Deformation Quantization of Vector Bundles on Lagrangian Subvarieties

UNIVERSITY OF CALIFORNIA, IRVINE

DISSERTATION

submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in Mathematics

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 L∞-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

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 ω. Assume that there exists a deformation quantization O~ of the structure sheaf OX which agrees with ω.

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 L∞-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 ω ∈ H (X, ΩX ), and assume that we are given a deformation quantization O~ of the structure sheaf OX which agrees with ω.

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 ∈ H (X, Λ TX ) is the Poisson bivector obtained from ω via the isomorphism

1 TX → ΩX induced by the same ω.

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 ω. 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)) ∈ 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 [ω] + ~ω1 + ~ ω2 + ~ ω3 + ··· 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 [ω] + ω1 + ~ω2 + ~ ω3 + ··· ~ 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, [ω] restricts to zero on Y . The class of c(L(O~)) in

2 ∗ HF (Y ) is a canonical lift of the restriction j ω1 of ω1 under the closed embedding

∗ j : Y → X. We will abuse notation and write j ω1 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 ω1 aﬀects the choice of E via the identity

1 c(L+(O )) = c (K ) + j∗ω , ~ 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. A rank r vector bundle E on a smooth Lagrangian subvariety j : Y → X admits a deformation quantization, i.e., there exists a complete ﬂat left O~-module E~ such that E~/~E~ ' j∗E if and only if the following conditions hold:

∗ 2 (1) j ωk = 0 in HDR(Y ) for k > 2;

(2) the projectivization P(E) admits a ﬂat algebraic connection;

2 (3) the refined first Chern class in HF (Y ) satisfies

1 1 c (E) = c (K ) + j∗ω ; r 1 2 1 Y 1

∗ 2 for the canonical lift of j ω1 to HF (Y ) representing the class of the Picard

O~ algebroid L(O~) = T or1 (OY , OY ).

If nonempty, the set of equivalence classes of all rank r deformation quantization on Y for various E has a free action of the group G of isomorphism classes of

∗ OY [[~]] -torsors with a flat algebraic connection. The set of orbits for this action may be identified with the space of all P GLr(C[[~]]) bundles with a flat algebraic connection.

5 In the C∞ case when a choice of connection is available and the cotangent bundle of Y serves as a good local model of X near Y , one can say more. In fact, Theorem 6.1 in [8] gives an explicit and canonical quantized action on half-densities (thus, r = 1), of a speciﬁc quantization of functions on T ∗Y .

To the best of our knowledge, quantization for square roots of the canonical bundle has been discovered (without proof) by M. Kshiwara in [17] in the framework of complex analytic contact geometry. Later D’Agnolo and Schapira [12] established a similar result for Lagrangian submanifolds of a complex analytic symplectic manifold. In the C∞ context, some closely related constructions can be found in the work of Nestand Tsygan [19]. Obstructions to deformation quantization have been also studied by Bordemann in [7]. The case of complex tori and quantization of arbitrary sheaves has been studied by Ben-Bassat, Block and Pantev in [5]. The case of an arbitrary line bundle and Lagrangian Y was considered in [1]. For general rank r, results on deformations modulo ~3 were obtained in [20] where the relation with projectively ﬂat algebraic connections has been discussed.

The outline of the thesis is as follows. In Chapter 2, we study the algebraic properties of formal Weyl algebra. We calculate the 2-cocycle pair of the non-abelian Lie algebra extension given by the Lie algebra of derivations. In Chapter 3, we give a deﬁnition of Harish-Chandra pairs and introduce the main Harish-Chandra pair for the deformation problems on sympletic varieties. We also introduce the deﬁnition of Harish-Chandra torsors and reformulate some deformation problems as lifting problems for transitive Harish-Chandra torsors. In Chapter 4, we give a complete answer to the deformation quantization problem of vector bundles on Lagrangian subvarieties. We discuss the relevant details on

6 Lie algebroids and modules over them. We solve the quantization problem of the Lagrangian case by lifting transitive Harish-Chandra torsors in three steps. In Chapter 5, we use the same “formal geometry” method to reformulate the deformation problem of coisotropic case into the lifting problem of torsors. We give the deﬁnition of curved L∞-algebra and curved Maurer-Cartan equation. Finaly, we expect a deformation quantization of line bundles on coisotropic subvarieties is equivalent to a solution of curved Maurer-Cartan equation.

Much of this thesis is based on the paper [2].

7 Chapter 2

Formal Weyl algebra

In this chapter, we consider the algebraic properties of formal Weyl algebra which is the deformation quantization of a formal power series. In theory, formal Weyl algebra is the prototype of deformation quantization problems. The Lie algebra of derivations and group of automorphisms of Weyl algebra will also play important roles in our study of deformation quantization of vector bundles on symplectic varieties.

Deﬁnition 2.1. The formal Weyl algebra (of dimension 2n) is the associative algebra

D = C[[x1, ··· , xn, y1, ··· , yn, ~]] generated by elements x1, ··· , xn, y1, ··· , yn, ~ with relations

[xi, xj] = [yi, yj] = [xi, ~] = [yi, ~] = 0, [yj, xi] = δij~, ∀ 0 6 i, j 6 n.

8 Notice that any f ∈ D can be written in the form of

P s1 sn t1 tn f = as1,··· ,sn,t1,··· ,tn (~)x1 ··· xn y1 ··· yn ,

where as1,··· ,sn,t1,··· ,tn (~) ∈ C[[~]]. Deﬁne

s s1 sn t t1 tn s = (s1, ··· , sn), t = (t1, ··· , tn), x = x1 ··· xn , y = y1 ··· yn .

P s1 sn t1 tn Under those notations, f = as1,··· ,sn,t1,··· ,tn (~)x1 ··· xn y1 ··· yn = P s t as,t(~)x y .

Proposition 2.1. [D, D] ⊂ ~D.

Proof. For any f, g ∈ D, let

P s t P s0 t0 f = as,t(~)x y , g = bs0,t0 (~)x y ,

0 0 0 0 0 0 where s = (s1, ··· , sn), t = (t1, ··· , tn).

[f, g] = fg − gf

P s t P s0 t0 P s0 t0 P s t = ( as,t(~)x y )( bs0,t0 (~)x y ) − ( bs0,t0 (~)x y )( as,t(~)x y )

P s t s0 t0 P s0 t0 s t = as,t(~)x y bs0,t0 (~)x y − bs0,t0 (~)x y as,t(~)x y

P s t s0 t0 P s0 t0 s t = as,t(~)bs0,t0 (~)x y x y − as,t(~)bs0,t0 (~)x y x y

P s t s0 t0 s0 t0 s t = as,t(~)bs0,t0 (~)(x y x y − x y x y )

P s t s0 t0 s s0 t t0 s s0 t t0 s0 t0 s t = as,t(~)bs0,t0 (~)(x y x y − x x y y + x x y y − x y x y )

P s t s0 t0 s s0 t t0 s0 s t0 t s0 t0 s t = as,t(~)bs0,t0 (~)(x y x y − x x y y + x x y y − x y x y )

P s t s0 t0 s0 s t0 t = as,t(~)bs0,t0 (~)(x [y , x ]y + x [x , y ]y )

9 t s0 s t0 Since [yj, xi] = δij~, ∀ 0 6 i, j 6 n,[y , x ], [x , y ] ∈ ~D. Therefore, [f, g] ∈ ~D.

2.1 Ideals and modules

Let A be the algebra of formal power series C[[x1, ··· , xn, y1, ··· , yn]]. Since A = D/~D, D can be viewed as a deformation quantization of A.

Let I be the ideal of A generated by (y1, ··· , yn), J be the preimage of I in D via the morphism D A. J is a double-sided ideal of D and isomorphic to I ⊕ ~D as a vector space. One can describe J as

n o X s t J = as,t(~)x y as,0(~) ∈ ~C[[~]] , where 0 = (0, ··· , 0).

Proposition 2.2. [J , J ] ⊂ ~J .

Proof. For any f ∈ D, since J is an ideal of D,[f, J ] ⊂ J , i.e., [~f, J ] ⊂ ~J .

s t s0 t0 0 Consider the bracket [x y , x y ] with some ti and ti > 0.

[xsyt, xs0 yt0 ] = xsytxs0 yt0 − xs0 yt0 xsyt

= xsytxs0 yt0 − xsxs0 ytyt0 + xsxs0 ytyt0 − xs0 yt0 xsyt

= xsytxs0 yt0 − xsxs0 ytyt0 + xs0 xsyt0 yt − xs0 yt0 xsyt

= xs[yt, xs0 ]yt0 + xs0 [xs, yt0 ]yt

10 0 0 Since [D, D] ⊂ ~D,[xsyt, xs yt ] ∈ ~J .

Notice that every element of J is the sum of terms that are either ~f for some

s t f ∈ D or x y with some ti > 0. We get the conclusion [J , J ] ⊂ ~J .

Let M = C[[x1, ··· , xn, ~]] be a D-module, where ~, xi ∈ D act by natural ∂ multiplication and yj by . M can also be understood as ~ ∂xj

M = D/D(y1, ··· , yn)

where D(y1, ··· , yn) is the left ideal of D generated by y1, ··· , yn.

⊕r Let Mr denote M , the rth direct sum of M. Mr can be view as a D-module, on which f ∈ D acts on each copy of M on the left.

2.2 Lie algebra of derivations

Let Der(D) be the set of all C[[~]]-derivations of D. Der(D) is a Lie algebra under the Lie bracket:

0 0 0 0 [dD(·), dD(·)] = dD(dD(·)) − dD(dD(·)), ∀ dD, dD ∈ Der(D).

Proposition 2.3. Every derivation in D is almost inner, i.e.,

Der(D) = {[ 1 f, ·]|f ∈ D}. ~

Furthermore, for f, g ∈ D, [ 1 g, ·] = [ 1 f, ·] if and only if f = g + p( ) for some ~ ~ ~ p(~) ∈ C[[~]].

11 Proof. By Proposition 2.1, [D, D] ⊂ D, thus [ 1 f, ·] is a derivation of D for any ~ ~ f ∈ D. For any derivation dD of D, since dD(1) = 0 and dD is C[[~]]-linear, dD(~) = ~dD(1) = 0.

[dD(xi), yj] + [dD(−yj), xi] = dD(xi)yj − yidD(xi) − dD(yi)xi + xidD(yi)

= dD(xiyj − yjxi) = dD(−δij~) = 0

Therefore, f can be recovered by taking the antiderivative of dD(xi) with respect to yi and antiderivative of dD(yj) with respect to xj.

