UNIVERSITY OF CALIFORNIA, IRVINE Deformation Quantization of Vector Bundles on Lagrangian Subvarieties DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics by Taiji Chen Dissertation Committee: Professor Vladimir Baranovsky, Chair Associate Professor Li-Sheng Tseng Associate Professor Jeffrey D. Streets 2018 c 2018 Taiji Chen TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii CURRICULUM VITAE iv ABSTRACT OF THE DISSERTATION v 1 Introduction 1 2 Formal Weyl algebra 8 2.1 Ideals and modules . 10 2.2 Lie algebra of derivations . 11 2.3 Non-abelian Lie algebra extensions . 13 3 Harish-Chandra torsors 18 3.1 Harish-Chandra pairs . 18 3.2 Group torsors . 21 3.3 Transitive Harish-Chandra torsors . 23 4 Quantization of Vector Bundles on Lagrangian Subvarieties 26 4.1 Modules over a Lie algebroid . 27 4.2 The construction of the torsor . 32 4.3 The Harish-Chandra extension . 35 4.4 Lifting the torsor by steps . 38 4.5 Remarks and open questions . 45 5 Quantization of Line Bundles on Coisotropic Subvarieties 49 5.1 Formal geometry . 50 5.2 L1-algebra and Maurer-Cartan equation . 52 5.3 Lifting the torsor by solving curved Maurer-Cartan equation . 54 Bibliography 66 ii ACKNOWLEDGMENTS I would like to express sincere gratitude to my advisor Vladimir Baranovsky, for his constant guidance, limitless patience and amazing benefaction. Without his excellent guidance I would not have been able to finish this thesis. I would also like to thank my wife Yi He, for all of her love, help and encouragement. She sacrificed her life on taking care of our newborn daughter Hosanna Chen throughout this journey. iii CURRICULUM VITAE Taiji Chen EDUCATION Doctor of Philosophy in Mathematics 2018 University of California Irvine, CA Bachelor of Science in Mathematics 2012 Capital Normal University Beijing, China TEACHING EXPERIENCE Teaching Assistant 2013{2018 University of California Irvine, CA iv ABSTRACT OF THE DISSERTATION Deformation Quantization of Vector Bundles on Lagrangian Subvarieties By Taiji Chen Doctor of Philosophy in Mathematics University of California, Irvine, 2018 Professor Vladimir Baranovsky, Chair We consider a smooth subvariety Y in a smooth algebraic variety X with an algebraic symplectic form !. Assume that there exists a deformation quantization O~ of the structure sheaf OX which agrees with !. When Y is Lagrangian, for a vector bundle E on Y , we establish necessary and sufficient conditions for the existence of the deformation quantization of E, i.e., an O~-module E~ such that E~=~E~ ' E: If the necessary conditions hold, we describe the set of equivalence classes of such quantizations. In the more general situation when Y is coisotropic, we reformulate the deformation problem into the lifting problem of torsors. We expect a deformation quantization of a line bundle on a coisotropic subvariety is equivalent to a solution of curved Maurer-Cartan equation of a curved L1-algebra. v Chapter 1 Introduction Deformation quantization theory can be viewed as a mathematical interpretation of the connection between the classical mechanical system and the quantum mechanical system. A deformation quantization problem usually starts with a classical system described by a commutative algebra A. By using the information contained in the classical system, the deformation quantization of A turns it into a non-commutative algebra B which depends on a parameter ~. So that for ∼ ~ = 0, it becomes A, i.e., B=~B = A. For a more concrete example, consider a smooth algebraic variety X with an 0 2 algebraic symplectic form ! 2 H (X; ΩX ), and assume that we are given a deformation quantization O~ of the structure sheaf OX which agrees with !. This means that O~ is a Zariski sheaf of flat associative C[[~]]-algebras on X, for which we can find local C[[~]]-module isomorphisms η : O~ 'OX [[~]] such that its product ∗ satisfies 1 a ∗ b ≡ ab + P (da; db) (mod 2) 2~ ~ 1 where a; b are local sections of OX (viewed as local sections of O~ using η) and 0 2 P 2 H (X; Λ TX ) is the Poisson bivector obtained from ! via the isomorphism 1 TX ! ΩX induced by the same !. Given such data and a coherent sheaf E of OX -modules, we could look for a deformation quantization of E as well. Thus we want a Zariski sheaf E~ of Oh-modules which is flat over C[[~]], complete in ~-adic topology and such that the O~-action reduces modulo ~ to the original action of OX on E. The usual questions are: does E~ exist at all, if yes then