and U a cover as in 2.1.2. Let us choose an objectx i 2S(U i ) for each i. For each pair i;j we have objects x i j U ij and x j j U ij , and therefore, an isomorphism ij :x i j U ij !x j j U ij u ij (2.1.5) for some ij 2 3;cl . The collection (x i ; ij ; ij ) is called a trivialization ofS. We will denote by the same letter ij all of its translates ij u id :x i j U ij u !x j j U ij u ( ij + ) for 2 3;cl (U i ). For each triple i;j;k consider the composition (over U ijk =U i \U j \U k ) jk ij 1 ik : x k 1 ik ! x i u ( ik ) ij ! x j u ( ij ik ) jk ! x k u ( ij + jk ik ) and denote by ijk the element of 2 (U ijk ) such that jk ij 1 ik = exp( ijk ) (2.1.6) One checks that d C ijk = 0; d DR ( ijk ) =d C ( ij ); d DR ( ij ) = 0 (2.1.7) 23 so that the pair ( ij ; ijk ) is an element of Z 2 (U; 2 ! 3;cl ). By denition, the class ofS, cl(S), is the class of ( ij ; ijk ) in H 2 (X; 2 ! 3;cl ). One has the following classical result (cf., e.g., [GMS1] for a proof). Proposition 2.5 S(X) is nonempty if and only if cl(S) = 0.N 2.1.4 The stackCExt h;i L As an example, and for future use, we recall the construction of a trivialization of the stack CExt h;i L given in [Bre]. Let us choose a cover U =fU i g such thatT U i is free, choose connections (O X -linear sections of the anchor map) r i :T U i !L U i and identifyL U i 'T U i g U i viar i . Dene c i =c(r i )2 2;cl U i O g U i to be the curvature of the connectionr i , i.e. c i (;) = [r i ();r i ()]r i ([;]) Recall the following Theorem 2.6 [Bre] Let UX andr :T U !L U is any connection. Then the categoryCExt h;i L (U) is nonempty if and only if the form 1 2 hc(r)^c(r)i is exact. Assume that 1 2 hc(r i )^c(r i )i =dH i for some H i 2 3 . Then one can construct a Courant extensionQ r i ;H i , which is equal to 24 L U i 1 U i as a sheaf ofO U -modules, and satises [;] = [;] L + H i ; ;2T U i ; (2.1.8) < g; 1 U >=< g;r i (T U )>= 0; (2.1.9) [;g] = [r i ();g] L h c(r i );gi: (2.1.10) For each i;j dene A ij =r i r j 2 1 U ij g U ij Theorem 2.7 [Bre] There exists an isomorphism inCExt h;i L (U ij ) ij :Q r i ;H i !Q r j ;H j u ij (2.1.11) given by 7! +A ij () 1 2 hA ij ();A ij i g7!ghg;A ij i !7!! (2.1.12) where ij =hc(r i )^A ij i 1 2 h[r i ;A ij ];A ij i + 1 6 h[A ij ;A ij ];A ij i +H i H j (2.1.13) The collection (Q r i ;H i ; ij ; ij ) is a trivialization of the gerbeCExt h;i L . On triple intersections U ijk =U i \U j \U k the isomorphisms ij satisfy ([Bre]) jk ij 1 ik = exp(<A ij ^A jk >) N (2.1.14) 25 Dene ijk =<A ij ^A jk >. Then ( ij ; ijk ) is a cocycle in Z 2 (U; 2 X ! 3;cl X ). The corresponding cohomology class was identied in [Bre] with minus one half of the rst Pontryagin class p 1 (L;h;i) of (L;h;i . Theorem 2.8 [Bre] This class is the class of the stackCExt h;i L : cl(CExt h;i L ) = 1 2 p 1 (L;h;i): 2.2 Linear algebra In this section we describe the main tool in the proof of the classication result: we dene linear algebra-like operations on various algebroids. The main technical result to be proved in this section is as follows. Theorem 2.9 Let U be suitable for chiralization. Then there exist a functor :CExt h;i L (U)CDO(U)!VExt h;i L (U) (Q;D)7!QD and a functor :VExt h;i L (U)CDO(U)!CExt h;i L (U) (A;D)7!AD such that for a xedD2CDO(U) the functors D : VExt h;i L (U)!CExt h;i L (U) 26 and D : CExt h;i L (U)!VExt h;i L (U) are mutually inverse equivalences ofGr( [2;3> (U))-torsors. In fact, the functor was dened in [Bre], together with several versions of dened for various algebroids. Our is just an extension of Bressler's denition. 2.2.1 The functor LetQ be a Courant extension ofL andD a cdo. We describe how to dene a vertex extension ofL which can be though of as a \sum" of these two structures; the construction parallels that of the Baer sum of two extensions. First, consider the pullbackA :=Q T D so that a section ofA is a pair (q;x), q2Q, x2D with (q) =(x). Dene operations (1) :O X A!A and (0) ; (1) :AA!A as follows: a (1) (q;x) := (aq;a (1) x) (2.2.1) (q;x) (0) (q 0 ;x 0 ) := ( [q;q 0 ] Q ;x (0) x 0 ); (2.2.2) (q;x) (1) (q 0 ;x 0 ) := hq;q 0 i +x (1) x 0 ; (2.2.3) ((q;x)) := (q) =(x); (2.2.4) @a = (@a; 0) (2.2.5) Note thatA contains two copies of 1 , one fromQ and the other fromD. Let us deneQD to be the pushout ofA with respect to the addition map + : 1 1 ! 1 so that one has the following 0! 1 1 ! A ! L ! 0 ? ? y + ? ? y 0! 1 ! QD ! L ! 0 27 Alternatively,QD ts into the diagram 0! ~ g 1 ! A ! T X ! 0 ? ? y + ? ? y 0! ~ g ! QD ! T X ! 0 where the rows are exact and the left square is a push-out square. Theorem 2.10 The operations (2.2.1 - 2.2.5) make sense onQD and give it the structure of a vertex algebroid Proof. The verication is straightforward. As an example, let us show that (1.2.5) is satised. For f2O X , q2Q, v2D, one has: (f (1) (q;v)) (1) (q 0 ;v 0 ) = (fq;f (1) v) (1) (q 0 ;v 0 ) =hfq;q 0 i + (f (1) v) (1) v 0 =fhq;q 0 i +f(v (1) v 0 )(v)(v 0 )(f) =f((q;v) (1) (q 0 ;v 0 ))((q;v))((q 0 ;v 0 ))(f) N Note that the assignment (Q;D)7!QD is naturally a functor :CExt h;i L (U)CDO(U)!VExt h;i L (U) Indeed, let f2 Hom CExt (Q;Q 0 ), f2 Hom CDO (D;D 0 ). In particular, f andg are maps over T , so (f;g) takesQ T DQD toQ 0 T D 0 . Since f and g act as identity on the subsheaf 1 , (f;g) gives a well-dened map between the pushoutsQD!Q 0 D 0 that will be denotedfg. Finally, it remains to note that the composition is \coordinate-wise": (fg)(f 0 g 0 ) =ff 0 gg 0 (2.2.6) which implies that (f;g)7!fg is a functor. 28 Let us note, among the elementary properties of this functor, the following: 1) for any 2 3;cl U ,Q2CExt h;i L (U),D2CDO U one has the equalities (QD)u =Q (Du) = (Qu)D (2.2.7) (by denition ofu the three parts of the equation have underlying sheafQD, one only has to check that the operations coincide). 2) one has the equality exp() id D = exp() = id Q exp() in Hom VExt (QD; (QD)ud); more generally, exp( 0 ) exp( 00 ) = exp( 0 + 00 ) (2.2.8) 2.2.2 The functor LetA is a vertex extension ofL andD a cdo. In [Bre] it is described how to dene a Courant algebroidAD. Let us recall this construction. LetQ :=A T D so that a section ofQ is a pair (v;x),v2A,x2D with(v) =(x). Dene operations :O X Q!Q; [; ] :QQ!Q;h;i :QQ!O X ; :Q!T; and @ :O X !Q as follows: a (v;x) := (a (1) v; a (1) x) (2.2.9) (v;x); (v 0 ;x 0 ) := (v (0) v 0 ; x (0) x 0 ) (2.2.10) h(v;x); (v 0 ;x 0 )i := v (1) v 0 x (1) x 0 (2.2.11) ((v;x)) := (v) =(x) (2.2.12) @a = (@a; 0) (2.2.13) 29 DeneAD to be the pushout ofQ with respect to the subtraction map : 1 1 ! 1 . One can show that all operations dened above make sense onAD. One has Theorem 2.11 ([Bre], Lemma 5.6) The sheafAD with the operations dened above is a Courant algebroid 2.2.3 Compatibility of and Theorem 2.12 The functors D : CExt h;i L (U)!VExt h;i L (U) and D : CExt h;i L (U)!VExt h;i L (U) are mutually inverse equivalences ofGr( [2;3> (U))-torsors. Proof. The compatibility ofD andD withGr( [2;3> (U))-action follows from prop- erties (2.2.7 - 2.2.8) and their obvious analogs for. Let us construct the natural isomor- phisms A :A' (AD)D whereA is a vertex extension ofL andD is a cdo. Dene A (v) = ((v;x);x) where x2D is arbitrary. To show A is well-dened note that for any x;y2D with (x) =(y) =(v) we have xy2 1 and ((v;x);x) = ((v; (yx) +y);x) = ((v;y) + (yx);x) = ((v;y);x + (yx)) = ((v;y);y) 30 To verify A is a morphism we check a (1) ((v;x);x) = (a(v;x);a (1) x) = ((a (1) v;a (1) x);a (1) x) = A (a (1) v) ((v;x);x) (0) ((v 0 ;x 0 );x 0 ) = ([(v;x); (v 0 ;x 0 )];x (0) x 0 ) = ((v (0) v 0 ;x (0) x 0 );x (0) x 0 ) = A (v (0) v 0 ) ((v;x);x) (1) ((v 0 ;x 0 );x 0 ) =h(v;x); (v 0 ;x 0 )i +x (1) x 0 =v (1) v 0 x (1) x 0 +x (1) x 0 =v (1) v 0 To check that A is an isomorphism, one can check that the map : (AD)D!A; ((v;x);y)7!v +(yx): is a well-dened inverse to . (Note that every section ((v;x);y) of (AD)D can be written as ((v;x);y) = ((v +(yx);x+(yx));y) = ((v +(yx);y);y) with v + (yx) independent of the choice of representative ((v;x);y)). The construction of the natural isomorphisms 0 Q : Q!QDD is analogous and left to the reader.N The constructions of sections 2.2.1, 2.2.2 and Theorem 2.12 furnish the proof of Theorem 2.9. 