[ 1 f, ·] = [ 1 g, ·] ⇐⇒ [ 1 (f − g), ·] = 0 ~ ~ ~

The recovery by xi and xi shows that it is unique up to a polynomial p(~) with respect to ~.

Deﬁne Der(D)J = {d ∈ Der(D) | d(J ) ⊂ J }. We have the following property.

1 Proposition 2.4. Der(D)J = {[ f, ·]|f ∈ J } ~

1 Proof. By Proposition 2.2, Der(D)J ⊃ {[ f, ·]|f ∈ J }. ~

On the other hand, since

1 s1 sn s1 si−1 sn [ x ··· x , yi] = −six ··· x ··· x ∈/ J ~ 1 n 1 i n

1 s any element f ∈ D such that [ f, ·] ∈ Der(D)J does not contain terms x with ~ 1 1 some si > 0. Therefore, Der(D)J = {[ f, ·]|f ∈ J ⊕ } = {[ f, ·]|f ∈ J } ~ C ~

12 Now deﬁne n o ˜ X s t J = as,t(~)x y a0,0(~) = 0 , where 0 = (0, ··· , 0).

We have

1 1 ˜ Der(D)J = {[ f, ·]|f ∈ J } = {[ f, ·]|f ∈ J} ~ ~

˜ 1 and there is a bijection from the set J to Der(D)J sending f to [ f, ·]. ~

2.3 Non-abelian Lie algebra extensions

Theorem 2.5. There exists a non-abelian Lie algebra extension

0 → gl~(r) → Der(D, Mr) → Der(D)J → 0.

where gl~(r) = glr(C[[~]]).

Proof. Since all derivations commute with the ~-action on D and Mr, and J can be viewed as the annihilator of Mr/~Mr, the image of the forgetful map

F : Der(D, Mr) → Der(D)

(dD, dMr ) 7→ dD

is contained in Der(D)J .

The kernel of the forgetful map consists of elements which map from Mr to itself

13 and commute with the D-action, i.e., module endomorphisms. Therefore,

ker(F ) = EndD(Mr) = gl~(r).

One has a left exact sequence

0 → gl~(r) → Der(D, Mr) → Der(D)J .

Moreover, for any f ∈ J˜, g ∈ D,

1 1 1 f(gm) = f, g m + g fm ~ ~ ~

1 f· : M → M is a derivation of M with respect to the derivation [ 1 f, ·]. Hence ~ ~ 1 1 1 ([ f, ·], f·) is a lifting of the derivation [ f, ·] ∈ Der(D)J to Der(D, M). The ~ ~ ~ sequence is therefore exact on the right.

According to the proof of Theorem 2.5, for any f ∈ J˜, we can deﬁne a section

s : Der(D)J → Der(D, M)

[ 1 f, ·] 7→ ([ 1 f, ·], 1 f·) ~ ~ ~

In general, notice that

1 1 1 [f, g] f, · , g, · = , · ~ ~ ~ ~ and [f,g] may not be in J˜, the above map doesn’t preserve Lie bracket. ~

14 ˜ P s t P s t For any f, g ∈ J , let f = as,t(~)x y and g = bs,t(~)x y . Deﬁne

n ! Pn X Y mi pf,g(~) = mi! (a0,m(~)bm,0(~) − am,0(~)b0,m(~))~ i=1 m1,··· ,mn i=1

where 0 = (0, ··· , 0), m = (m1, ··· , mn).

Lemma 2.6. n n ˜ P P s t ˜ (i) For any f ∈ J , if si > 0 and ti > 0, then [f, x y ] ∈ ~J . i=1 i=1

t s ˜ (ii) If si 6= ti for some i, then [y , x ] ∈ J .

n Pn m m Q i=1 mi ˜ (iii) Let m = (m1, ··· , mn), [y , x ] − ( i=1 mi!) ~ ∈ ~J .

Proof. n n s t ˜ P P (i) By Proposition 2.2, [f, x y ] ∈ ~J . Since si > 0, ti > 0 and i=1 i=1

[f, xsyt] = xs[f, yt] + [f, xs]yt, there are no ~i terms in [f, xsyt].

m m ˜ 0 (ii) Notice that when m > 1, [yi , xi], [yi, xi ] ∈ J , and for any m 6= m

m m0 m−1 m0 m−1 m0 m−1 m0 m−1 0 m0−1 [yi , xi ] = [yi , xi ]yi + yi [yi, xi ] = [yi , xi ]yi + yi (m ~xi )

m−1 m0 0 m0−1 m−1 m−1 m0−1 = [yi , xi ]yi + ~m (xi y + [y , xi ]).

m m0 ˜ By induction, [yi , xi ] ∈ J .

In general, if si 6= ti for some i,

t s t s s t t1 s1 tn sn s t [y , x ] = y x − x y = (y1 x1 ) ··· (yn xn ) − x y

15 s1 t1 t1 s1 sn tn tn sn s t ˜ = (x1 y1 + [y1 , x1 ]) ··· (xn yn + [yn , xn ]) − x y ∈ J

m m m ˜ ˜ (iii) By induction, one has [yi , xi ] − m!~ ∈ ~J . In ~D/~J ,

m m m m m m m1 m1 mn mn m m [y , x ] = y x − x y = (y1 x1 ) ··· (yn xn ) − x y

m1 m1 m1 m1 mn mn mn mn m m = (x1 y1 + [y1 , x1 ]) ··· (xn yn + [yn , xn ]) − x y

m1 m1 m1 mn mn mn m m = (x1 y1 + m1!~ ) ··· (xn yn + mn!~ ) − x y n Y Pn m = mi!~ i=1 i i=1

Proposition 2.7. For any f, g ∈ J˜,

[f, g] − p ( ) f,g ~ ∈ J˜. ~

Furthermore, pf,g(~) ∈ [[ ]]. ~2 C ~

Proof. In ~D/~J˜, by Lemma 2.6,

X m m X m m [f, g] = [a0,m(~)y , bm,0(~)x ] + [am,0(~)x , b0,m(~)y ] m1,··· ,mn m1,··· ,mn

X m m X m m = a0,m(~)bm,0(~)[y , x ] + am,0(~)b0,m(~)[x , y ] m1,··· ,mn m1,··· ,mn

= pf,g(~).

˜ [f,g]−pf,g(~) ˜ Therefore, [f, g] − pf,g( ) ∈ J , i.e., ∈ J . ~ ~ ~

˜ pf,g(~) Since f, g ∈ J , am,0, bm,0 ∈ [[ ]]. Therefore, 2 ∈ [[ ]]. ~C ~ ~ C ~

16 Theorem 2.8. The 2-cocycle pair of the non-abelian Lie algebra extension

0 → gl~(r) → Der(D, Mr) → Der(D)J → 0. is (χ, ψ), where

χ : Der(D)J ∧ Der(D)J → gl~(r)

0 0 0 dD ∧ dD 7→ [s(dD), s(dD)] − s([dD), dD])

ψ : Der(D)J → Der(gl~(r))

dD 7→ [s(dD), ·].

˜ For any dD ∈ Der(D)J , ψdD = 0. And for any f, g ∈ J , χ satisﬁes the equation:

1 1 pf,g(~) χ f, · ∧ g, · = Ir ~ ~ ~2

where Ir is the r × r identity matrix.

Proof. For any f, g ∈ J˜, by Proposition 2.7,

1 1 1 1 1 [f, g] − pf,g(~) pf,g(~) pf,g(~) χ f, · ∧ g, · = f·, g· − · = · = Ir ~ ~ ~ ~ ~ ~ ~2 ~2

˜ For any f ∈ J ,A ∈ gl~(r),

s 1 1 ψh 1 i(A) = f·,A = fIr,A = 0 f,· ~ ~ ~

17 Chapter 3

Harish-Chandra torsors

In this chapter, we give the deﬁnition of Harish-Chandra pairs and Harish-Chandra torsors. They were ﬁrst introduced by A. Beilinson and J. Bernstein [4]. In the paper [6], R. Bezruavnikov and D. Kaledin introduced a way to reformulate deformation quantization problems as lifting problems for transitive Harish-Chandra torsors. We are going to use the same technique in Chapter 4 to solve the deformation quantization problem of vector bundles on Lagrangian subvarieties.

3.1 Harish-Chandra pairs

Deﬁnition 3.1. A Harish-Chandra pair hG, hi over C is a pair consisting of a connected (pro)algebraic group G over C, a Lie algebra h over C with a G−action and an embedding g = Lie(G) ⊂ h such that the adjoint action of g on h is the diﬀerential of the given G-action.

18 A module V over a Harish-Chandra pair hG, hi is a representation V of the Lie algebra h whose restriction to g ⊂ h is integrated to an algebraic representation of the group G.

Example 3.1. Let V be a vector space which can be viewed as an additive algebraic group. The Lie algebra of this group is identiﬁed with V . The pair hV,V i is a Harish-Chandra pair.

Example 3.2. Let Aut(A) be the group of C-linear automorphisms of the formal power series A = C[[x1, ··· , xn, y1, ··· , yn]]. Then Aut(A) and the Lie algebra of C-linear derivations Der(A) form a Harish-Chandra pair hAut(A), Der(A)i.

Let Aut(D) be the group of C[[~]]-linear automorphism of the formal Weyl algebra D. Then Aut(D) and Der(D) form a Harish-Chandra pair hAut(D), Der(D)i.

The algebra projection D D/~D = A induces a canonical projection

hAut(D), Der(D)i hAut(A), Der(A)i of Harish-Chandra pairs.

P Furthermore, if we equip A with the symplectic form ω = dxi ∧ dyi, one can insert a Harish-Chandra pair hSympA, Hi between hAut(D), Der(D)i and hAut(A), Der(A)i, where H ⊂ Der(A) is the Lie subalgebra of Hamiltonian vector ﬁelds and SympA is induced by H. In other words, there is an injective morphism

hSympA, Hi ,→ hAut(A), Der(A)i. and a canonical morphism

hAut(D), Der(D)i → hSympA, Hi.

19 Example 3.3. Let Aut(D)J , resp. Der(D)J be the subset of Aut(D), resp.

Der(D) formed by the maps f : D → D such that f(J ) ⊂ J . Aut(D)J and

Der(D)J form a Harish-Chandra pair hAut(D)J , Der(D)J i, which is a subpair of hAut(D), Der(D)i.

Let Aut(D, M) be the group of automorphisms of the pair (D, M), Der(D, M) be the Lie algebra of derivations of the pair (D, M). Aut(D, M) and Der(D, M) form a Harish-Chandra pair hAut(D, M), Der(D, M)i.