how many such sheaves can we find? In full generality, this is a difficult problem. One possible simplification is to assume that E is a direct image of a vector bundle on a closed smooth subvariety j : Y ! X. We will denote this bundle by E as well (i.e., abusing notation we think of any sheaf on Y as a sheaf on X using the direct image functor j∗). In general, E~ will not exist at all. The first observation is that Y must be coisotropic with respect to the symplectic form !. In other words, the bivector P projects to a zero section of Λ2N , where N is the normal bundle. See Proposition 2.3.1 in [3] for the explanation why Y has to be coisotropic (this also follows from the proof of Gabber's Integrability of Characteristics Theorem). If we assume 1 that Y is Lagrangian (i.e. isotropic of dimension 2 dimC X). Then P induces a perfect pairing between the tangent bundle TY and the normal bundle N of ∗ Y . and hence an isomorphism N 'TY . The case when E has rank r = 1 is considered in [1] and we deal with general r in Chapter 4. The second observation is that E must carry a structure somewhat similar to a flat algebraic connection. One could say that the \quasi-classical limit" of E~ is 2 given by E together with this additional structure, and it is this quasi-classical limit which is being deformed, not just E. More details are given in Section 4.1, and the brief account follows here. A convenient language to use is that of Picard algebroids on Y , cf. [4] that is , those Lie algebroids L which fit a short exact sequence 0 !OY !L!TY ! 0 (the trivialization of the sheaf on the left is chosen and deforms a part of the structure). Such algebroids are classified by their characteristic class c(L) with values in the truncated de Rham cohomology 2 2 1 2 HF (Y ) := H (Y; 0 ! ΩY ! ΩY !··· ) O~ One example of such a sheaf is L(O~) = T or1 (OY ; OY ). Next, E itself gives an Atiyah Lie algebroid L(E) with its exact sequence 0 ! EndOY (E) !L(E) !TY ! 0: 2 A choice of deformation quantization E~, or even the isomorphism class of E~=~ E~ 2 as a module over O~=(~ ), gives a morphism of Zariski sheaves γ : L(O~) !L(E) which agrees with the Lie bracket but not the OY -module structure. One can change the module structure on the source of γ, also changing its characteristic 2 + class in HF (Y ), to obtain a new Picard algebroid L (O~) and a morphism of + + Zariski sheaves γ : L (O~) !L(E) which now agrees with both the bracket and the OY -module structure. It also embeds OY ! EndOY (E) as scalar 3 endomorphisms and descends to identity on TY . In this situation, following + [4] we say that E is a module over L (O~). Existence of such γ+ is a non-trivial condition on E. We will see in Section 4.1, and it is only a slight rephrasing of Section 2.3 in [4], that in this case the projectivization P(E) has a flat algebraic connection and the refined first Chern 2 class c1(E) = c(L(det E)) 2 HF (Y ) safisfies the identity 1 c (E) = c(L+(O )): r 1 ~ + Existence of a full deformation quantization for an L (O~)-module E is formulated in terms of the non-commutative period map of [6]-a particular choice of O~ gives a class 2 3 [!] + ~!1 + ~ !2 + ~ !3 + ··· in the algebraic de Rham cohomology of X. We will mostly treat the period map as a black box, appealing to rank 1 result of [1] that will serve as a bridge between the definitions in [6] and our argument. One remark, for which we are grateful to Alexander Gorokhovsky, is that it is slightly better to divide this class by ~ and work with 1 2 [!] + !1 + ~!2 + ~ !3 + ··· ~ Since in this case the formulation of our result below will also become invariant under automorphism of C[[~]] of the form ~ 7! ~ + ~2 ··· In the Lagrangian condition, [!] restricts to zero on Y . The class of c(L(O~)) in 2 ∗ HF (Y ) is a canonical lift of the restriction j !1 of !1 under the closed embedding ∗ j : Y ! X. We will abuse notation and write j !1 for that lift as well (note however that in a number of cases of interest, such as X and Y being complex 4 2 2 projective, HF is a subspace of HDR so equations in the truncated de Rham cohomology may be viewed as equations in the usual de Rham cohomology). The class !1 affects the choice of E via the identity 1 c(L+(O )) = c (K ) + j∗! ; ~ 2 1 Y 1 established in Proposition 4.3.5 and Lemma 5.3.5(ii) of [1]. The restrictions of the remaining classes are also an important ingredient in our main result on the Lagrangian case: Theorem 1.1.