2.3 Classication 2.3.1 Local existence Let U be suitable for chiralization and supposer :T U !L U is a connection. Theorem 2.13 Then the following are equivalent: 1) The categoryVExt h;i L (U) is nonempty 2) The categoryCExt h;i L (U) is nonempty 3) The Pontryagin form 1 2 hc(r)^c(r)i is exact. Proof. Since U is suitable for chiralization, there exists a CDOD on X. Then (1) and (2) are equivalent due to the addition / subtraction operations: given a vertex extension 31 A there exists a Courant extensionQ =AD and vice versa, givenQ one can produce a vertex extensionA =QD. Finally, the equivalence of (2) and (3) is the content of Theorem 2.6.N 2.3.2 The obstruction Theorem 2.14 SupposeVExt h;i L is nonempty. Then its class is equal to cl(VExt h;i L ) = 1 2 p 1 (L;h;i) +ch 2 ( 1 X ) where p 1 (L;h;i) is the rst Pontryagin class of a Lie algebroidL with pairingh;i. Proof. What we will be proving is the following: cl(VExt h;i L ) =cl(CExt h;i L ) +cl(CDO(X)) This is indeed sucient, in view of Theorem 2.8 and the fact that cl(CDO(X)) =ch 2 ( 1 X ) [GMS1, Bre] Let U be a cover of X by open subsets U suitable for chiralization. SinceVExt h;i L is nonempty, so isCExt h;i L . Suppose we are given a trivialization of the gerbeCDO and that ofCExt h;i L . In other words, we are given a CDOD i and a Courant extensionQ i on eachU i , as well as isomorphisms ij : D i j U ij !D j j U ij u ch ij and ij :Q i j U ij !Q j j U ij u Q ij where Q ij ; ch ij 2 3;cl (U ij ), such that on triple intersections U ijk =U i \U j \U k one has ch ij + ch jk = ch ik ; Q ij + Q jk = Q ik ; 32 and jk ij 1 ik = exp( ch ijk ); jk ij 1 ik = exp( Q ijk ); (2.3.1) for some ch ijk ; Q ijk 2 2 (U ijk ) Then ( ch ij ; ch ijk ) and ( Q ij ; Q ijk ) are cocycles representing the classes of the gerbesCDO X andCExt h;i L respectively. Now let us construct a trivialization of the gerbeVExt h;i L . Dene A i =Q i D i 2VExt h;i L (U i ): One has the following isomorphisms: Q i j U ij D i j U ij ij ij ! (Q j u Q ij )j U ij (D j u ch ij )j U ij Q j j U ij D j j U ij u ( Q ij + ch ij )) the latter being the identity on the level of vector spaces, by denition of ?u (cf. sect. 2.1.1). Thus ij ij :A i !A j u ( Q ij + ch ij ): The collection (A i ; ( Q ij + ch ij ); ij ij ) is a trivialization of the gerbeVExt h;i L . Let us compute its class. On triple intersections U ijk =U i \U j \U k we have (cf. (2.3.1), (2.2.6), (2.2.8) ) ( jk jk )( ij ij )( ik ik ) 1 = jk ij 1 ik jk ij 1 ik = exp( Q ijk ) exp( ch ijk ) = exp( Q ijk + ch ijk ) (2.3.2) (here, again, we slightly abuse the notation by writing ij for any of its translates under the action ofGr( [2;3> )). 33 It follows that ( Q ij + ch ij ; Q ijk + ch ijk ) is a cocycle representing the class of the gerbe VExt h;i L .N 34 Chapter 3: Twisted CDO and their deformations 3.1 Twisted chiral dierential operators In this section we recall the denition of the sheaf D ch;tw X of twisted chiral dierential operators (TCDO) corresponding to a given CDOD ch on a smooth projective variety X. 3.1.1 The universal Lie algebroidT tw The Lie algebroidT tw underlying TCDO is a \family of all TDO". More precisely, the universal enveloping algebraD tw X ofT tw possesses the following property: for every 2 H 1 (X; 1 X ! 2;cl X ) there exists an ideal m D tw X such that the quotientD tw X =m is iso- morphic to the tdoD X corresponding to the class . Let us sketch the construction. SinceX is projective,H 1 (X; 1 X ! 2;cl X ) is nite-dimensional, and there exists an ane cover U so that H 1 (U; 1 X ! 2;cl X ) =H 1 (X; 1 X ! 2;cl X ). Let = H 1 (U; 1 X ! 2;cl X ). We x a lifting H 1 (U; 1 X ! 2;cl X )! Z 1 (U; 1 X ! 2;cl X ) and identify the former with the subspace of the latter dened by this lifting. Thus, each 2 is a pair of cochains = (( (1) ij ); ( (2) i )) with (1) ij 2 1 (U i \U j ); (2) i 2 2;cl (U i ); satisfying d DR (1) ij =d C (2) i and d C 1 ij = 0: For = ( (1) ij ; (2) i )2 denoteD the corresponding sheaf of twisted dierential opera- tors. One can considerD as an enveloping algebra of the (Picard) Lie algebroidT =D 1 35 [BB2]. As anO X -module,T is an extension 0!O X 1!T !T X ! 0 given by ( (1) ij ). The Lie algebra structure onT U i is given by [;] T = [;] +i i 2 i 1: and [1;T U i ] = 0. Letf i g andf i g be dual bases of and respectively. Denote by k the dimension of . DeneT tw to be an abelian extension 0!O X !T tw X !T X ! 0 such that [ ;T tw ] = 0 and there exist connectionsr i :T U i !T tw U i satisfying r j ()r i () = X r (1) r (U ij ) r (3.1.1) [r i ();r i ()]r i ([;]) = X r (2) r (U i ) r (3.1.2) It is clear that the pair (T tw ;O X ,!T tw ) is independent of the choices made. We call the universal enveloping algebraD tw X =U O X (T tw ) the universal sheaf of twisted dierential operators. 3.1.2 A universal twisted CDO Let ch 2 (X) = 0 and x a CDOD ch X . To each such sheaf one can attach a universal twisted CDO,D ch;tw X , a sheaf of vertex algebras whose "underlying" Lie algebroid is T tw X . Let us place ourselves in the situation of the previous section, where we had a xed ane cover U =fU i g of a projective algebraic manifold X, dual basesf i g2 H 1 (X; [1;2> X ), f i g2H 1 (X; [1;2> X ) , and a lifting H 1 (X; [1;2> X )!Z 1 (U; [1;2> X ). We can assume that U i are suitable for chiralization. Let us x, for each i, an abelian 36 basis (i) 1 ; (i) 2 ;::: of (U i ;T X ). Then the CDOD ch is given by a collection of 3-forms (i) 2 (U i ; 3;cl X ) (cf. sect. 1.2.4, Theorem 1.1) and transition maps g ij :D ch U j j U i \U j !D ch U i j U i \U j : Let us as well x splittingsT U i ,!D ch U i and view g ij as maps g ij : (T U j 1 U j )j U i \U j ! (T U i 1 U i )j U i \U j The universal sheaf of twisted chiral dierential operatorsD ch;tw X corresponding toD ch X is a vertex envelope of theO X -vertex algebroidA tw determined by the following: A tw is a vertex extension of (T tw X ; 0); there are embeddingsT U i ,!A U i such that (i) l (0) (i) m = (i) l (i) m (i) + X (i) l (i) m (2) k (U i ) k the transition function from U j to U i is given by g tw ij () =g ij () X (1) k (U i \U j ) k (3.1.3) The reader is referred to [AChM] for details of the original construction. We will present another construction of TCDO based on Baer arithmetic for vertex and Courant algebroids developed in section 2.2. Alternative construction First, let us remark that any transitive Lie algebroidL admits a Courant extensionQ L that induces a trivial pairingh;i = 0 on its kernel of the anchor g. Namely, one denesQ L to be Q L = 1 X L 37 as anO X -module, with the operations determined by the following [x;!] Q = Lie (x) ! (3.1.4) hx;!i = (x) ! (3.1.5) hx;yi = 0 (3.1.6) [x;y] Q = [x;y] L (3.1.7) for x;y2L, !2 1 X . We will callQ L the canonical Courant extension ofL. Then one can dene the sheaf of TCDOD ch;tw corresponding to a CDOD ch to be D ch;tw =Q tw D ch whereQ tw =Q T tw is the canonical Courant extension of the Lie algebroidT tw . 3.1.3 Locally trivial twisted CDO Observe that there is an embedding H 1 (X; 1;cl X ),!H 1 (X; [1;2> X ) (3.1.8) The space H 1 (X; 1;cl X ) classies locally trivial twisted dierential operators, those that are locally isomorphic toD X . Thus for each 2 H 1 (X; 1;cl X ), there is a unique up to isomorphism TDO D X such that for each suciently small openUX, D X j U is isomorphic toD U . Let us see what this means at the level of the universal TDO. In terms of Cech cocycles the image of embedding (3.1.8) is described by those ( (1) ; (2) ), see section 3.1.1, where (2) = 0, and this forces (1) to be closed. Picking a collection of such cocycles that represent a basis of H 1 (X; 1;cl X ) we can repeat the constructions of sections 3.1.1 and 3.1.2 to obtain sheaves T tw X and D ch;tw X . The latter is glued of pieces 38 isomorphic (as vertex algebras) toD ch U i H X with transition functions as in (3.1.3); here H X is the vertex algebra of dierential polynomials on H 1 (X; 1;cl X ). We will call the sheaf D ch;tw X the universal locally trivial sheaf of twisted chiral dierential operators. 