Forgetting the action on M yields a morphism of Harish-Chandra pairs:

F = hFAut, FDeri : hAut(D, M), Der(D, M)i → hAut(D), Der(D)i

Lemma 3.1. There is a short exact sequence of Harish-Chandra pairs:

∗ F 1 → hC [[~]], C[[~]]i → hAut(D, M), Der(D, M)i → hAut(D)J , Der(D)J i → 1.

Example 3.4. In general, Aut(D, Mr) and Der(D, Mr) form a Harish-Chandra pair hAut(D, Mr), Der(D, Mr)i. One can generalize the short exact sequence given by the previous theorem, but the extension is no longer abelian.

Let GL~(r) = GLr(C[[~]]) and gl~(r) = glr(C[[~]]).

Theorem 3.2. There is a short exact sequence of Harish-Chandra pairs:

F 1 → hGL~(r), gl~(r)i → hAut(D, Mr), Der(D, Mr)i → hAut(D)J , Der(D)J i → 1.

1 Moreover, Der(D, M1) ' J where the right hand side is considered with ~ commutator bracket.

20 ⊕r Proof. Since Mr 'M and we have the map Aut(D, M1) → Aut(D, Mr)

⊕r sending (Φ, Ψ) to Φ, Ψ . Hence any lift for Φ ∈ Aut(D)J to Aut(D, M1) also gives a lift of the same Φ to Aut(D, Mr). Therefore the right arrow is surjective.

Its kernel is the group of automorphisms of Mr as a D-module. Every such automorphism is uniquely determined by its value on generators, and we can choose a set of generators on which yi acts by zero. Then their images are independent on xi, which means that the automorphism is given by an invertible r × r matrix with entries in C~.

The proof in the case of Lie algebras is entirely similar.

3.2 Group torsors

Roughly speaking, a group torsor is like a group that has forgotten its identity element.

Deﬁnition 3.2. Let G be a group acting on a set S, i.e., there is a map G×S → S, written as hg, si 7→ g(s), such that for any g, h ∈ G, (gh)(s) = g(h(s)). We say that S is a G-torsor under the action of G if there exists an element s0 ∈ S such that the mapping g 7→ g(s0) is a bijective mapping of G to S. Note that if there exists one such s0 ∈ S, then the same is true for any other element s1 ∈ S. Thus S is a G-torsor if and only if it satisﬁes the following two conditions:

(1) For every s ∈ S the induced mapping g 7→ g(s) is bijective from G to S;

(2) S is nonempty.

We have the following examples.

21 Example 3.5. Every group G can be viewed as a G-torsor under the action of left multiplication.

Example 3.6. If A is the aﬃne space underlying a vector space V , then A is a V -torsor under the action of additive group of translations of vectors in V .

Example 3.7. If F is a ﬁeld and V is an F -vector space of dimension n. Let

X be the set of all ordered bases of V . Then X is a GLn(F )-torsor under the action of changing basis.

We can generalize the deﬁnition of the group torsor to a topological space.

Deﬁnition 3.3. Let X be a topological space and G be a topological group. A topological space P is said to be a G-torsor if P consists of the following data:

(1) A continuous map

π : G × P → X

(g, p) 7→ gp

such that π(p) = π(gp), and

(2) the G-action is locally trivial, i.e., there is an open cover {Ui} of X such

−1 −1 that π (Ui) = Ui × G and G acts trivially on π (Ui). More precisely, for

−1 any Ui, there is a homeomorphism π (Ui) → Ui ×G such that the diagram below induced by the G-action commutes.

−1 −1 G × π (Ui) π (Ui)

G × (Ui × G) Ui × G.

Example 3.8. A GLr-torsor on a topological manifold X is just a vector bundle of rank r on X.

22 0 Example 3.9. Let 1 → H → G → G → 1 be a split group extension. If PH

0 and PG are H-torsor and G-torsor over a space X, then PH × PG is a G -torsor over X.

Based on the above deﬁtion, the set of group torsors over X can be described by Cechˇ cohomology.

Theorem 3.3. H1(X,G) is the set of G-torsors over X. [18]

Finally, we give a deﬁnition of the group torsor in the language of algebraic geometry.

Deﬁnition 3.4. Let X be a scheme and G be a group. A scheme P is said to be a G-torsor if P is faithfully ﬂat over X and equipped with an action map G × P → P which commutes with the projection to X and induces and isomorphism

G × P → P ×X P.

In our applications, all torsors will be locally trivial in Zariski topology.

3.3 Transitive Harish-Chandra torsors

The concept of a G-torsor over a smooth variety Y can be extended to Harish-Chandra pairs. We will only need the special case of a transitive Harish-Chandra torsor.

Deﬁnition 3.5. A transitive Harish-Chandra torsor over a Harish-Chandra pair hG, hi on a smooth variety Y is a G-torsor P over

23 0 Y together with a Lie algebra morphism h 7→ H (P,TP ) which induces an isomorphism of vector bundles h ⊗ OP ' TP .

This data can be rephrased in terms of the Atiyah algebra L(P) of G-invariant vector ﬁelds on P, viewed as a Lie algebroid on Y

0 → Ad(P) = Pg → L(P) → TX → 0

For a transitive Harish-Chandra torsor P we have an isomorphism of locally free sheaves on Y :

A : L(P) 'Ph

which restricts to identity on Ad(P) = Pg. This isomorphism does not agree with the Lie bracket but instead Ph has a ﬂat algebraic connection such that

A[x, y] − [Ax, Ay] = ∂xAy − ∂Y AX

where x 7→ ∂x is the anchor maps of L(P).

Example 3.10. Let X be a smooth variety and for any point x ∈ X let

ObX,x denote the completion of the local ring at x. A choice of a formal coordinate system at x is equivalent to a choice of a topological C[[~]]-algebra isomorphism η : ObX,x 'A. The pairs (x, η) form the set of (closed) points of a

Harish-Chandra torsor PX over the pair hAut(A), Der(A)i.

Theorem 3.4. Let X be a 2n-dimentional symplectic variety. There is a one-to-one correspondence between symplectic structures on X and reductions of the hAut(A), Der(A)i-torsor PX to hSympA, Hi ⊂ hAut(A), Der(A)i. [6]

Furthermore, equipping X with a symplectic structure ω, the deformation

24 quantization of the stucture sheaf OX of X compatible with the sympletic structure can also be represented by lifting of Harish-Chandra torsor.

Theorem 3.5. Let (OX , ω) be an algebraic symplectic variety and let Ps be the hSympA, Hi-torsor corresponding to the symplectic structure ω on X.

Then there exists a natural bijection between the set Q(OX , ω) of isomorphism

1 classes of quantizations of OX with ω, and the set HPs (X, hAut(D), Der(D)i) of the isomophism classes of liftings of the hSympA, Hi-torsor Ps to a hAut(D), Der(D)i-torsor with respect to the canonical map of Harish-Chandra pairs hAut(D), Der(D)i → hSympA, Hi.

[6]

Theorem 3.6. Let X be a 2n-dimentional symplectic variety over C with a closed

0 2 2 non-degenerated form ω ∈ H (X, ΩX ). Denote by [ω] ∈ HDR(X) the cohomology class of the symplectic form. Let Q(OX , ω) be the set of isomorphism classes of quantizations of OX compatible with the form ω. Then there exists a natural bijection map

2 Per : Q(OX , ω) ,→ HDR(X)[[~]] called the non-commutative period map. Moreover, for every quantization

2 q ∈ Q(OX , ω), the power series Per(q) ∈ HDR(X)[[~]] has a constant term [ω].

[6]

25 Chapter 4

Quantization of Vector Bundles on Lagrangian Subvarieties

Let X be a 2n-dimensional symplectic variety over C with a symplectic form

ω. Assume that a deformation quantization O~ of OX is given, such that its non-commutative period is

2 2 ω(t) = [ω] + tω1 + t ω2 + · · · ∈ HDR(X) ⊗ C[[~]].

Assume that a smooth Lagrangian subvariety Y ⊂ X is given, with a rank r vector bundle E on Y . E can be considered as a sheaf of OX -modules via the direct image functor.

In this chapter, we give some necessary and suﬃcient conditions for the existence of the deformation quantization of E, i.e., a O~-module E~ such that

E~/~E~ ' E.

26 4.1 Modules over a Lie algebroid

2 ∗ Denote by I ⊂ OX the ideal sheaf of Y . There is an isomorphism I/I 'N of coherent sheaves on Y. Since Y is Lagrangian,

∗ N 'TY .

Let I~ ⊂ O~ be the preimage of I in O~, with respect to the quotient map

O~ → OX . Assume that a deformation quantization E~ of E is given and ﬁxed. We will work locally on Y modulo ~2, assuming that local splitting

2 2 O~/(~ ) 'OY + ~OY , E~/~ E~ ' E + ~E

are given. For O~ these exist by [21] and for E~ the argument is similar.

Choosing a local section a of OX and a local section e of E, we write the deformed actions as

a ∗ e = ae + ~γ1(a, e)

2 If a ∈ I we see that a ∗ E~ ⊂ ~E~. Moreover, if a1, a2 ∈ I then modulo ~ we can write 1 a a = a ∗ a − ω(da , da ) 1 2 1 2 ~2 1 2

Since ω(da1, da2) ∈ I by the Lagrangian assumption on Y , this implies that

2 2 2 2 I ∗ E~ ⊂ ~ E~. Therefore (~OY + I/I ) sends E 'E~~E~ to E ' ~E~~ E~, with

~a + b sending e to ae + γ1(b, e). Writing out associativity equations

a ∗ (b ∗ e) = (a ∗ b) ∗ e; b ∗ (a ∗ e) = (b ∗ a) ∗ e

27 and comparing them we get the two conditions

1 γ (b, ae) − aγ (b, e) = ω(db, da)e; γ (ab, e) − aγ (b, e) = ω(db, da)e. 1 1 1 1 2

2 Observe also that b 7→ ω(db, ·) is exactly the isomorphism I/I 'TY . Therefore

~a+b acts on E by a ﬁrst order diﬀerential operator with scalar principal symbol and we obtain (locally, at this moment) a map

2 γ : ~OY + I/I → L(E) with values in the sections of the Atiyah algebroid of E. This map agrees with Lie brackets if its source is given the bracket induced by (a, b) 7→ ω(db, da), (b1, b2) 7→

2 ω(db1, db2). If we don’t start with E~, just with a deformation modulo ~ , we need to assume existence of its extension modulo ~3 to ensure agreement with the bracket. The map γ is not OY -linear but satisﬁes

1 γ(f( a + b)) − fγ( a + b) = ω(db, f). ~ ~ 2

To globalize this consider

O L(O ) = I /(I ∗ I ) 'I ⊗ O 'T or ~ (O , O ) ~ ~ ~ ~ Y O~ Y 1 Y Y

2 and observe that ~O~/~I~ and I/(~O~ + I~ ∗ I~) 'I/I 'TY . In other words,

L(O~) is a Picard algebroid on Y in the sense of Section 2 in [4] with the bracket that descends from (a, b) 7→ 1 (a ∗ b − b ∗ a). Our local computation above gives ~ a morphism of Zariski sheaves

γ : L → L O~ E

28 which agrees with the Lie bracket but satisﬁes

1 γ(fx) − fγ(x) = x¯(f) 2 where f is a locally deﬁned function andx ¯ is the image of the local section x of

L(O~) in TY . So γ fails to be a morphism of OY -modules.

To repair the situation we use the fact that Picard algebroids form a vector space, that is, for two algebroids L1, L2 and any pair of scalars λ1, λ2 there is a Picard algebroid L = λ1L1 + λ2L2 and a morphism of sheaves

sλ1,λ2 : L1 ×TY L2 → L,

cf. Section 2.1 in [4] which on the subbundle copies of OY is given by (a1, a2) 7→

λ1a1 + λ2a2, and on the quotient copies of TY it is given by the identity. The other fact that we use is that the Atiyah algebroid of the canonical bundle L(KY ) has a non-OY -linear splitting sending a vector fields ∂ to the Lie derivative l(∂) on top degree differential forms, which satisfies

l(f∂) − fl(∂) = ∂(f), cf. Section 2.4 in [4]. So we consider the algebroid

1 L+(O ) = L(O ) + L(K ) ~ ~ 2 Y

where L(KY ) is the Aityah algebroid of the canonical bundle KY . The expression x 7→ s 1 (x, l(¯x)) deﬁnes an isomorphism of Zariski sheaves 1, 2

+ L(O~) → L (O~)

29 which agrees with the bracket, descends to identity on TY but fails to be OY -linear in exactly the same way as it happens for L(O~) → L(E). Composing the inverse isomorphism with γ we obtain a morphism of sheaves of OY -modules

+ + γ : L (O~) → L(E) which is now a morphism of Lie algebroids on Y .

Existence of γ+ imposes strong restrictions on E. For instance, composing with

+ the trace morphism EndOY (E) → OY we get a morphism L (O~) → L(det(E)) to the Atiyah algebroid of the determinant bundle det(E). By construction, this map is multiplication by r = rk(E) on the subbundles OY and identity on the quotient bundles TY . In particular, the map is an isomorphism of Lie algebroids. Since by [4] both Picard algebroids have characteristic classes in the truncated de Rham cohomology, we get

+ c1(E) = c1(det(E)) = r · c(L (O~))

2 in HF (Y ). Furthermore, taking the quotient of L(E) by the subbundle OY ⊂

EndOY (E) we obtain the Atiyah algebroid L(P(E)) of the associated P GL(r) bundle P(E). By construction, γ descends to a Lie morphism TY → L(P(E)) which lifts the identity on TY . In other words, we have a ﬂat algebraic connection on P(E).

The above discussion can also be reversed (we are adjusting the argument in

2 Section 2.3 of [4]): assume that the equation on c1 in HF (Y ) is satisﬁed and we are given a ﬂat connection on the P GL(r)-bundle P(E) associated to E. Then

30 we have a commutative diagram

+ OY L (O~) TY

0 OY ⊕ End (E) L(det(E)) ⊕ L(P(E)) TY ⊕ TY

Here the lower row is understood as the sum of Atiyah algebras of det(E) and

P(E). The map into the Atiyah algebra of det(E) is as before (identity on TY and multiplication by r on OY ). The map into the Atiyah algebra of P(E) is the composition of the projection to TY and the splitting TY → L(P(E)) given by the ﬂat connection. Observe that the quotient map TY → TY ⊕ TY is just the diagram morphism ∂ 7→ (∂, ∂).

Now a local section of the Atiyah algebra of E can be interpreted as an invariant vector ﬁeld on the total space of the principal GL(r)-bundle of E and its direct image gives vector ﬁelds on the total spaces of the C∗-bundle and the P GL(r)-bundle, corresponding to det(E) and P(E), respectively. Indeed, these total spaces are obtained as quotients by the SL(r) action and the C∗ action, respectively. This gives an embedding of sheaves

L(E) → L(det(E) ⊕ LP(E)) which agrees with brackets and has image equal to the preimage of the diagonal

+ in TY ⊕ TY . This means that the earlier morphism from L (O~) lands to L(E), as required. We can summarize the discussion of this section as the following.

Theorem 4.1. A ﬂat deformation quantization of E modulo ~2 which admits an extension to a deformation modulo ~3, gives E a structure of a module over the

+ 1 ∗ 2 Lie algebroid L (O~), with the class 2 c1(KY ) + j ω1 in HF (Y ). For an arbitrary

31 vector bundle E on Y such structure is equivalent to a choice of an isomorphism

+ L (O~) 'L(det(E)) as Lie algebroids on Y which satisﬁes f 7→ rf on functions, and a ﬂat P GL(r) connection on the P GL(r)-bundle P(E) associated to E.

4.2 The construction of the torsor

Assume that x ∈ Y is a point and in some neighborhood of x ∈ X we are given a deformation quantization E~ of E.

The stalk O~,x is a non-commutative ring with a maximal ideal mx, which is the preimage of the maximal ideal nx in the commutative local ring O~,x/~O~,x '

OX,x. Let Ob~,x and Eb~,x be completions of stalks with respect to this maximal ideal.

Recall that D = C[[x1, ··· , xn, y1, ··· , yn, ~]] is the formal Weyl algebra of dimension 2n, and Mr is the rth direct sum of the D-module M =

C[[x1, ··· , xn, ~]].

Lemma 4.2. There exist isomorphisms

η : Ob~,x 'D, µ : Eb~,x 'Mr

of C~-modules which are compatible with ﬁltrations and action of rings on corresponding modules. Moreover, the ﬁrst isomorphism may be chosen in such a way that the images of y1, ··· , yn in ObX,x come from a regular sequence in OX,x generating the ideal Ix of functions vanishing on Y .

32 Proof. First, we can ﬁnd an isomorphism

ObX,x = C[[x1, ··· , xn, y1, ··· , yn]]

such that y1, ··· yn are the images of a regular sequence deﬁning Y in the neighborhood of x. This is due to the formal Weinstein Lagrangian Neighborhood Theorem, the proof of which, cf. Section 7,8 on [11], may be used without changes. The key point here is that the Moser Trick in Section 7.2 of loc.cit. works in the completion of the local ring, although not the local ring itself.

Next, both D and Ob~,x can be viewed as deformation quantizations of the algebra P ∂ ∂ [[x1, ··· , xn, y1, ··· , yn]] with the Poisson bivector equal to ∧ + C ~ i ∂xi ∂yi ~2α(~). By general deformation theory, the bivector α(~) is a Maurer-Cartan solution for the DG Lie algebra of polyvector fields with the non-trivial differential given by the bracket with the bivector P ∂ ∧ ∂ . If we identify i ∂xi ∂yi polyvector fields with differential forms, this bracket will become the standard de Rham differential. Since for the algebra of formal power series the de Rham complex is exact and the Maurer-Cartan groupoid is invariant under quasi-isomorphisms, all deformation quantizations are equivalent and D is isomorphic to the completion of O~,x. This settles the assertions about the algebras.

As for the module part, since (E~/~E~)x is free over OY,x we can ﬁnd an isomorphism of this module with

⊕r Mr = Mr/~Mr ' C[[x1, ··· , xn]]

. Now we repeat the argument of lemma 2.3.5 on [1] to produce an isomorphism

33 of Eb~,x and Mr.

Given data (X,Y, O~) we can consider the proalgebraic scheme PJ parameterizing the pairs (x, η), where x ∈ Y and η is an isomorphism as above.

Proposition 4.3. PJ is a transitive hAut(D)J , Der(D)J i-torsor.

Proof. The group Aut(D)J acts on PJ by changing the isomorphism η. Locally in the Zariski topology on X we can denote by A, B the rings of sections of

OX , O~ over an aﬃne subset, and then the pair (x, η) can be described by two ring homomorphisms

x : A → k; η : lim(B/bn) 'D n where b ⊂ B is the preimage in B of the ideal Ker(x) ⊂ A = B/~B. Given a derivation ∂ : D → D, we can consider its composition with D → k = D/m which descends to a linear function on m/m2, and using the isomorphism η and

~-linearity of ∂, to a linear function z : Ker(x)/(Ker(x))2 → k. Thus we can write extensions of x and η over the ring k[]/(2) of dual numbers

x(a) = x(a) + z(a); η(b) = η(b) + ∂(η(b)).

This means that Der(D)J maps to vector fields on the torsor PJ . The defining properties of transitive Harish-Chandra torsors (agreement with the group action and the bracket on vector fields) are immediate from the definitions.

Theorem 4.4. A choice of a vector bundle E of rank r on Y and its deformation quantization E~ is equivalent to a choice of a lift of the torsor PJ to a transitive

Harish-Chandra hAut(D, Mr), Der(D, Mr)i-torsor PMr along the extension of

34 pairs

1 → hGL~(r), gl~(r)i → hAut(D, Mr), Der(D, Mr)i → hAut(D)J , Der(D)J i → 1.

Proof. Given E~, we can construct the transitive Harish-Chandra torsor P for which the ﬁber at x ∈ Y represents all pairs (η, µ) which identify completions of

O~ and E~ at x with the “formal local models” D and Mr. It has the transitive

Harish Chandra structure similarly to PJ . By construction, the trosor PJ is induced from P by forgetting the isomorphism µ.

Conversely, given a lift P of the transitive Harish-Chandra torsor PJ , we have

a vector bundle PMr associated via the action of hAut(D, Mr), Der(D, Mr)i on

Mr. It carries a ﬂat algebraic connection as any vector bundle associated to a transitive Harish-Chandra torsor. As in the Lemma 3.4 of [6], the Zariski sheaf

E~ can be recovered as the sheaf of ﬂat sections of PMr .

4.3 The Harish-Chandra extension

In order to lift the transitive Harish-Chandra torsor along the extension of Harish-Chandra pairs

1 → hGL~(r), gl~(r)i → hAut(D, Mr), Der(D, Mr)i → hAut(D)J , Der(D)J i → 1.