3.1.4 TCDO on ag manifolds TCDO on P 1 Let us see what our constructions give us if X = P 1 . We have P 1 = C 0 [C 1 , a cover U =fC 0 ;C 1 g, where C 0 is C with coordinate x, C 1 is C with coordinate y, with the transition function x7! 1=y overC =C 0 \C 1 . Dened over C 0 and C 1 are the standard CDOs,D ch C 0 andD ch C1 . The spaces of global sections of these sheaves are polynomials in @ n (x), @ n (@ x ) (or @ n (y), @ n (@ y ) in the latter case), where @ is the translation operator, so that, cf. sect. 1.2.4, (@ x ) (0) x = (@ y ) (0) y = 1: There is a unique up to isomorphism CDO onP 1 ,D ch P 1 ; it is dened by gluingD ch C 0 andD ch C1 overC as follows [MSV]: x7! 1=y; @ x 7! (@ y ) (1) (y 2 ) 2@(x): (3.1.9) The canonical Lie algebra morphism sl 2 ! (P 1 ;T P 1); (3.1.10) where e7!@ x ; h7!2x@ x ; f7!x 2 @ x ; (3.1.11) 39 e;h;f being the standard generators of sl 2 , can be lifted to a vertex algebra morphism V 2 (sl 2 )! (P 1 ;D ch P 1 ); (3.1.12) where e (1) j0i7! @ x ; h (1) j0i7!2(@ x ) (1) x; f (1) j0i7!(@ x ) (1) x 2 2@(x): (3.1.13) The twisted version of all of this is as follows ([AChM]). Since dimP 1 = 1, H 1 (P 1 ; 1 P 1 ! 2;cl P 1 ) =H 1 ( 1;cl P 1 ); so all twisted CDO onP 1 are locally trivial. Furthermore,H 1 (P 1 ; 1;cl P 1 ) =C and is spanned by the cocycleC 0 \C 1 7!dx=x. We haveH P 1 =C[ ;@( );::::]. LetD ch;tw C 0 =D ch C 0 H P 1, D ch;tw C1 =D ch C1 H P 1 and deneD ch;tw P 1 by gluingD ch;tw C 0 ontoD ch;tw C1 via 7! ; x7! 1=y; @ x 7!(@ y ) (1) y 2 2@(y) +y (1) : (3.1.14) Morphism (3.1.12) \deforms" to V 2 (sl 2 )! (P 1 ;D ch;tw P 1 ); (3.1.15) e (1) j0i7!@ x ;h(1)j0i7!2(@ x ) (1) x + ;f (1) j0i7!(@ x ) (1) x 2 2@(x) +x (1) : (3.1.16) Furthermore, consider T = e (1) f (1) +f (1) e (1) + 1=2h (1) h2 V 2 (sl 2 ). It is known that T2 z(V 2 (sl 2 )), the center of V 2 (sl 2 ), and in fact, the center z(V 2 (sl 2 )) equals the 40 commutative vertex algebra of dierential polynomials in T . The formulas above show T7! 1 2 (1) @( )2H P 1: (3.1.17) All of the above is easily veried by direct computations, cf. [MSV]. The higher rank analogue is less explicit but valid nevertheless. Let G be a simple complex Lie group, B G a Borel subgroup, X = G=B, the ag manifold, g = LieG the corresponding Lie algebra, h a Cartan subalgebra. One has a sequence of maps h !H 1 (X; 1;cl X )!H 1 (X; 1 X ! 2;cl X ): (3.1.18) The leftmost map attaches to an integral weight 2 P h the Chern class of the G- equivariant line bundleL =G B C , and then extends thus dened mapP!H 1 (X; 1;cl X ) to h by linearity. The rightmost one is engendered by the standard spectral sequence converging to hypercohomology. It is easy to verify that both these maps are isomorphisms. Therefore, h !H 1 (X; 1;cl X ) !H 1 (X; 1 X ! 2;cl X ); (3.1.19) and each twisted CDO on X is locally trivial. Note thatL being G-equivariant, there arises a map from Ug to the algebra of dier- ential operators acting onL or, equivalently, [BB2], Ug!D X : A moment's thought shows that this map is a polynomial in ; hence it denes a universal map Ug!D tw X : (3.1.20) Constructed in [MSV] is a (unique up to isomorphism [GMS2]) CDOD ch X . We arrive at the universal twisted CDOD ch;tw X locally isomorphic toD ch U H X , where H X is the vertex 41 algebra of dierential polynomials on h . Embedding of ane vertex algebra Constructed in [MSV] { or rather in [FF1], see also [F1] and [GMS2] for an alternative approach { is a vertex algebra morphism V h _(g)! (X;D ch X ): (3.1.21) Furthermore, it is an important result of Feigin and Frenkel [FF2], see also an excellent presentation in [F1], that V h _(g) possesses a non-trivial center, z(V h _(g)), which, as a vertex algebra, isomorphic to the algebra of dierential polynomials in rkg variables. Lemma 3.1 [AChM] Morphism (3.1.21) \deforms" to : V h (g)! (X;D ch;tw X ): Moreover, (z(V h _(g)))H X : Sketch of Proof. For each x2 g, (x) can be written, schematically, as follows (x) = (classical) + (chiral) + (classical) ; where (classical) are those terms that appear in the image of the canonical mapUg!D X , (classical)+(classical) are those that appear in the image of the Beilinson-Bernstein map (3.1.20), and (chiral) is the rest; note that equivalently (classical) + (chiral) is the image of map (3.1.21). We have to verify that (x) (1) (y) =h _ hx;yi and (x) (0) (y) =([x;y]). Only terms (classical) + (chiral) contribute to the former; that their contribution is as needed is the content of assertion (3.1.21). Given the former, the latter becomes precisely the classical construction of the morphism Ug!D tw X . 42 The assertion on the image of the center was actually veried in [FF2, F1]. Indeed, since H X is the space of global sections of the constant sheaf H X , it is enough to verify the assertion for the composition of with the embedding of (X;D ch;tw X ) in (X e ;D ch;tw X ), where X e X is the big cell. The space (X e ;D ch;tw X ) is a Wakimoto module, and it is the properties of thus dened morphism from V h (g) to the Wakimoto module that were studied in [FF2, F1].N 3.2 A deformation 3.2.1 Motivation: Wakimoto modules Let X =G=B be a ag variety and U =NB X the big cell of X. In virtue of Lemma 3.1, the sections (U;D ch;tw X ) become a V h _(g)-module, hence a g-module at the critical level. Following [FF1, F2], we call (U;D ch;tw X ) a Wakimoto module of highest weight (0;h _ ), to be denoted W 0;h _. By construction W 0;h _ =D ch (U) H X . In fact, Feigin and Frenkel proved [FF1] that there exists a whole family of g-modules W 0;kh _ =D ch (U) H k where H k is the Heisenberg vertex algebra associated to the space h with bilinear pairing kh;i 0 , i.e., k times the canonically normalized Killing form. The g-module structure is dened by a vertex algebra morphism V kh _(g)!D ch (U) H k and thus, W 0;kh _ is a g-module of level kh _ . When the level is critical, W 0;h _ = (U e ;D ch;tw X ) One might ask whether sheaves with an analogous property exist for Wakimoto modules at a non-critical level. To be more precise, we are interested in a sheafV of vertex algebras such that: 43 its sections on the big cellU and itsW -translates are isomorphic to the tensor product of vertex algebrasD ch (A dimg=b ) H k , for nonzero k; the associated Lie algebroid ofV is the universal tdoT tw . In other words,V is a vertex extension of the pair (T tw G=B ;kh;i 0 ) We show that such sheaves do indeed exist onG=B; moreover, the construction is rather general and can be carried out for any variety. We call the obtained sheaves the deformations of TCDO or deformed TCDO; deformations because they depend onh;i as a parameter, withh;i = 0 corresponding to a TCDO. 3.2.2 Denition The discussion above suggests the following denition. LetX be a smooth projective variety andT tw the Lie algebroid underlying the universal TDO (cf. section 3.1.1). Recall thatT tw X ts into an exact sequence 0!O X !T tw X !T X ! 0 where =H 1 (X; 1 ! 2;cl ): Let us x a symmetric bilinear pairingh;i : ! C and extendO X -linearly to O X . Denition 3.2 We will say that a sheafV is ah;i-deformation of TCDO ifV is a vertex extension of the pair (T tw X ;h;i). Without specifyingh;i, a deformation of TCDO is just a vertex extension of the Lie algebroid T tw X . Being vertex extensions,h;i-deformations form a stack, to be denoted TCDO h;i X :=VExt h;i T tw 44 3.2.3 Classication of deformations We apply the results of sections 2.3.2. Theorem 2.14 implies that, whenTCDO h;i X is locally nonempty, its class is equal to cl(TCDO h;i X ) =cl(CExt h;i T tw ) +ch 2 ( 1 X ) We are going to use the description of cl(CExt h;i T tw ) given in section 2.1.4. Let us work in the setup of sections 3.1.1, 3.1.2. Thus, we pick a basis f r g of H 1 (X; 1 X ! 2;cl X ), a dual basisf r g in H 1 (X; 1 X ! 2;cl X ), and a lifting H 1 (X; 1 X ! 2;cl X )! Z 1 (X; 1 X ! 