We need to have a deeper understanding of this extension in both group and Lie algebra level.

1 Recall that Der(D, M1) ' J , it will help the reader to understand the Lie ~

35 algebra 1 J if we introduce the grading in which ~

deg ~ = 3, deg yj = 2, deg xi = 1.

The Lie algebra can be understood as an inﬁnite direct product, not a direct sum, of its homogeneous components. The degree −1 component is spanned by

1 1 the elements ad(y1), ··· , ad(yn), and this gives the non-integrable part of the ~ ~

Lie algebra. The degree > 0 part is the tangent Lie algebra of Aut(D, M1). The degree 0 component is spanned by 1 and xiyj , which gives the tangent Lie ~ ∗ algebra of the reductive part C ×GL(n). The degree > 1 part is the pro-nilpotent subalgebra with contains, in particular, the elements of ~C~.

Similar grading exists on Der(D, Mr) if we place gl(r) in degree zero.

The following diagram for Lie algebras and its group analogue will be fundamental in our analysis of the main Harish-Chandra pair hAut(D, Mr), Der(D, Mr)i:

1 0 J Der(D)J 0 C~ ~

0 gl~(r) Der(D, Mr) Der(D)J 0 where the middle arrow sends α ∈ 1 J to the pair (φ, ψ) with φ = [α, ·] and ψ = ~ α(...). In both cases the maps are well deﬁned, that is, a possible denominator

1 cancels out. Since the images of J and gl (r) commute in Der(D, Mr), the ~ ~ following lemma is immediate,

Lemma 4.5.

1 1 Der(D, Mr) ' J ⊕ gl (r) /C~ ' J ⊕ pgl (r). ~ ~ ~ ~

36 In the group case we have a diagram

1 ∗ Aut(D, M ) Aut(D) 1 C~ 1 J

1 GL~(r) Aut(D, Mr) Aut(D)J 1

but the corresponding splitting fails in the constant term with respect to ~:

∗ Aut(D, Mr) has a subgroup GL(r) which does not split as P GL(r) × C . However, there is a slightly diﬀerent splitting on the group level, and the interplay between the two splittings is key to our later arguments on deformation quantization.

Lemma 4.6. We have a splitting

∗ Aut(D, Mr) ' [Aut(D, M1)/C ] × [GL(r) n exp(~ · pgl~(r))]

where exp(~ · pgl~(r)) is the pro-unipotent kernel of the evaluation map

P GL~(r) → P GL(r)

~ 7→ 0

The two splittings (of the group and the Lie algebra) agree modulo hC∗, Ci:

1 hAut(D, M ), Der(D, M )i /h ∗, i ' Aut(D, M )/ ∗, J / ∗ ×hP GL (r), pgl (r)i . r r C C 1 C 2 C ~ ~

Proof. The ﬁrst statement follows from the diagram before the lemma and the

fact that the images of GL~(r) and Aut(D, M1) in Aut(D, Mr) commute. To prove the splitting mod C∗, observe that the pro-algebraic groups on both sides are semidirect products of ﬁnite dimensional reductive subgroups and inﬁnite

37 dimensional pro-unipotent groups. The reductive part on the left hand side is

GL(r) × GL(n) where GL(r) acts on the generators of Mr and GL(n) on the

∗ variables x1, ··· , xn ∈ D, with the dual action on y1, ··· , yn. The copy of C acts by scalar automorphisms on Mr only. Thus, the reductive part of the quotient on the left hand side is P GL(r) × GL(n). The same argument repeated for r = 1 shows that the reductive part on the right is the same. The isomorphism of tangent Lie algebras on the pro-unipotent parts follows from the provious lemmas. Since we are taking quotients by the central copy of C∗ which acts trivially on the Lie algebras, the semidirect products match as well

Informally we could say that on the group level there is an extra copy of C∗ ⊂ GL(r) which on the level of Lie algebras migrates to the other factor ⊂ 1 J' C ~

Der(D, M1) but after taking the quotient by these, the two splittings match.

4.4 Lifting the torsor by steps

In this section we study the problem of transitive Harish-Chandra torsor lifting by considering the chain of surjections

hG, hi → hG2, h2i → hG1, h1i → hG0, h0i

At the two ends we have pairs

hG, hi = hAut(D, Mr), Der(D, Mr)i, hG0, h0i = hAut(D)J , Der(D)J i

38 and the intermediate pairs both have product decompositions

hG1, h1i = hAut(D)J , Der(D)J i × hP GL~(r), pgl~(r)i

∗ ∗ 1 hG2, h2i = hG, hi/hC Ci = hAut(D, M1)/C , J /Ci × hP GL~(r), pgl (r)i. ~ ~

Proposition 4.7. The groupoid category of lifts of PJ to a transitive

Harish-Chandra torsor P1 over hG1, h1i is equivalent to the category of

P GL~(r)-bundles on Y with a ﬂat algebraic connection.

Proof. This is true due to the splitting of the Harish-Chandra pair hG1, h1i. We are looking for a torsor P1 over G1 and a G1-equivariant form h1-valued form on the total space of P1, which lifts a similar form on the total space of PJ . Due to the splitting of G1 such P1 would be a ﬁber product over Y , of the original PJ and a P GL~(r)-torsor Q. The connection form on P1, due to the G1-equivariant splitting of h1, would also have to be a sum of the h0-valued connection on PJ and a pgl~(r)-valued connection on Q. The zero curvature condition on P1 (i.e., agreement with the Lie bracket) implies that the connection on Q is ﬂat.

Proposition 4.8. Given a choice of P1, the category of its lifts to a transitive Harish-Chandra torsor P2 over hG2, h2i is equivalent to the category of lifts of the original torsor PJ to a transitive Harish-Chandra torsor over

∗ 1 2 hAut(D, M1)/ , J / i. The latter is non-empty if and only if in H (Y )[[ ]] C ~ C DR ~ one has

∗ 2 3 j (~ ω2 + ~ ω3 + ··· ) = 0.

Proof. We use the splitting of the Harish-Chandra pair hG2, h2i and the fact that the flat bundle Q is already defined when lifting PJ to P1. Therefore, P2 would be a fiber product of Q and a transitive Harish-Chandra torsor R over

39 ∗ 1 hAut(D, M1)/ , J / i lifting PJ . Comparing the deﬁnitions we see that a C ~ C transitive Harish-Chandra structure on P2 of the required type is equivalent to a similar structure on R lifting that of PJ . Existence of R follows by a combination of Proposition 2.7 in [6] and Lemma 5.2.2 of [1].

Proposition 4.9. Given a choice of P2, the groupoid category of its lifts to a transitive Harish-Chandra torsor P over hG, hi is equivalent to the category of

+ rank r modules E over the Lie algebroid L (O~), equipped with an isomorphism of P(E) and the ﬂat P GL(r)-bundle induced from P1 via the homomorphism

P GL~(r) → P GL(r). Such lifts exist if and only if the following equality holds

2 in HF (Y ): 1 1 c (E) = c (K ) + c(L(O )) = c(L+(O )). r 1 2 1 Y ~ ~

Proof. Consider the short exact sequence

∗ 1 → hC , Ci → hG, hi → hG2, h2i → 1.

First consider the group side. Due to the splitting of Lemma 4.6 on the level of usual torsors, we just need to lift a P GL~(r) torsor Q to a torsor ˜ Q over GL(r) n exp(~ · pgl~(r)). The required Harish-Chandra structure is an isomorphism of bundles TY ' h ⊗C OP which is G-equivariant and has

∗ the zero curvature condition. Since the action of C ⊂ GL(r) on ~ · pgl~(r) and h is trivial, we can take C∗ invariant direct image of both sheaves under

∗ P → P2 = P/C where they have a G2-equivariant structure, and look for a

G2-equivariant isomorphism on the total space of P2 instead.

∗ The tangent bundle of P turns into the Atiyah algebroid of the C -torsor P → P2 and the trivial bundle with ﬁber h also gets a Lie algebroid structure since on

40 P2 we have identiﬁed TP2 with the trivial bundle with ﬁber h2 ' h/C. We need to construct a G2-equivariant isomorphism of two bundles with an additional property that corresponds to agreement of brackets (zero curvature) on P. The

G2-equivariant structure means that the projection to TP2 ' h2 ⊗C OP1 has a partial section over the smaller sub-bundle g2 ⊗C OP1 . Taking the quotient of both algebroids by the image of this sub-bundle, and then taking G2-equivariant direct image to Y , we reduce to the question of isomorphism of two Picard

2 algebroids on Y . Both are classiﬁed by a cohomology class in HF (Y ), hence the isomorphism exists precisely when the two classes are equal.

To compute the class for the algebroid obtained from h⊗OP observe that instead of taking the equivariant descent with respect to G2, we can ﬁrst take the

∗ descent by P GL~(r) and then by Aut(D, M1)/C . The ﬁrst step will lead to an equivariant Picard algebroid on R with the ﬁber corresponding to the middle term of the Lie algebra extension

0 → C → Der(D, M1) → Der(D, M1)/C → 0.

By Proposition 4.3.5 in [1] its equivariant descent to Y gives a Picard algebroid

1 ∗ with the class 2 c1(KY ) + j ω1.

For the Atiyah algebroid of P → P2 we will ﬁrst take the descent with respect

∗ to Aut(D, M1)/C , then with respect to exp(~ · pgl~(r)). This leads to a GL(r)-torsor E on Y lifting a P GL(r)-torsor P(E). The equivariant descent of the Atiyah algebroid of P → P2 to a Picard algebroid on Y is just the quotient of the Atiyah algebroid of E by the trace zero part in End(E) ⊂ L(E).

1 2 1 Therefore, it has class r c1(E) and the desired equation in HF (Y ) is r c1(E) = 1 ∗ 2 c1(KY ) + j ω1.

41 Proposition 4.10. For a given choice of P1, the groupoid category of its lifts to a transitive Harish-Chandra torsor P over hG1, h1i–if non empty–is equivalent

∗ to the groupoid category OY [[~]] -torsor with a ﬂat algebraic connection. More

∗ precisely, for any fixed choice of P and a flat OY [[~]] -torsor L there is a well-defined transitive Harish-Chandra torsor L ? P which gives an equivalence of categories.

Proof. We interpret the transitive Harish-Chandra torsor lifting in the language of gerbes, following Chapter 5 of [3], Consider the central extension

∗ 1 → hC , Ci → hG, hi → hG1, h1i → 1.

The lifting of a transitive Hairsh-Chandra torsor P1 = Q ×Y PJ to P, can be splitted as follows. First, choose a lifting on the group level. As we have seen before, globally on Y this may not be possible but local lifts form a gerbe over

∗ OY [[~]] , see Deﬁnition 5.2.4 of [9]. If a (local) lift is chosen, we can look for an h-valued connection on the lift, and all possible choices of such a connection form a connective structure on the gerbe in the sense of Deﬁnition 5.3.1 of loc.cit. Moreover, whenever we can choose a connection, this leads to the curvature

1 dΩ + 2 Ω ∧ Ω where Ω is the h-valued form on the total space of the torsor which describes the connection. This gives a curving of the connective structure, see Deﬁnition 5.3.7 of loc.cit. As in Theorem 5.3.17 of loc.cit, this leads to a gerbe over the C~-version of Deligne complex

∗ 1 2 Del~(Y ) := OY [[~]] → ΩY [[~]] → ΩY [[~]] → · · ·

df where the ﬁrst arrow sends f to f . Since we want to trivialize the gerbe over this

42 complex (i.e., the lifting torsor must exist and it must admit a connection and the connection must have zero curvature), the category of such trivializations-if nonempty-is equivalent to the category of torsor over the same complex. By a

C~ version of Theorem 2.2.11 in loc.cit. the latter is equivalent to the category

∗ of OY [[~]] -torsor with a ﬂat algebraic connection.

In particular the set of equivalence classes of deformation quantization with a ﬁxed projectivization Q is in bijective correspondence with the set of

∗ isomorphism classes of OY [[~]] -torsor with a ﬂat algebraic connection.

A more explicit, if a bit less conceptual version of this step is as follows. We look at the short exact sequence

>1 ∗ Ω (Y )[[~]] → Del~(Y ) → OY [[~]] and the piece of the associated long exact sequence in cohomology:

1 ∗ 2 2 2 ∗ · · · → H (Y, OY [[~]] ) → HF (Y )[[~]] → H (Y, Del~(Y )) → H (Y, OY [[~]] ) → · · ·

If a hG1, h1i-torsor and its transitive Harish-Chandra structure can be trivialized on an covering {Ui} of X then we can ﬁrst look at G-valued liftings ψ : Uj ∩Uj →

G of the cocycle deﬁning P2. Then we can look at those h-valued connections

∇i on the trivial G-bundles on each Ui which lift the given h1-valued connection on P1.

The functions ψij lead to an expression aijk = ψijψjkψki on Ui∩Uj∩Uk which takes

∗ values in OY [[~]] since its projection to G2 is trivial. Moreover, the difference of ∇i − ∇j on Ui ∩ Uj is given by a Ch-valued differential 1-form bij and a quick comparison of definitions shows that on triple intersections (bij + bjk + bki) =

43 daijk/aijk. On each Ui the curvature of ∇i is given by a 2-form ci which takes values in the Lie subalgebra C~ ⊂ h since the projection of the connection to h2 satisﬁes the zero curvature condition. On the double intersections we have

(ci − cj) = dbij, the usual comparison of curvatures for two diﬀerent connections.

The two identities relating bij with aijk and ci with bij mean that the three

2 groups of sections deﬁne a single cohomology class in H (Y, Del~(Y )).

2 ∗ The projection of this class to H (Y, OY [[~]] ) is represented by the cocycle aijk.

In the case when the class is trivial we can readjust the choice of ψij by the cochain resolving aijk and ensure aijk ≡ 1. This means that ψij’s do deﬁne a lift

2 of P1 to a G-torsor P. Now the obstruction too is lifted from H (Y, Del~(Y )) to

2 HF (Y )[[~]]. The fact that such lift is only well deﬁned up to an element in the

1 ∗ image of H (Y, OY [[~]] ) reﬂects the fact that P is only well deﬁned only up to a twist by a torsor over O [[ ]]∗. Since P → P is a G -equivariant torsor over ∗ Y ~ 1 1 C~ on P1 we can descend its Lie algebroid to Y and obtain a class of this algebroid

2 in HF (Y )[[~]] which must vanish if we want to lift the transitive Harish-Chandra structure to P.

To sum up, we have the following theorem for deformation quantization of vector bundle E on Lagrangian subvarieties Y .

Theorem 4.11. A rank r vector bundle E on a smooth Lagrangian subvariety j : Y → X admits a deformation quantization, i.e., there exists a complete ﬂat left O~-module E~ such that E~/~E~ ' j∗E if and only if the following conditions hold:

∗ 2 (1) j ωk = 0 in HDR(Y ) for k > 2;

(2) the projectivization P(E) admits a ﬂat algebraic connection;

44 2 (3) the refined first Chern class in HF (Y ) satisfies

1 1 c (E) = c (K ) + j∗ω ; r 1 2 1 Y 1

∗ 2 for the canonical lift of j ω1 to HF (Y ) representing the class of the Picard

O~ algebroid L(O~) = T or1 (OY , OY ).

If nonempty, the set of equivalence classes of all rank r deformation quantization on Y for various E has a free action of the group G of isomorphism classes of

∗ OY [[~]] -torsors with a flat algebraic connection. The set of orbits for this action may be identified with the space of all P GL(r, C[[~]]) bundles with a flat algebraic connection.

4.5 Remarks and open questions

When Y is aﬃne, a straightforward argument following Section 3 and Appendix A from [21] or, alternatively, an imitation of the arguments in Section 5 of [10], shows that we can assume that E~ is isomorphic to E[[~]] as a Zariski sheaf of

k C~-modules. This is established inductively, considering E~/~ E~ and observing

k+1 k that obstructions to trivializing E~/~ E~, given a trivialization of E~/~ E~, live in an Ext group which vanishes since Y is aﬃne and E is locally free on Y.

The result of Lagrangian case makes sense in the category of complex manifolds where de Rham cohomology groups correspond to the holomorphic de Rham complex. In fact, the arguments of this chapter carry over to the the case of etale topology, and should hold in the case of smooth Deligne-Mumford stacks as well.

45 For the case of real manifolds more work is needed to adapt our arguments but one expects that the output will be similar in spirit to the arguments in [19]

Perhaps the most celebrated example of a vector bundle with a ﬂat projective connection is the bundle of conformal blocks on the moduli space M of (marked) curves with the Hitchin connection. It would be interesting to see whether M may be realized as a Lagrangian subvariety (actually, sub-orbifold) in a larger sympletic variety, so that the bundle of conformal blocks admits a deformation quantization. Maybe the approach to M via representattions of π1(C) in

PSL(2, R) (which are embedded into PSL(2, C) representations) could give something here.

In theory, our methods should work for vector bundles on smooth coisotropic subvarieties. We will talk about this situation in detail in Chapter 5. In this case the conormal bundle N ∗ embeds as a null-foliation sub-bundle of the tangent bundle:

N ∗ ' TF ,→ TY .

Consequently, the full de Rham complex will be repaced by the normal de Rham

∗ complex built from exterior powers of TF 'N . For second order deformations this has been studied in [20].

It has been noted in [20] that for a given bundle E on Y its deformation quantization problem is described by a sheaf of curved dg Lie algebras. If we assume that all ωi restrict to zero on Y , we have an interesting situation: at the ﬁrst step of the deformation here is an obstruction (coming from the curvature element) but if it is resolved, there are no further obstructions. One way to understand it is to view E~ as a deformation of an O~-module E over the power series ring C[[~]]. Not every deformation is a deformatinon quantization

46 though, and one way to formulate the condition is to require that the ﬁrst order deformation is given by an element of Ext1 (E,E) which projects to identity O~ under the map

1 1 Ext (E,E) → HomOX (T or (OX ,E),E) = HomOX (E,E) O~ O~ provided by the Change of Rings Spectral Sequence for the homomorphism

O~ → OX . This implies that the identity IdE must be closed with respect to this spectral sequence differential Hom (E,E) → Ext2 (E,E). Once this OX OX obstruction vanishes, any higher order extension of the first order deformation as an O~-module, is automatically a deformation quantization of E and the first order adjustment to the differential of the deformation complex in [20] removes the curvature.

+ Perhaps the category of vector bundles with a structure of a L (O~)-modules deserves a closer attention. The condition of having a ﬂat structure on P(E) and

2 an equation in HF (Y ) 1 1 c (E) = c (K ) + j∗ω r 1 2 1 Y 1 is stable under direct sums, and taking the tensor product of E with a ﬂat vector bundle F . For any pair of bundles E1,E2 with these conditions the bundle

HomOY (E1,E2) is ﬂat. Indeed, for a local section ϕ of these bundles we can attempt to take its derivation along a vector ﬁeld ∂ by lifting it to a section of

+ + L (O~) and then using the action of L (O~) of E1 and E2. The lift of ∂ is only

+ well-deﬁned up to a section of OY ⊂ L (O~) but ϕ is O-linear so its derivative will not depend on the choice of this lift.

In particular, if we can ﬁnd a line bundle L which satisﬁes the above equation

47 + with r = 1, then we can write any L (O~)-module in the form E = F ⊗ L where

1 ∗ F has ﬂat algebraic connection. Of course, when 2 c1(KY ) + j ω1 = 0 we can ∗ take L = OY . Another instance is when j ω1 = 0, in which case the category

∗ admits an involution E 7→ E ⊗ KY . When, in addition, we can ﬁnd L such that

⊗2 KY ' L this involution corresponds to dualization of the local system F . It would be interesting to compare the de Rham cohomology of the local system Hom (E ,E ) and the groups Ext (E , E ) for deformation quantization of OY 1 2 O~ 1,~ 2,~

E1,E2, respectively.

The original motivation of [3] was to relate deformation quantization to the Kapustin-Rozansky 2-category of the original algebraic symplectic variety (X, ω),

2 cf. [16]. At least when the class ω1 ∈ HDR(X) vanishes, it is natural to expect

+ that an L (O~)-module E on a Lagrangian Y should deﬁne an object (Y,E) in this category. We further expect that for the same Y and diﬀerent E1,E2 the cyclic homology of the 1-category Hom((Y,E1), (Y,E2)) has something to do

with the de Rham cohomology of the local system HomOY (E1,E2).

+ This also suggests a connection between L (O~)-modules and generalized complex branes of Gualtieri, [14]. Indeed, those are deﬁned as modules over a Lie algebroid which appears after restriction to a subvariety. On the other hand, counting dimensions we see that L(O~) cannot come from an exact Courant algebroid on X, but perhaps one should work with some algebroid on X × Spec(C[[~]]). By an earlier remark in this section one expects a more general construction for vector bundles on smooth coisotropic subvarieties when connections and similar structures are only deﬁned along the null-foliation.