2;cl X ), so that each r is a pair of cochains ( (1) r ; (2) r )2 Q 1 (U ij ) Q 2;cl (U i ). By construction, the Lie algebroidT tw X admits connectionsr i :T U i !T tw U i such that A ij :=r i r j = k (1) k (U ij ) (3.2.1) (summation over repeated indices is assumed) and c(r i ) = k (2) k (U i ) (3.2.2) Theorem 3.3 Lethi6= 0 be a symmetric bilinear form onO X . Then: (1) hi-deformations exist locally on X if and only if the 4-form h r ; s i (2) r (U i )^ (2) s (U i ) (3.2.3) is exact; (2) Assume (1) and pick, for every i, a 3-form H i such that 2dH i =h r ; s i (2) r (U i )^ (2) s (U i ). Denote ij = 1 2 h r ; s i (2) r (U i ) + (2) r (U j ) ^ (1) s (U ij ) +H i H j 45 and ijk =h r ; s i (1) r (U ij )^ (1) s (U jk ) Then a globalh;i-deformation exists if and only if the class of the cocycle ( ij ; ijk ) in H 2 (X; 2 X ! 3;cl X ) is equal toch 2 ( X ) (minus second graded piece of Chern character of 1 X ). Proof. (1) Follows from Theorem 2.13, since the 4-form (3.2.3) is just the Pontryagin form 1 2 hc(r i )^c(r i )i for the Lie algebroidT tw X . (2) Using the connectionsr i (and formulas (3.2.1), (3.2.2)) in the construction of the section 2.1.4 one veries that the cocycle ( ij ; ijk ) represents the class ofCExt h;i T tw . The statement follows immediately from Theorem 2.14 and the fact that cl(CDO) = ch 2 ( 1 X ) [Bre].N Remark 3.4 In the presence of CDO, the classication problem for deformed TCDO be- comes one for Courant extensions of (T tw X ;h;i), as any CDOD ch denes an equivalence of stacks over X ?D ch :CExt h;i T tw !TCDO h;i : 3.2.4 Deformations of locally trivial TCDO Recall from section 3.1.3 that locally trivial TCDO are constructed in the same way as TCDO by consistently replacing H 1 (X; 1 ! 2;cl ) with H 1 (X; 1;cl ). In particular we construct a Lie algebroid T tw . We dene the corresponding versions of deformations as follows. A locally trivial de- formed TCDO is a vertex extension of T tw . A locally trivialh;i-deformation of TCDO is a vertex extension of ( T tw ;h;i): The locally trivialh;i-deformations form a stackTCDO h;i;lt . Theorem 3.3 has the following analogue in the locally trivial case: 46 Theorem 3.5 Lethi6= 0 be a symmetricT -invariant bilinear form onO X . Then: (1) hi-deformations exist locally on X. (2) everyh;i-deformationA tw;lt h;i is locally isomorphic toD ch U H h;i whereD ch U is a CDO and H h;i is a Heisenberg vertex algebra associated to the space H 1 (X; 1;cl ) with the bilinear formh;i. (3) Denote ijk =h r ; s i 1 r (U ij )^ 1 s (U jk ) and let [(0; ( ijk ))] stand for the class of (0; ( ijk )) in H 2 ( 2 ! 3;cl ). Then the class ofTCDO h;i;lt in H 2 ( 2 ! 3;cl ) is given by cl(TCDO h;i;lt ) =ch 2 ( 1 X ) + [(0; ( ijk ))] Proof. (1) By construction, the Lie algebroid T tw admits at connectionsr i :T U i ! T tw X j U i , which implieshc(r i )^c(r i )i = 0: The local existence now follows from Theorem 2.13. (2) Supposer is a at connection on an open set U X, and letQ =Q r;H be a Courant extension of T tw over U (cf. 2.1.4). ThenQ'T U (O U H 1 (X; 1;cl )) 1 U and sincec(r) = 0 one immediately observes from (2.1.9) and (2.1.10) that the constant subsheaf H 1 (X; 1;cl ) \decouples". It is clear from the construction, that it stays decoupled inQD, for any cdoD on U. It has a structure of a Courant (equivalently, vertex) algebroid over Spec(C) whose vertex envelope is the algebra H h;i . (3) The proof is identical to that of Theorem 3.3, Part (2).N 3.3 Deformed TCDO on ag varieties 3.3.1 Deformed TCDO on P 1 Let us put ourselves in the situation of Example 3.1.4 (TCDO on P 1 ). 47 Thus, we are using standard coordinate charts U 0 and U 1 so that P 1 = U 0 [U 1 with 02U 0 ,12U 1 and coordinate functions x :U 0 !C and y :U 1 !C with x = 1 y . Denote = dy y = dx x a cocycle representative of a generator of 1-dimensional H 1 (P 1 ; 1;cl ). By denition, T tw U i =T U i O U i ; i = 0; 1; (3.3.1) with Lie bracket dened by [;] L = [;], [;a ] =(a) . Letr i :T U i !T tw U i , i = 0; 1 be the canonical inclusions. The formula (3.2.1) in this case reads as r 1 r 0 = dy y ; (3.3.2) which dictates the following gluing map g 01 :T tw 1 j C !T tw 0 j C 7! +i (3.3.3) 7! In the chosen coordinates, it is @ y =x 2 @ x +x : The deformed TCDO We wish to construct a vertex extension of (T tw P 1 ;h)i, whereh;i is a symmetricT tw -invariant O-bilinear pairing on g(T tw ) =O X H 1 (X; 1 ! 2 ) =O . In this case it is deter- mined by a number k2 C assigned toh j i. Let us x k and assume k6= 0 (k = 0 corresponds to the usual TCDO). Since dimP 1 = 1, i = 0 for i > 1, in particular H i (P 1 ; 2 ! 3;cl ) = 0 for all i. Therefore there exists a unique vertex extension for any pair (L;h;i). LetA tw h;i denote the 48 vertex extension of (T tw P 1 ;h;i). Denote byH h;i P 1 the Heisenberg vertex algebra generated by a led satisfying (1) = h ; i; (n) = 0;n6= 1: Theorem 3.5 describesA tw h;i locally: one has isomorphisms of vertex algebras (A tw h;i ) U i 'D ch U i H h;i P 1 ; i = 0; 1: Some global information is provided by the following Theorem 3.6 (1) There are isomorphisms i :A tw h;i j U i !D ch U i H h;i P 1 , i = 0; 1, such that 0 1 1 (@ y ) =x 2 @ x 2dx +x + 1 2 h ; idx (3.3.4) 0 1 1 ( ) = h ; ix 1 dx (3.3.5) (2) The anchor map ofA tw h;i induces a vector space isomorphism H 0 (P 1 ;A tw h;i )'H 0 (P 1 ;T P 1): Proof. (1) The construction of section 2.2.1 and the results of section 2.1.4 imply that the most general gluing formula is as follows: 7!g ij () +A() 1 2 hA();Ai + (3.3.6) g7!ghg;Ai (3.3.7) whereg ij is a transition function for a CDO,2 2 U i \U j ,A =r j r i , for some connectoions r i :T U i !L U i . Applying to our case and using (3.1.9) and (3.3.2), we see that @ y 7!x 2 @ x 2dx +x + 1 2 h ; idx (3.3.8) 49 and the map ~ gj U 1 ! ~ gj U 0 is given by 7! h ; dy y i = +k dx x (3.3.9) (2) The gluing formula (3.3.9) implies that the map H 0 (P 1 ;g)! H 1 (P 1 ; 1 P 1 ) in the long exact sequence associated to 0! 1 P 1 ! ~ g! g! 0 is an isomorphism. SinceH j (P 1 ; 1 ) =H k (P 1 ;g) = 0 forj6= 1,k6= 0, one can conclude thatH i (P 1 ; ~ g) = 0 for all i. In turn, the long cohomology sequence associated to the sequence 0! ~ g!A tw h;i !T P 1! 0 shows that H i (P 1 ;A tw h;i )'H i (P 1 ;T ).N 3.3.2 Embedding of ane sl 2 . For 2 C letA (sl 2 ) denote the vertex algebroid over C equal to sl 2 as a space, with bracket g (0) g 0 = [g;g 0 ] and pairing g (1) g 0 =hgjg 0 i wherehji is the canonically normalized invariant form (for sl 2 , it is 1 4 hji Killing ). Let e =@ x h =2@ x(1) x + f =@ x(1) x 2 2dx +x + 1 2 h j idx (3.3.10) Lemma 3.7 The elements e, f, h given by the formulas (3.3.10) (1) satisfy the relations of b sl 2 () where = h j i 2 2; (2) belong to H 0 (P 1 ;A tw h;i ) Proof. Restricted to the big cell, the statement of Part (1) goes back to Wakimoto [W]; see also [F1]. 50 The rest follows from the following equalities over U 0 \U 1 : @ x =@ y(1) y 2 2dy +y + 1 2 h j idy 2@ x(1) x + = 2@ y(1) y @ x(1) x 2 2dx +x + 1 2 h j idx =@ y (3.3.11) N Corollary 3.8 The formulas (3.3.10) dene an isomorphism of vertex algebroids over k A (sl 2 )'H 0 (P 1 ;A tw h;i ) (3.3.12) that extends to the vertex algebra embedding V (sl 2 )!H 0 (P 1 ;U(D ch;tw h;i )) (3.3.13) Proof. The map dened by (3.3.10) is clearly injective and the rst statement follows by dimension count. The restriction of the second map to the big cell was shown in [F1] to be injective.N 3.3.3 The case of a general ag variety Recall that we have an identication : h 'H 1 (X; 1;cl )'H 2 (X;C) (3.3.14) In other words, the tdo on G=B are classied by h . The Lie algebroidT tw G=B is an extension 0!O G=B C h ! T tw G=B ! T G=B ! 0 A deformation of TCDO is therefore a vertex extension of (T tw G=B ;h;i) whereh;i is a 51 symmetric bilinear pairingh;i : hh!C We have the following Theorem 3.9 Let X =G=B. Then the class ofTCDO hi is equal to 0 if and only ifh;i is proportional to the restriction of the Killing form on h. Proof. First, we nd a convenient cocycle representation of the obstruction. Letf r g be the set of fundamental weights,L r the corresponding line bundles over X, D r algebras of tdo acting onL r and T r the corresponding Lie algebroids. Dene the cocycles r = ( ij r )2 Z 1 (X; 1;cl X ) corresponding to T r . Then the map (3.3.14) is the one taking r to the class of ( ij r ) in H 1 (X; 1;cl X ). Take r to be the basis of h dual to the basisf r g. Using Theorem 3.5 and the existence of CDO on X ([GMS2]), we conclude that the class ofTCDO hi is represented by a cocycleh r j s i ij r ^ jk s . Its image under the natural embedding H 2 (X; 2 ! 