48 Chapter 5

Quantization of Line Bundles on Coisotropic Subvarieties

In this chapter, we assume that X is a 2n-dimensional symplectic variety over

C with a symplectic form ω. Y is a smooth coisotropic subvariety of X with dimension 2n − k. For any point x ∈ Y , after formal completion, one can choose P local formal coordinates x1, ··· , xn, y1, ··· , yn, such that ω = dxi ∧ dyi and the ideal of functions vanishing on Y is generated by y1, ··· , yk.

In this case, the Lie algebra extension of the deformation quantization of a line bundle is not abelian anymore. Therefore, the curved Maurer-Cartan equations that describe the lifting of torsors are not automatically quasi-isomorphism invariant. For this reason, we need a more explicit description of the torsor lifting arising from a Maurer-Cartan solution. In this chapter, We will make some calculations of the L∞ structure deﬁned on the total complex of the sheaf of diﬀerential graded Lie algebras.

49 5.1 Formal geometry

Redeﬁne the D-module M to be C[[x1, ··· , xn, yk+1, ··· , yn, ~]], where

~, x1, ··· , xn, yk+1, ··· , yn ∈ D

∂ act by natural multiplication and yj ∈ D, 1 j k, by . In other words, 6 6 ~ ∂xj M can be understood as

M = D/D(y1, ··· , yk).

where D(y1, ··· , yk) is the left ideal of D generated by y1, ··· , yk.

We also redeﬁne the two-sided ideal J to be D(y1, ··· , yk, ~). Therefore, we have the following relation

D/J'M/~M.

Notice that when k = n, Y is the Lagrangian subvariety of X. The coisotropic case is the generalization of the Lagrangian case. Now deﬁne

∗ ∗ Ck,~ = C[[xk+1, ··· , xn, yk+1, ··· , yn, ~]], Ck,~ = C[[xk+1, ··· , xn, yk+1, ··· , yn, ~]]

In the Lagrangian case, = [[ ]], ∗ = ∗[[ ]]. Ck,~ C ~ Ck,~ C ~

By using the similar argument in Theorem 3.2, we have the following lemma.

Lemma 5.1. There is a short exact sequence of Harish-Chandra pairs:

∗ F 1 → hCk,~, Ck,~i → hAut(D, M), Der(D, M)i → hAut(D)J , Der(D)J i → 1.

50 Let L be a line bundle over Y . Denote the deformation quantization of L by L~. For any x ∈ Y , there exist isomorphisms

η : Ob~,x 'D, µ : Lb~,x 'M

where Ob~,x and Lb~,x are completions of stalks with respect to the point x.

Deﬁne PJ to be the proalgebraic scheme parameterizing the pair (x, η).

Therefore PJ is a transitive hAut(D)J , Der(D)J i-torsor.

Theorem 5.2. A choice of a line bundle L of on Y and its deformation quantization L~ is equivalent to a choice of a lift of the torsor PJ to a transitive

Harish-Chandra hAut(D, M), Der(D, M)i-torsor PM along the extension of pairs

∗ 1 → hCk,~, Ck,~i → hAut(D, M), Der(D, M)i → hAut(D)J , Der(D)J i → 1.

Proof. For any x ∈ Y , we have a chain of isomorphisms

L~,x/~L~,x ' Lx 'OY,x 'O~,x/(~O~,x + Jy) 'D/J'M/~M

Therefore, various choices of isomorphism µ : Lb~,x 'M for all x ∈ Y give the required lift of the torsor PJ to a transitive Hanrish-Chandra torsor PM over hAut(D, M), Der(D, M)i.

In the opposite direction, let PM be a lift of PJ . Then Aut(D, M) acts on

M, therefore one has a vector bundle MP associated to the Aut(D, M)-module

M and PM, viewed as an Aut(D, M)-torsor. Moreover, the Lie algebra action of Lie(Aut(D, M)) extends to the action of the full algebra Der(D, M) which

51 implies that the bundle MP admits a ﬂat algebraic connection. Now L~ may be recovered as the sheaf of ﬂat sections with respect to this connection.

5.2 L∞-algebra and Maurer-Cartan equation

∞ L Deﬁnition 5.1. An L∞-algebra is a graded vector space L = Ln equipped n=0 with brackets

n ln(−, −, ··· , −) := Λ L → L

of degree n − 2, such that for all v1, ··· , vn ∈ L:

X ij X (−1) χ(σ, v)lj+1(li(vσ(1), ··· , vσ(i)), vσ(i+1), ··· , vσ(n)) = 0 (5.1) i+j=n σ

In the above formula, σ is an (i, j)-unshuﬄe such that

σ(1) < ··· < σ(i), σ(i + 1) < ··· < σ(n), and the sign χ(σ, v) is deﬁned by

vσ(1) ∧ · · · ∧ vσ(n) = χ(σ, v)v1 ∧ · · · ∧ vn where ∧ is the graded skew-symmetric exterior product in L satisﬁes

v ∧ w = −(−1)|v||w|w ∧ v. for all homogeneous elements v, w ∈ L.

52 ln is graded anti-symmetric, i.e.,

ln(vσ(1), ··· , vσ(n)) = χ(σ, v) · ln(v1, ··· , vn).

A curved L∞-algebra is an L∞-algebra with an additional component:

0 l0 :Λ L = C → L which is uniquely determined by 1 ∈ C. The image of 1 in L is called the curvature of the curved L-algebra.

Remark 5.1.

1. When n = 2, denote l1 by ∂, formula 5.1 says ∂(∂(v1)) = 0, ∀v1 ∈ L.

2. When n = 3, denote l2 by [−, −], formula 5.1 says

|v1||v2| −[∂(v1), v2] − (−1) [∂(v2), v1] + ∂([v1, v2]) = 0

In other words,

|v1| ∂([v1, v2]) = [∂(v1), v2] + (−1) [v1, ∂(v2)]

3. When n = 3, formula 5.1 says

|v1|(|v2|+|v3|) |v2|(|v1|+|v3|) [[v1, v2], v3]+(−1) [[v2, v3], v1]+(−1) [[v1, v3], v2] = −∂([v1, v2, v3]).

4. If all higher brackets vanish, i.e., ln>2 = 0 then for n = 4:

|v1|(|v2|+|v3|) |v2|(|v1|+|v3|) [[v1, v2], v3] + (−1) [[v2, v3], v1] + (−1) [[v1, v3], v2] = 0

53 which is the graded Jacobi identity. Therefore, in this case, L∞-algebra is just a diﬀerential graded Lie algebra.

Deﬁnition 5.2. For an L∞-algebra L with brackets ln(−, −, ··· , −), a Maurer-Cartan element is an element a ∈ L satisﬁes the Maurer-Cartan equation: ∞ X 1 l (a, ··· , a) = 0. n! n n=1

If L is a curved L∞-algebra, the curved Maurer-Cartan equation is

∞ X 1 l (a, ··· , a) = 0. n! n n=0

5.3 Lifting the torsor by solving curved

Maurer-Cartan equation

In this section, we are going to deﬁne a curved L∞ algebra with respect to the Lie algebra extension and the structure sheaf OP of the torsor. We expect that the lifting of the torsor can be interpreted as the solution of the curved Maurer-Cartan equation.

We start with an abelian group extension

1 → A → Gˆ → G → 1 and assume that all groups are unipotent, i.e., there are Lie algebras a, gˆ, g such that A = exp(a), Gˆ = exp(gˆ),G = exp(g)

54 Therefore, the corresponding Lie algebra extension is:

0 → a → gˆ → g → 0.

Let s : g → gˆ be a C-linear lifting map from g to gˆ. s deﬁnes a lifting σ : Gˆ → G such that σ(exp(x)) = exp(s(x)), ∀x ∈ g.

We have a commutative diagram:

0 a gˆ g 0

exp exp σ exp

1 A Gˆ G 1

To relate the operation of group and Lie algebra, we have the Baker-Campbell-Hausdorﬀ formula.

Theorem 5.3 (Baker-Campbell-Hausdorﬀ formula). Let G be a Lie group with Lie algebra g and let exp : g → G

be the exponential map. Then for any X,Y ∈ g,

∞ X (−1)n−1 X [Xr1 ,Y s1 ,Xr2 ,Y s2 , ··· ,Xrn ,Y sn ] log(exp(X) exp(Y )) = Pn Qn n ( (ri + si)) · ri!si! n=1 ri+si>0 i=1 i=1

where the sum is performed over all nonnegative values of si and ri, and

[Xr1 ,Y s1 , ··· ,Xrn ,Y sn ] = [X, [X, ··· [X, [Y, [Y, ··· [Y, ··· [X, [X, ··· [X, [Y, [Y, ··· ,Y ]] ··· ]] | {z } | {z } | {z } | {z } r1 s1 rn sn

55 In particular, we can list the ﬁrst few terms of the Baker-Campbell-Hausdorﬀ formula:

1 log(exp(X) exp(Y )) =X + Y + [X,Y ] 2 1 1 + ([X, [X,Y ]] + [Y, [Y,X]]) − [Y, [X, [X,Y ]]] 12 24 1 − ([Y, [Y, [Y, [Y,X]]]] + [X, [X, [X, [X,Y ]]]]) 720 1 + ([X, [Y, [Y, [Y,X]]]] + [Y, [X, [X, [X,Y ]]]]) 360 1 + ([Y, [X, [Y, [X,Y ]]]] + [X, [Y, [X, [Y,X]]]]) + ··· 120

[13]

Let P be a (usual) G-torsor over X with the projection map:

π : P → X.

We are going to lift the G-torsor P to a Gˆ-torsor Pˆ along the group extension.

• Deﬁne Ωπ to be the relative de Pham cohomology along the ﬁber of π : P → X. Consider a sheaf of algebras:

• 0 1 2 Ωπ ⊗ a = Ωπ ⊗ a → Ωπ ⊗ a → Ωπ ⊗ a → · · · .

It can also be written as

• 2 Ωπ ⊗ a = OP ⊗ a → OP ⊗ Hom(g, a) → OP ⊗ Hom(∧ g, a) → · · ·

56 Let {Ui} be an open covering of P , i.e.,

[ P = Ui i and Uij = Ui ∩ Uj, Uijk = Ui ∩ Uj ∩ Uk.

n Denote Ln = OP ⊗ Hom(∧ g, a), we have a double complex

. .

d L δ . L2(Ui) . i d d L δ L δ L1(Ui) L1(Uij) ··· i ij d d d L δ L δ L δ L0(Ui) L0(Uij) L0(Uijk) ··· i ij ijk where the vertical “diﬀerential”

d : Hom(∧ng, a) → Hom(∧n+1g, a) is the Chevalley-Eilenberg “diﬀerential”.