3;cl )!H 4 (X;C) (cf. [GMS2]) equals to that of the element S =h r j s i r s 2S 2 h : which naturally corresponds to the formh;i : hh!C. By [BGG], S becomes zero in H 4 (X;C) if and only if S is W -invariant. Therefore, the formh;i has to be a multiple of the Killing form.N 3.4 Embedding of the ane vertex algebra In this section we show that ane Lie algebra embeds into the space of global sections of a deformed TCDO on a ag variety. More precisely we will prove the following result. Theorem 3.10 LetD tw;ch G=B;k be a deformed TCDO on G=B. Then the innitesimal action map a : g!T X lifts to a vertex algebroid embedding A k+k crit (g)! (G=B;D tw;ch G=B;k ); (3.4.1) 52 which is an isomorphism when k6= 0. 3.4.1 Cohomology of ag varieties Let X =G=B. Recall the following classical lemma Lemma 3.11 H i (G=B; j ) = 0 for i6=j Consider the exact sequence of sheaves ofO G=B -modules: 0! (0! 2;cl )! ( 1 ! 2;cl )! 1 ! 0 and the associated LES in cohomology: H 0 (X; 2;cl )!H 1 (X; 1 ! 2;cl )!H 1 (X; 1 )! H 1 (X; 2;cl )!H 2 (X; 1 ! 2;cl )!H 2 (X; 1 )!::: (3.4.2) Using the Lemma, and left exactness of H 0 , (H 0 (X; 2;cl ) H 0 (X; 2 ) = 0) one can rewrite 0!H 1 (X; 1 ! 2;cl )!H 1 (X; 1 )!H 1 (X; 2;cl )!H 2 (X; 1 ! 2;cl )! 0 (3.4.3) Now consider 0! 1;cl ! 1 !d 1 ! 0 0!d 1 ! 2 ! 2 =d 1 ! 0 Since H 0 (G=B;d k ) H 0 (G=B; k+1 ) = 0, the map H 1 (G=B; k;cl )! H 1 (G=B; k ), k 0, is injective, so H 1 (X; 2;cl ) = 0: and therefore H 1 (G=B; 1 ! 2;cl )'H 1 (G=B; 1 ) (3.4.4) 53 Recall the following classical Theorem 3.12 ForX smooth proper variety overC, the Hodge-de Rham spectral sequence degenerates at E pq 1 =H p (X; q X ). This means that the map H p (X; q )!H p (X; q+1 ) induced by d dR is zero, therefore H p (X; q ) =H p (X; q;cl ) at least for p = 0; 1. In particular, H 1 (G=B; 1;cl )'H 1 (G=B; 1 ) (3.4.5) 3.4.2 Cohomology of vertex and Courant algebroids Let g be a simple Lie algebra, t a Cartan subalgebra of g and letX be the ag variety of g. LetV be a transitive vertex algebroid over X. We denote by ~ h the kernel of its anchor 0! ~ h!V!T X ! 0 which is an extension 0! 1 ! ~ h! h! 0 where h is the kernel of the anchor of the associated Lie algebroidL =V= 1 , 0! h!L!T X ! 0 54 All of this may be expressed by the following exact diagram 0 0 ? ? y ? ? y 0! 1 ! ~ h ! h ! 0 ? ? y ? ? y 0! 1 ! V ! L ! 0 ? ? y ? ? y T X T X ? ? y ? ? y 0 0 WhenV is a TCDO, one has h =H 1 (X; 1 ! 2;cl ) O X =H 1 (X; 1;cl ) O X = t O X : Cohomology of the subsheaf ~ h First, let us recall a general observation due to Bressler. The pairing (1) :VV!O X restricted to ~ hV satises ~ h (1) 1 X = 0, and therefore induces a pairing h;i : ~ hL!O X (3.4.6) which isO X -bilinear. This pairing, in turn, inducesh;i : hh!O X . These pairings produce maps h;i : ~ h!L _ (3.4.7) h;i : h! h _ (3.4.8) 55 which t into the diagram 0! 1 ! ~ h ! h ! 0 h;i ? ? y ? ? y h;i 0! 1 ! L _ i _ ! h _ ! 0 (3.4.9) The following lemma is immediate Lemma 3.13 [Bre] The diagram (3.4.9) is commutative and ~ h =L _ h _ h Now assumeV is a (deformed) TCDO, i.e.L =T tw . The exact sequence 0! 1 ! (T tw ) _ i _ ! H 1 (X; 1 ! 2;cl ) O X ! 0 (3.4.10) gives the long exact sequence in cohomology 0!H 0 (X; 1 )!H 0 (X; (T tw ) _ )!H 1 (X; 1 ! 2;cl ) @ !H 1 (X; 1 ) Lemma 3.14 The map@ and the canonical map appearing in the sequence (3.4.3) coincide. Proof. Assume a cover U of X, as in denition ofT tw is given and isomorphisms T tw j U i = hj U i T U i are xed. Then the transition function g ij 2Aut((hT X )j U i \U j ) is g ij = 0 B @ id P ( (1) k ) ij k 0 id 1 C A Therefore, the transition functiong T ij 2Aut((h _ 1 )j U i \U j ) for the dualO X -module (T tw ) _ is g T ij = 0 B @ id 0 P ( (1) k ) ij k id 1 C A 56 where now P ( (1) k ) ij k is to be interpreted as the \projection" map H 1 (X; 1 ! 2;cl )! Z 1 (X; 1 ) taking = ( (1) ij ; (2) i ) to (1) ij .N Now let us compute H i (X; ~ h) for X =G=B. In this case the statement of Lemma 3.14 reduces to @ = id H 1 (X; 1 ) . Consider the LES in cohomology associated to 0! 1 ! ~ h! h! 0: 0!H 0 (X; 1 )!H 0 (X; ~ h)!H 0 (X;h)!H 1 (X; 1 )!H 1 (X; ~ h)!H 1 (X;h)!H 2 (X; 1 )!::: By the fact that H i (X;h) =H i (X;O X ) H 1 (X; 1 ! 2;cl ) one can see that H 0 (X;h) =H 1 (X; 1 ! 2;cl ) and H i (X;h) = 0; i 1. Lemma 3.11 implies that H i (X; ~ h) = 0; i 2 (3.4.11) Consider the rst terms of the LES: 0!H 0 (X; ~ h)!H 0 (X;h)!H 1 (X; 1 )!H 1 (X; ~ h)! 0 0!H 0 (X; ~ h) ! H 1 (X; 1 ! 2;cl ) ~ @ ! H 1 (X; 1 ) ! H 1 (X; ~ h)! 0 Now, the commutativity of the diagram (3.4.9) and lemma 3.14 imply that ~ @ is the composition of the map H 1 (X; 1 ! 2;cl )! H 1 (X; 1 ! 2;cl ) induced byh;i and the canonical map (3.4.4). Whenh;i is nondegenerate, ~ @ is an isomorphism. One obtains Lemma 3.15 SupposeV is a deformed TCDO over G=B with G simple. Then H i (X; ~ h) = 0; i 0: 57 Cohomology ofV Consider the sequence 0! ~ h!V!T X ! 0 and the LES in cohomology associated to it: 0!H 0 (X; ~ h)!H 0 (X;V)!H 0 (X;T X )!H 1 (X; ~ h)!H 1 (X;V)!H 1 (X;T X )! 0 (3.4.12) From Lemma 3.15 one immediately obtains Theorem 3.16 IfV is a deformed TCDO (i.e.h;i6= 0), then H 0 (X;V)'H 0 (X;T X ) (3.4.13) The same result applies to any Courant algebroid extendingT tw with a nondegenerate pairingh;i on h. 3.4.3 Proof of Theorem 3.10 Lemma 3.17 Suppose H 0 (X; 1 ) = 0 andV a vertex or Courant algebroid over X. Then H 0 (X;V) is a Lie algebra with respect to (0) and H 0 (X;V)! H 0 (X;T X ) is a Lie algebra map. Proof. The Jacobi identity is one of the axioms in the denition of either vertex or Courant algebroid; the violation of skew-symmetry, x (0) y +y (0) x lands in H 0 (X; 1 X ); the anchor map preserves brackets.N Corollary 3.18 Let X = G=B andV a deformed TCDO over X. Then the anchor map V!T X induces an isomorphism of Lie algebras H 0 (X;V)'H 0 (X;T X ): 58 Thus one may consider the composition a : g!H 0 (X;T X )!H 0 (X;V) which is a Lie algebra map lifting the map a. Now, the Lie algebra H 0 (X;V) is equipped with an invariant bilinear pairing (x;y)7! x (1) y2H 0 (X;O X ) =C. Denote byh;i a the pullback of this pairing to g; it must also be invariant. In this way a vertex algebroid embedding arises : A(g;h;i a )!H 0 (X;V) (3.4.14) To show that this map is what we need, it remains to convince ourselves thath;i a coincides with (k +k crit ) times the normalized Killing form. And here is one way to see it. Upon restriction on the big cell U, the space of sections ofD tw;ch G=B;k is the Wakimoto moduleD ch (U) H t;k , where t denotes the Cartan subalgebra of g. E. Frenkel showed [F1] the existence of vertex algebra map w k : V k+k crit (g)!D ch (U) H t;k This map restricted to graded component 1 is a morphism of vertex algebroids lifting the Lie algebra map a : g!T X (U), and so is the map (3.4.14). One concludes that the dierence between two is an element of Hom(g; 1 (U)). The following lemma nishes the proof: Lemma 3.19 Suppose a : g!T X , U X,h;i is an invariant bilinear form on g and ! : g! 1 (U) is such that hg;hi = a(h) !(g) + a(g) !(h) (3.4.15) Thenh;i = 0, ! = 0. 59 Proof. The invariance condition and (3.4.15) imply a(z) !([y;x]) + a(x) !([y;z]) = 0 (3.4.16) For x;y in the Cartan subalgebra t this yields a(x) !([y;z]) = 0; x;y2 t; z2 g (3.4.17) Since every E equals a commutator, (3.4.17) implies a(h) !(E ) = 0; h2 t; 2 (3.4.18) Now, by (3.4.15) and (3.4.18) a(E) !(h) =hE ;hi a(h) !(E ) = 0 for all h2 t and 2 . Since a(E ), 2 generateT G=B , this implies that !