Deﬁne:

2 ^ C : g → a

(g1, g2) 7→ [s(g1), s(g2)]ˆg − s([g1, g2]g)

∇ : g ⊗ a → a

57 (g, a) 7→ [s(g), a]ˆg

∇ is almost a Lie action. Therefore,

d(φ)(g1, ··· , gn+1)

X i+1 X i+j = (−1) gi · φ(g1, ··· , gˆi, ··· , gn+1) + (−1) φ([gi, gj], ··· , gˆi, ··· , gˆj ··· ))

X i+1 X i+j = (−1) ∇(gi, φ(g1, ··· , gˆi, ··· , gn+1)) + (−1) φ([gi, gj], ··· , gˆi, ··· , gˆj ··· ))

X i+1 X i+j = (−1) [gi, φ(g1, ··· , gˆi, ··· , gn+1)] + (−1) φ([gi, gj], ··· , gˆi, ··· , gˆj ··· ))

However, d is not an actual diﬀerential since for any a ∈ a,

d2(a) = [C, a].

The horizontal diﬀerential δ is the Cechˇ diﬀerential

M M δ : Ln(Ui0,...,ip ) → Ln(Ui0,...,ip+1 ) p+1 X k f 7→ (−1) f ˆ | . i0,...,ik,...,ip+1 Ui0,...,ip+1 k=0

In order to lift the G-torsor P to a Gˆ-torsor Pˆ, one can construct an H-torsor ˆ P over P by taking β ∈ L0(Uij) which satisﬁes the cocycle condition:

exp(βik) = exp(βij) exp(βjk)

To make Pˆ in to a Gˆ-torsor over X, we need to deﬁne a compatible Gˆ action on Pˆ by taking α ∈ L1(Ui). Since gˆ = g⊕a as vector spaces and Γ(TPˆ) ' (g⊕a)⊗OPˆ.

58 On each Ui, αi ∈ Hom(g, a) gives a Lie algebra homomorphism:

gˆ → Γ(TPˆ)

(g, a) 7→ (g, a + αi(g)) which induces a Gˆ action on Pˆ.

Therefore, the cocycle condition:

exp(βik) = exp(βij) exp(βjk) ⇐⇒ δβ = 0.

α deﬁnes a Lie algebra homomorphism means:

dα = C i.e.

[s(g1), αi(g2)] − [s(g2), αi(g1)] − αi([g1, g2]), ∀αi, ∀g1, g2 ∈ g

The group action deﬁned by α is compatible with the gluing if and only if

αi(g) − αj(g) = [s(g), a] ∀αi, αj, ∀g ∈ g. in other words δα = dβ.

In the case when X is a point ∗, the lifting of P is Pˆ = A × G. The left multiplication action of Gˆ on itself induces a inﬁnitesimal action

a ⊕ g → a ⊗ OA×P ⊕ g ⊗ OA×P

59 Set theoretically,

(A × G) × (A × G) → A × G

((a, g), (a0, g0)) 7→ (a(σ(g)a0σ(g)−1)(σ(g)σ(g0)σ(gg0)−1), gg0)

In particular,

(A × G) × (A × G) → A × G

((a, 1), (a0, g0)) 7→ (aa0, g0)

((1, g), (a0, g0)) 7→ ((σ(g)a0σ(g)−1)(σ(g)σ(g0)σ(gg0)−1), gg0)

Let a = exp(εα), a0 = exp(α0), g = exp(εx), g0 = exp(x0)

Since A is abelian, log(aa0) = α0 + εα

Therefore, the image of (α, 0) under the Lie morphism is (α ⊗ 1, 0), i.e.,

a ⊕ g → a ⊗ OA×P ⊕ g ⊗ OA×P

(α, 0) 7→ (α ⊗ 1, 0)

By using the Baker-Campbell-Hausdorﬀ formula, we have:

0 0 1 0 1 0 0 0 log(gg ) = εx + x + 2 [εx, x ] + 12 ([εx, [εx, x ]] + [x , [x , εx]]) + ···

0 1 0 1 0 0 2 = x + ε(x + 2 [x, x ] + 12 [x , [x , x]] + ··· ) + O(ε )

60 The Lie algebra element corresponding to σ(g)a0σ(g)−1 is

log(σ(g)a0σ(g)−1)

= log(exp(s(εx)) exp(α0) exp(s(εx))−1)

= log(exp(εs(x)) exp(α0) exp(−εs(x)))

0 1 0 1 0 0 2 = log(exp(α + ε(s(x) + 2 [s(x), α ] + 12 [α , [α , s(x)]] + ··· ) + O(ε ))) exp(−εs(x)))

0 1 0 1 0 0 1 0 0 0 0 =α + ε(s(x) + 2 [s(x), α ] + 12 [α , [α , s(x)]] − 720 [α , [α , [α , [α , s(x)]] + ··· )

1 0 1 0 0 1 0 0 0 0 2 + ε(−s(x) − 2 [α , s(x)] − 12 [α , [α , s(x)]] + 720 [α , [α , [α , [α , s(x)]] + ··· ) + O(ε )

=α0 + ε([s(x), α0]) + O(ε2).

Finally, the preimage of the cocycle of the group extension is

log(σ(g)σ(g0)σ(gg)−1)

= log(exp(s(εx)) exp(s(x0))σ(exp(εx) exp(x0))−1)

0 0 1 0 1 0 0 2 −1 = log(exp(s(εx)) exp(s(x ))σ(exp(x + ε(x + 2 [x, x ] + 12 [x , [x , x]] + ··· ) + O(ε ))) )

0 1 0 1 0 0 2 = log(exp(s(x ) + ε(s(x) + 2 [s(x), s(x )] + 12 [s(x ), [s(x ), s(x)]] + ··· ) + O(ε ))

0 1 0 1 0 0 2 · exp(−s(x ) − ε(s(x) + 2 s([x, x ]) + 12 s([x , [x , x]]) + ··· ) + O(ε )))

1 0 1 0 0 =ε((s(x) + 2 [s(x), s(x )] + 12 [s(x ), [s(x ), s(x)]] + ··· )

1 0 1 0 0 − (s(x) + 2 s([x, x ]) + 12 s([x , [x , x]]) + ··· ))

1 0 1 0 1 0 0 − 2 [s(x ), s(x) + 2 s([x, x ]) + 12 s([x , [x , x]]) + ··· ]

1 1 0 1 0 0 0 − 2 [s(x) + 2 [s(x), s(x )] + 12 [s(x ), [s(x ), s(x)]] + ··· , s(x )]

1 0 0 1 0 1 0 0 − 12 [s(x ), [s(x ), s(x) + 2 s([x, x ]) + 12 s([x , [x , x]]) + ··· ]]

1 0 0 1 0 1 0 0 + 12 [s(x ), [s(x ), s(x) + 2 [s(x), s(x )] + 12 [s(x ), [s(x ), s(x)]] + ··· ]]

+ ··· ) + O(ε2)

1 0 0 =ε( 2 ([s(x), s(x )] − s([x, x ])

1 0 0 0 0 0 0 + 12 (−s([x , [x , x]]) + 3[s(x ), s([x , x])] − 2[s(x ), [s(x ), s(x)])

61 + ··· ) + O(ε2)

Therefore, the image of (0, x) under the Lie morphism is

g → a ⊗ OA×P

1 x 7→ [s(x), ·] + 2 ([s(x), s(·)] − s([x, ·])

1 + 12 (−s([·, [·, x]]) + 3[s(·), s([·, x])] − 2[s(·), [s(·), s(x)]) + ···

g → g ⊗ OA×P

1 1 x 7→ x + 2 [x, ·] + 12 [·, [·, x]] + ···

Where “ · ” is the G coordinate of A × P.

To sum up, we have the following theorem:

L L Theorem 5.4. Let Ln,p = Ln(Ui0,...,ip ). The total complex { n+p=m Ln,p}:

L0,0 → L0,1 ⊕ L1,0 → L0,2 ⊕ L1,1 ⊕ L2,0 → · · ·

is a curved L∞-algebra, where l0 = C, l1 is the total diﬀerential of the double complex, l2 = [ , ].

ˆ ˆ (α, β) ∈ L0,1 ⊕ L1,0 defines a lifting of the G-torsor P to G-torsor P iff (α, β) satisfies the curved Maurer-Cartan equation:

1 l + l (α, β) + l (α, β) = 0 0 1 2 2

Furthermore, if (α, β) and (α0, β0) describe the same Gˆ-torsor upto equivalence,

62 then there is a γ ∈ L0(Ui) such that

α0 = α + dγ

β0 = β + δγ.

In general, if the group extension 1 → H → Gˆ → G → 1 is not abelian, the cocycle condition for β is :

exp(βik) = exp(βij) exp(βjk).

L And the vertical complex { i Ln(Ui)}:

d d d L0(Ui) → L1(Ui) → L2(Ui) → · · ·

is a curved L∞-algebra with l0 = C, l1 = d, l2 = [ , ].

α defines a Lie algebra homomorphism iff it satisfies the curved Maurer-Cartan equation: 1 C + dα + [α, α] = 0 2

Furthermore, if we deﬁne a new diﬀerential dα by

dαi = d + [αi, · ]

L then dα will be an actual diﬀerential on the vertical complex { i Ln(Ui)}

63 satisfying

2 dα = 0

The group action deﬁned by α is compatible with the gluing if and only if

exp(βij)dαi exp(−βij) = dαj this can also be denoted by the gauge equivalent relation:

αi ∼ αj βij

We have the following conjecture:

L L Conjecture 5.5. Let Ln,p = Ln(Ui0,...,ip ). The total complex { n+p=m Ln,p}:

L0,0 → L0,1 ⊕ L1,0 → L0,2 ⊕ L1,1 ⊕ L2,0 → · · ·

is a curved L∞-algebra, where l0 = C, l1 is the total diﬀerential of the double complex, l2 = [ , ].

ˆ ˆ (α, β) ∈ L0,1 ⊕ L1,0 defines a lifting of the G-torsor P to G-torsor P iff (α, β) satisfies the curved Maurer-Cartan equation:

1 l + l (α, β) + l (α, β) = 0 0 1 2 2

Furthermore, two solutions (α, β) and (α0, β0) are equivalent to each other iﬀ there is a γ ∈ L0,0 such that:

0 (i) αi ∼ αi; γi

64 0 (ii) βij = βij + γi − γj;

(iii) exp(βij) = exp(γi) exp(βij) exp(−γj).

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