(h) = 0 for h2 t, which in turn implies thathh;h 0 i = 0 for all h;h 0 2 t. Since a nonzero invariant pairing is nondegenerate on t,h;i must be zero.N. The torsor of local embeddings LetX be aG-variety. The action ofG onX induces the innitesimal action of g onX, i.e. a Lie algebra map a : g!T X . Let us assume that there exists a Lie algebra morphism : g!T tw X (3.4.19) lifting the morphism a. This is true, for example, in the case of X =G=B. LetA k (g) X denote the constant sheaf with sections equal to g, equipped with the 60 structure of aC X -vertex algebroid as follows: x (0) y = [x;y] x (1) y =khxjyi = 0; @ = 0 (3.4.20) LetA be a (locally trivial)h;i-deformation of TCDO. Consider the sheaf of homomorphisms of vertex algebroids Hom (A k (g) X ;A) (3.4.21) that lift the morphism . Proposition 3.20 Suppose the image of a : g!T X generatesT X as anO X -module. Then the sheaf (3.4.21), if locally nonempty, is an 2;cl X -torsor. Proof. Let us work locally on a subset UX small enough to admit an identication Aj U 'T tw X j U 1 U . Let w;w 0 2Hom (A k (g) X ;A)(U). Then w 0 (g) =w(g) +!(g) for some ! : g! 1 , since theT tw -component is xed. Let us show that ! must be given by !(g) = (g) where 2 2;cl X . First, observe that a(h) !(g) + a(g) !(h) = 0 (3.4.22) (this follows from w(g) (1) w(h) =w 0 (g) (1) w 0 (h)). Since the image ofa generatesT X , this easily implies the following: ifa(g 0 ) = P f i a(g i ) for g 0 ;g i 2 g, then !(g 0 ) = P f i !(g i ): 61 Conversely, it is straightforward to check that the result of adding () to any mor- phism inHom (A k (g) X ;A)(U) is inHom (A k (g) X ;A)(U) again.N When dimX = 1 the torsor (3.4.21) is trivial, therefore the existence of local embeddings implies the existence of a global one. 62 Chapter 4: Half-integrable modules over transitive vertex algebroids In this chapter we prove the existence of a ltration for modulesM that satisfy certain integrability condition and use it to prove a generalization of Theorem 5.2. in [AChM]. We work in a setup that is slightly more general than that of TCDO with an intention to apply the results to deformations of TCDO dened in Section 3.2.2. LetV be a sheaf of vertex algebras. We will call a sheaf of vector spacesM aV- module if for each open U X M(U) is aV(U)-module and the restriction morphisms M(U)!M(U 0 ), U 0 U, areV(U)-module morphisms, with theV(U)-module structure onM(U 0 ) given by pull-back. Recall that to each graded vertex algebra V one can associate LieV , the Borcherds Lie algebra of V (cf. section 1.1). We say that aV-moduleM is half-integrable i each point x2X has a neighbourhood U withM(U) a Lie (V(U)) + -integrable module, that is to say, Lie (V ) + acts onM by locally nilpotent operators. We call the center ofV the subsheafZ(V)V with stalksZ X;x = Z(V X;x ), x2 X where Z(V ) denotes the center of V . Dene the categoryM int + (V) as the full subcategory of the category ofV-modulesM such that 1) M is half-integrable 63 2) h n m = 0; for all h2Z(V);m2M: Our goal in this section is the proof of the Theorem below. We work in the analytic topology. In section 4.1.3 we present a version of this result which works in Zariski topology. Theorem 4.1 SupposeV is a vertex envelope of a transitive vertex algebroid A whose associated Lie algebroid L = A= 1 X is locally free of nite rank and ts into an exact sequence 0! h!L!T X ! 0 (4.0.23) in which h is an abelianO X -Lie algebra. LetM2M int + (V). Then (1) M is generated by the subsheaf SingM =fm2M : v n m = 0 for all v2V; n> 0g (2) There is a ltrationfM i g i0 onM withM 0 = SingM, compatible with the grading ofV. 4.1 Proof of Theorem 4.1 4.1.1 The center First of all, let us describe the center of any suchV. For that, let us look more closely at (4.1.21). Since h is abelian, it possesses the structure of aD X -module. It is locally free as anO X -module and therefore, is of the form h =O X C h r where h r denotes the subsheaf of horizontal sections of h. The pairing (1) induces anO X -bilinearL-invariant symmetric pairingh;i : hh!O X which restricts toh;i : h r h r !C X . Let z = h2 h r : hh;h r i = 0 . Thus z is the kernel ofh;i restricted to h r . 64 Lemma 4.2 There exists a lifting s : z!A, i.e. s = id z , such that s(h), h2 z generate a subalgebra in (X;V) that is central in every (U;V), U X. Moreover, such a lifting is unique. Proof. Let s 0 : z!A be any (local) lifting. Fix a basis i inT (U) and the dual basis f! i g in 1 (U). Fix arbitrary lifts ~ i of i inA. For s 0 (h) to be central, s 0 (h) (1) ~ i must be zero for all i. This may fail, and so we are forced to change s 0 . It is clear that s(h) =s 0 (h) (~ i(1) s 0 (h))! i ; h2 z satises the desired condition s(h) (1) ~ i = 0 for all h2 z and all basis elements i ; further- more, the latter condition determines s(h) uniquely. Therefore, s is unique and extends globally. It remains to show that ~ i(0) s(h) = 0 for all i Now, [ i ;h] = 0 inL (since z h r ). Therefore ~ i(0) s(h) is in 1 . However, for any 1j dimX (~ i(0) s(h)) (1) ~ j = ~ i(0) (s(h) (1) ~ j )s(h) (1) (~ i(0) ~ j ) = 0 Thus ~ i(0) s(h) must be zero.N LetUX be suitable for chiralization (cf. 1.2.4) Choose an abelian basisf i g ofT (U), the dual basisf! i g in 1 (U) and a basisfh k g of h r (U). Choose any liftingL!A extending that of Lemma 4.2 and identify i andh k with the corresponding lifts. Let V be the vertex enveloping algebra ofA(U). V is generated byO U , vector elds f i g and elements h k . 65 The following relations hold in V . [! i;n ;! j;l ] = 0 (4.1.1) [ i;n ;! j;l ] = n i;j n+l;0 id (4.1.2) i;n ; j;l = n+l + X k f k(1) h k n+l (4.1.3) [h k;n ;h l;m ] = nhh k ;h l i n+m;0 id (4.1.4) [ i;n ;h r;m ] = n+m (4.1.5) where 2 1 (U), f k 2O(U) and 2 1 (U) may depend on i , j and h r resp. Using (4.1.2 - 4.1.5), it is easy to show that the subalgebra generated by (U;s(z)), see Lemma 4.2, is all of Z(V ). 4.1.2 Proof of (1) Let us now describe the strategy of proving Theorem 4.1, (1). The statement is local and we continue working on a suitable for chiralization subset U. Let M =M(U). Denote SingM =fm2M :v n m = 0 for all v2V;n> 0g Introduce the ltration: 0M ;h M M (4.1.6) where M =fm2M :! i;n m = 0 for all n> 0;ig M ;h = m2M :h r;n m = 0 for all n> 0;r Let [SingM] be the submodule of M generated by SingM. We show, step by step, that each of the terms in the ltration (4.1.6) is generated by SingM. 66 Step 1 We show that [SingM] contains the subspace M ;h . Let us introduce a ltration on M ;h indexed by functions d : f1; 2;:::; dimXgZ + !Z + vanishing at all but nitely many pairs (i;n). Call any such function a degree vector. For a degree vector d dene M ;h (d) = \ i;n>0 ker d(i;n)+1 i;n : (4.1.7) It is clear from denitions that M ;h = S M ;h (d) and M ;h (0) = SingM. Let us show that each M ;h (d) is a subspace of [SingM] using the induction on the lengthjdj = P i;n d(i;n) of d. The base of induction is established in the line above. Let d6= 0 be a degree vector. SupposeM ;h (d 0 ) [SingM] for all d 0 of smaller length. Fix (i;n) such thatk :=d(i;n)> 0 and let d 0 be equal to d everywhere except at (i;n) where d 0 (i;n) = d(i;n) 1. By induction assumption, M ;h (d 0 ) [SingM]. Let m2M ;h (d). Since m2 ker k+1 i;n , one has 0 =! i;n k+1 i;n m = k i;n (nkm +! i;n i;n m) (4.1.8) Introduce the elements m 0 = i;n m; m 00 =nkm! i;n m 0 : (4.1.9) Then m = 1 nk (m 00 +! i;n m 0 ) (4.1.10) 67 Hence, to show m2 [SingM] it suces to show that m 0 , m 00 lie in [SingM]. Let us show m 0 2M ;h (d 0 ). One has k i;n m 0 = k+1 i;n m = 0. To show d(j;l)+1 j;l i;n m = 0 it suces to show [ j;l ; i;n ] q j;l m = 0; q 0: But that follows from the fact that j;l M ;h M ;h , which is a consequence of (4.1.2), (4.1.5). Finally, to see thatm 00 is inF d 0 , we need to check that d 0 (j;l)+1 j;l m 0 = 0 for all (j;l). For (j;l)6= (i;n) this follows immediately from (4.1.2) and the fact that m 0 2 F d 0 . The case (j;l) = (i;n) is clear due to (4.1.8). Remark 4.3 Note that (4.1.8) is a particular case of the following observation used in Steps 2 and 3 as well and originating in Kashiwara's lemma. Let A and B be linear operators on a space V such that [A;B] commutes with A. Suppose A n+1 m = 0 for m2V , n 0. Then A n (n[B;A] +BA)m = 0 (4.1.11) Indeed, 0 =BA n+1 m = [B;A n ]Am +A n BAm =A n (n[B;A] +BA)mN Steps 2 and 3 are proved in essentially the same way. The reader may safely skip the rest of the proof. However, for the sake of completeness we will keep the same level of detail. Step 2 We show M [SingM]. In the caseh;i = 0 the assumptions of the theorem imply h r;n = 0 on M so that M =M ;h and there's nothing to prove. 68 Let us prove the claim in the case z is a proper subset of h r . Letfb r g be an orthonormal basis for some complement h r nondeg to z in h r . In particular [b k;n ;b l;m ] =n k;l n+m;0 id (4.1.12) Introduce the ltration M (d) = n m2M : (b r ) d(r;n)+1 n m = 0;8n> 08r o where d :f1:: dimh r nondeg gZ + !Z + is a degree vector. Then, clearly, M (0) =M ;h and M = S d M (d). Suppose that d6= 0 is a degree vector and suppose M (d 0 ) is a subspace of [SingM] for all d 0 of smaller length. Let us show M (d) [SingM] Fix some i;n such that d(i;n) > 0 and dene d 0 by d 0 (j;l) = d(j;l) ij nl . Let k = d(i;n). Let m2M (d). We have (cf. (4.1.11)) 0 = (b i;n ) k (nkm +b i;n b i;n m) (4.1.13) Introduce the elements m 0 =b i;n m; m 00 =nkm +b i;n m 0 (4.1.14) Since!'s commute withb r 's, these elements are inM wheneverm is. We wish to show that they are in fact in M (d 0 ) and therefore, in [SingM]. This would imply that m = 1 nk (m 00 +b i;n m 0 ) (4.1.15) 69 is in [SingM] as well. It is clear from (4.1.12) that m 0 2M (d 0 ). Let us show that m 00 2M (d 0 ). We need to show m 00 2 ker(b s;m ) d 0 (s;m)+1 for all s and all m > 0; but this follows from (4.1.12) in case (s;m)6= (i;n) and from (4.1.13) in case (s;m) = (i;n). Thus m 00 2M (d 0 ). Step 3 We complete the proof by showing M [SingM]. Introduce a ltration onM indexed by degree vectors d : f1; 2;:::; dimXgZ + !Z + where for each d we dene M(d) = \ i;n>0 ker(! d(i;n)+1 i;n ): (4.1.16) Clearly, M = S M(d) and M(0) =M . In order to show that M(0) [SingM] implies M(d) [SingM] for all d we will use induction on the lengthjdj = P i;n d(i;n). Thus, we suppose that d6= 0 is a degree vector and M(d 0 ) is a subspace of [SingM] for all d 0 of smaller length. Let us show M(d) [SingM] Fix some i;n such that d(i;n) > 0 and dene d 0 by d 0 (j;l) = d(j;l) ij nl . Let k = d(i;n). Let m2M(d). Since m2 ker! k+1 i;n , one has (cf. (4.1.11) ) 0 = i;n ! k+1 i;n m =! k i;n (nkm + i;n ! i;n m) (4.1.17) It follows from (4.1.1 -4.1.3) and (4.1.17) that the elements m 0 =! i;n m; m 00 =nkm i;n m 0 (4.1.18) 70 belong to M(d 0 ) [SingM]. Therefore, m = 1 nk (m 00 + i;n m 0 ) (4.1.19) is also in [SingM].N 4.1.3 Proof of (2) Let us work locally as in the proof of part (1) and keep all the notation used there. Thus we have UX, M =M(U), V is the vertex envelope ofA(U). Dene Lie (V ) + to be the subalgebra of the Borcherds Lie algebra Lie (V ) spanned by v t n with positive n; and let Lie (V ) 0 be the image of V C[t 1 ] in Lie (V ). Then Lie (V ) = Lie (V ) + Lie(V ) 0 The enveloping algebras U(Lie (V )), U(Lie (V ) + ) have a natural grading dened by degv n 1 :::v n k =n 1 + +n k . Dene the following subspaces of M: M n =fm : U(Lie (V ) + ) k m = 0; for all k>ng (4.1.20) It is clear that M n M n+1 , M 1 =f0g and M 0 = SingM. The decomposition U(Lie (V ))' U(Lie (V ) ) U(Lie (V ) + ) implies that M n = ~ M n where ~ M n =fm2M : U(Lie (V )) k m = 0 for all k>ng Note that ~ M n is compatible with the grading ofV , i.e. v k ~ M n ~ M nk : Hence, the submodule [SingM] generated by SingM =M 0 is a subset of the union of ~ M n . But [SingM] =M by part (1) and thus, M n is an (exhaustive) ltration. SinceV is a graded sheaf, the formula (4.1.20) makes sense globally, giving the desired ltrationM n .N 71 The algebraic case Theorem 4.1 remains true in Zariski topology, as long as one imposes one additional re- striction. Since algebraicD-modules may have non-algebraic solutions, h r may fail to have "right size". Thus, one has to demand that h be equal toO X h r . To be more precise, one has: Theorem 4.4 SupposeV is a vertex envelope of a transitive vertex algebroid A whose associated Lie algebroidL =A= is locally free of nite rank and ts into an exact sequence 0! h!L!T X ! 0 (4.1.21) in which h =O X h r for a locally constant sheaf h r such that [L;h r ] = 0. LetM2M int + (V). Then: (1) M is generated by the subsheaf SingM =fm2M : v n m = 0 for all v2V; n> 0g (2) There is a ltrationfM i g i0 onM withM 0 = SingM, compatible with the grading ofV. The proof is identical to that of Theorem 4.1. 4.2 Equivalence Now we prove a generalization of Theorem 5.2 in [AChM] in the setup of the previous section. This is a result establishing the equivalence between certain subcategories ofM int + (V) and categories of modules over sheaves of tdo, based on a version of Zhu correspondence. The result is true in both analytic and Zariski topology. LetV be as in Theorem 4.1 and assume further that h =O X h r with h r a constant sheaf. By abuse of language we denote its ber by h r and write (h r ) and z for vector 72 space duals of h r and z :=fh2 h r : hh;si = 0;s2 h r g Let us x 2 (h r ) ; = X n2Z n z n1 2 z ((z)): Dene the categoryM int + ;(z) (V) to be the full subcategory ofM int + (V) consisting of modulesM satisfying: (1) for all h2 h r , m2M h 0 m =(h)m; (4.2.1) (2) for all c2 z, n2Z c n m = n (c)m; (4.2.2) ForM int + ;(z) (V) to be nonzero, an evident compatibility condition has to be satised: j z = 0 (4.2.3) Furthermore, the half-integrability condition dictates n = 0; n> 0 (4.2.4) for any nonzero module inM int + ;(z) (V). In other words, (z) has regular singularity at 0. LetM2M int + ;(z) (V) be arbitrary. By Theorem 4.1,M is aZ 0 -ltered module over a sheaf of (graded) vertex algebrasV. Thus,M 0 = SingM is equipped with an action of the Zhu algebra ofV, or, rather, a sheaed version of it. Furthermore, the action ofZhu(V) onM 0 = SingM factors throughZhu(V)=I where I is the ideal generated by h(h), h2 h r . AsZhu(V) =U(L) ( [AChM], Theorem 3.1), the quotient algebraZhu(V)=I is a TDO, to be denotedD X . Thus we have a functor Sing : M int + ;(z) (V)!M(D X ) (4.2.5) 73 Theorem 4.5 Assume that the conditions (4.2.3, 4.2.4) are satised. Then the functor (4.2.5) is an equivalence of categories. This theorem is a generalization of Theorem 5.2., [AChM] and has an almost identical proof. For the sake of completeness, however, we will provide all the details. 4.2.1 Proof of Theorem 4.5. In order to prove Theorem 4.5 we will construct the left adjoint to (4.2.5) and show that it is a quasi-inverse of (4.2.5). The left adjoint to (4.2.5) We begin by constructing the left adjoint functor locally. DenoteM int + ;(z) (V(U)) the category of lteredV(U)-modules dened by analogy with M int + ;(z) (V). The functorM7!M 0 fromM int + ;(z) (V(U)) toModD X (U) admits a left adjointZhu . It is constructed as follows. Let F be a (U;D X )-module. In particular, it is a (U;U O X (L))-module, by pullback. Recall ([R] , see also [AChM], section 3.5.1) that if V is a vertex algebra and Zhu(V ) its Zhu algebra, one can dene a functor Zhu V : Zhu(V )mod! Vmod, left adjoint to M7!M 0 . Let us dene ~ F =Zhu V (F ) (4.2.6) where V =V(U). Then ~ F is aV-module such that ~ F 0 is a (U;U O X (L))-module on which h2 h r act via . For a graded (U;V)-module N denote K (N) to be the subspace spanned by vectors of the form c (n) m n (c)m; where m2N, n1, c2 z. It is easy to see that K (N) is a submodule of N. 74 Dene Zhu (F ) = ~ F=K ( ~ F ): (4.2.7) It is not hard to see that Zhu (F ) is an object ofM int + ;(z) (V(U)). AnyD X (U)-module map f :F!F 0 extends uniquely to a map Zhu(f) :Zhu (F )! Zhu (F 0 ). Therefore, the functor Zhu is the left adjoint to the functor M!M 0 . Now we proceed to dene a sheaf version of Zhu . IfM is aD X -module, let us denote byZhu (M) the sheaf associated to the presheaf U7!Zhu ;U M(U) (4.2.8) with restriction maps extended uniquely from those ofM. It is clear thatZhu (M) belongs toM int + ;(z) (V). Since maps extend uniquely, this extends to a functor Zhu :ModD X !M int + ;(z) (V) (4.2.9) left adjoint to the functor (4.2.5). The quasi-inverse property We have to show the following two functor isomorphisms Zhu ! Id ModD X ; (4.2.10) Zhu ! Id M int + ;(z) (V) ; (4.2.11) The rst is obviously true, because by construction the functors are actually equal: Zhu = Id ModD X . Let us now prove (4.2.11). Let UX be a suitable for chiralization open subset of X, A = (U;O X ),f@ i g be an abelian basis forA-module (U;T X ),f! i g the dual basis of (U; 1 X ), andfl r g an orthogonal basis of a complement to z in h r . 75 Let V = (U;V), M a V -module IdentifyL U withT U h 1 U . by xing a connectionr :T U !L U and choosing a section s : h U !V 1 j U as in Lemma 4.2. We will denote the kth mode of @ i (resp. ! i , l r ) by @ i;k (resp. ! i;k , l r;k .) LetP denote the polynomial algebra in variablesD i n , i n ,L r n n> 0, 1i dimX. Dene the map a :P! EndM, a(D i n ) =@ i;n , a( i n ) =! i;n , a(L r n ) =l r;n . Choose any total order on the set of variables that satises D i n L r k j m 1 for all m> 0, k> 0, n> 0, i;j;r, and A n B m if n>m and both A and B are either D i , or L k , or j . Dene :P M 0 !M as follows: x 1 x 2 :::x k m 7! a(x 1 )a(x 2 ):::a(x k )m (4.2.12) where x 1 x 2 x k ; for k = 0 set to be the identity map of M 0 . Lemma 4.6 Suppose M is a ltered (U;D ch;tw X )-module generated by M 0 , on which H X acts via the character (z)2 z [[z]]z 1 Then: (A) The map (4.2.12) is a vector space isomorphism. (B) If NM is a non-zero submodule, then N\M 0 is also non-zero. Proof of Lemma 4.6. (A) It is clear from the assumptions that the map (4.2.12) is surjective. To prove injectivity, extend to the lexicographic order on the set of monomials x 1 x 2 :::x k m. Let 2 Ker and = 0 + , where 0 is the leading (w.r.t. the lexicographic ordering) non-zero term. Write 0 =x 1 x 2 :::x k m. Then y 1 y 2 y k ( ) = 0; where we choose y j to be 1 n @ i;n if x j = i n , 1 n ! i;n if x j =D i n , and 1 nhlr;lri l r;n if x j =L r n . The relations (4.1.1 { 4.1.5) and the fact thatM 0 = SingM is the lowest component of 76 M imply that y 1 y 2 y k ( ) = @ @x 1 @ @x 2 @ @x k (x 1 x 2 x k ) (m): Therefore (m) = 0, but the restriction of to M 0 being the identity, m has to be zero, hence = 0, as desired. Proof of item (B) is very similar: one has to pick a non-zero 2N of the lowest degree, and then apply to the highest degree term, 0 , an appropriate y 1 y 2 y k so as to produce a non-zero element of N\M 0 .N Theorem 4.5 follows from Lemma 4.6 easily. We have the adjunction morphism Zhu ! Id M int + ;(z) (V) ; (4.2.13) hence Zhu (M)!M: (4.2.14) for eachV-moduleM. The restriction of (4.2.14) toM 0 is the identity. By construc- tion,Zhu (M) is generated byM 0 = (M). Therefore, due to Lemma 4.6, it is an isomorphism, hence (4.2.13) is a functor isomorphism. This proves (4.2.11).N As an example, consider the case of TCDO, V = D ch;tw X . In this case L = D tw , h r = H 1 (X; [1;2i X ) andh;i is trivial, so that z = H 1 (X; [1;2i X ) and (4.2.3) translates into the condition = 0 2 H 1 (X; [1;2i X ). Thus, we can simplify the notation and write M int + (z) (D ch;tw X ) instead ofM int + ;(z) (D ch;tw X ). Theorem 4.5 specializes to the following variant of [AChM], Theorem 5.2. Theorem 4.7 Suppose that (z)2 H 1 (X; [1;2i X )((z)) has regular singularity. Then the functor Sing :M int + (z) (D ch;tw X )!D 0 X mod is an equivalence. 77 WhenV is a CDO, i.e. h = 0, one obtains the equivalence between half-integrable modules over a CDOV and the category of D-modules over X. 78 Bibliography [AChM] T. Arakawa, D. Chebotarov, F. Malikov, Algebras of twisted chiral dierential operators and ane localization of g-modules, Sel. Math. Vol. 17, no. 1 (2011), 1-46 [BB1] A. Beilinson, J. Bernstein, Localisation de g-modules. (French) C. R. Acad. Sci. Paris S er. I Math. 292 (1981), no. 1, 15{18. [BB2] A. Beilinson, J. Bernstein, A proof of Jantzen conjectures. I. M. Gelfand Seminar, 1{50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993. [BD1] A. Beilinson, V. Drinfeld, Chiral algebras. American Mathematical Society Collo- quium Publications, 51. American Mathematical Society, Providence, RI, 2004. vi+375 pp. ISBN: 0-8218-3528-9 [Bo] A. Borel, ed., Algebraic D-Modules, Perspectives in Mathematics, 2, Boston, MA: Academic Press, 1987. ISBN 978-0-12-117740-9 [BGG] J. Berstein, I.M. Gelfand, S.I. Gelfand. Shubert cells and the cohomology of the ag manifold (Russian). Func.Analysis and Applications, vol.7, no. 1, (1973), 64-65. [Bre] P. Bressler, The rst Pontryagin class. Compositio Mathemarica, 143 (2007), 1127- 1163 [D] P. Deligne, La formule de dualit e globale, 1973. SGA 4 III, Expos e XVIII. [F1] E. Frenkel, Wakimoto modules, opers and the center at the critical level. Adv. Math. 195 (2005), no. 2, 297{404. 79 [F2] E. Frenkel, Langlands correspondence for loop groups, Cambridge University Press, 2007 [FBZ] E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves, 2nd edition, Mathe- matical Surveys and Monographs, v.58, AMS, 2004 [FF1] B. Feigin, E. Frenkel, Representations of ane Kac-Moody algebras and bosonization, in: V.Knizhnik Memorial Volume, L.Brink, D.Friedan, A.M.Polyakov (Eds.), 271-316, World Scientic, Singapore, 1990 [FF2] B. Feigin, E. Frenkel, Ane Kac-Moody algebras at the critical level and Gelfand- Dikii algebras, in: Innite Analysis, eds. A.Tsuchiya, T.Eguchi, M.Jimbo, Adv. Series in Math. Phys. 16 197-215, Singapore, World Scientic, 1992 [GMS1] V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral dierential operators. II. Vertex algebroids, Invent. Math. 155 (2004), no. 3, 605-680. [GMS2] V. Gorbounov, F. Malikov, V. Schechtman, On chiral dierential operators over homogeneous spaces. Int. J. Math. Math. Sci. 26 (2001), no.2, 83{106. [HTT] R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and represen- tation theory, Progress in Mathematics, 236, Boston, MA: Birkh auser Boston, 2008. MR2357361, ISBN 978-0-8176-4363-8. [KV1] M. Kapranov, E. Vasserot, Vertex algebras and the formal loop space. Publ. Math. Inst. Hautes Etudes Sci. 100 (2004), 209-269. [KV2] M. Kapranov, E. Vasserot, Formal loops IV: chiral dierential operators, preprint math.AG/0612371. [LWX] Z.-J. Liu, A. Weinstein, and P. Xu, Manin triples for Lie bialgebroids. J. Dierential Geom. Volume 45, Number 3 (1997), 547-574. [MSV] F. Malikov, V. Schechtman, A. Vaintrob, Comm. in Math. Phys. 204 (1999), 439-473 80 [R] M. Rosellen, A course in vertex algebra, arXiv:math/0607270. [W] M. 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Abstract (if available)

Abstract In this dissertation we accomplish the following: ❧ 1) We present a classification of transitive vertex algebroids on a smooth variety X carried out in the spirit of Bressler's classification of Courant algebroids. In particular, we compute the class of the stack of transitive vertex algebroids. ❧ 2) We define deformations of sheaves of twisted chiral differential operators (TCDO) introduced in [AChM] and use the classification result to describe and classify such deformations. As a particular case, we obtain a localization of Wakimoto modules at non-critical level on flag manifolds. ❧ 3) We study representation theory of TCDO and their deformations. In particular, we show an equivalence between certain categories of modules over (deformed) TCDO and categories of twisted D-modules, thus extending a result of [AChM].

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Creator Chebotarov, Dmytro (author)

Core Title Classification of transitive vertex algebroids

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 08/03/2011

Defense Date 05/23/2011

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag algebroids,classification,OAI-PMH Harvest,representation theory,vertex algebras

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Malikov, Fedor (committee chair), Friedlander, Eric M. (committee member), Pilch, Krzysztof (committee member)

Creator Email chebotar@usc.edu,chebotarov@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c127-647776

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