Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry (EMS Series of Lectures in Mathematics)

EMS Series of Lectures in Mathematics Edited by Andrew Ranicki (University of Edinburgh, U.K.) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim, Uhlenbeck Compactness Torsten Ekedahl, One Semester of Elliptic Curves Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups

Damien Calaque Carlo A. Rossi

Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

Authors: Damien Calaque ETH (D-MATH) Rämistrasse 101 8092 Zürich Switzerland

Carlo A. Rossi Max Planck Institute for Mathematics Vivatsgasse 7 53111 Bonn Germany

[email protected]

[email protected]

2010 Mathematics Subject Classification: 13D03, 17B56, 14F43 Key words: Lie algebra, Hochschild cohomology, complex manifolds, deformation theory, Kontsevich’s graphical calculus, Atiyah class, Duflo isomorphism, Todd class

ISBN 978-3-03719-096-8 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2011 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Druckhaus Thomas Müntzer GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface

Since the fundamental results by Harish-Chandra and others, it is now well known that the algebra of invariant polynomials on the dual of a Lie algebra of a particular type (solvable [18], simple [24] or nilpotent) is isomorphic to the center of the corresponding universal enveloping algebra. This fact was generalized to an arbitrary finite-dimensional real Lie algebra by M. Duflo in 1977 [19]. His proof is based on Kirillov’s orbits method that parametrizes infinitesimal characters of unitary irreducible representations of the corresponding Lie group in terms of co-adjoint orbits (see e.g. [28]). This isomorphism is called the Duflo isomorphism. It happens to be a composition of the well-known Poincaré–Birkhoff–Witt isomorphism (which is only an isomorphism at the level of vector spaces) with an automorphism of the space of polynomials (which descends to invariant polynomials), whose definition involves the power series j.x/ ´ sinh.x=2/=.x=2/. In 1997 Kontsevich [29] proposed another proof, as a consequence of his construction of deformation quantization for general Poisson manifolds. Kontsevich’s approach has the advantage of working also for Lie super-algebras and extending the Duflo isomorphism to a graded algebra isomorphism on the whole cohomology. The inverse power series j.x/1 D .x=2/=sinh.x=2/ also appears in Kontsevich’s claim that the Hochschild cohomology of a complex manifold is isomorphic as an algebra to the cohomology ring of holomorphic poly-vector fields on this manifold. We can summarize the analogy between the two situations in the following table: Lie algebra symmetric algebra universal enveloping algebra taking invariants Chevalley–Eilenberg cohomology

Complex geometry sheaf of algebra of holomorphic poly-vector fields sheaf of algebra of holomorphic poly-differential operators taking global holomorphic sections sheaf cohomology

These lecture notes provide a self-contained proof of the Duflo isomorphism and its complex geometric analogue in a unified framework, and gives in particular a unifying explanation of the reason why the series j.x/ and its inverse appear. The proof is strongly based on Kontsevich’s original idea, but actually differs from it (the two approaches are related by a Koszul type duality recently pointed out in [39] and proved in [8], this duality being itself a manifestation of Cattaneo–Felder constructions for the quantization of a Poisson manifold with two coisotropic submanifolds [12]). Note that the series j.x/ also appears in the wheeling theorem by Bar-Natan, Le and Thurston [4] which shows that two spaces of graph homology are isomorphic as

vi

Preface

algebras (see also [31] for a completely combinatorial proof of the wheeling theorem, based on Alekseev and Meinrenken’s proof [1], [2] of the Duflo isomorphism for quadratic Lie algebras). Furthermore this power series also shows up in various index theorems (e.g. Riemann–Roch theorems). Throughout these notes we assume that k is a field with char.k/ D 0. Unless otherwise specified, algebras, modules, etc. are over k. Each chapter consists (more or less) of a single lecture. Acknowledgements. The authors thank the participants of the lectures for their interest and excitement. They are responsible for the very existence of these notes, as well as for improvement of their quality. The first author is grateful to G. Felder who offered him the opportunity to give this series of master course lectures in the fall semester of the academic year 2007–08 at ETH Zurich. He also thanks M. Van den Bergh for his kind collaboration in [9] and many enlightening discussions about this fascinating subject. His research is fully supported by the European Union thanks to a Marie Curie Intra-European Fellowship (contract number MEIF-CT-2007-042212).

Contents

Preface

v

1

. . . .

1 2 4 6 8

2

Hochschild cohomology and spectral sequences 2.1 Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Application: Chevalley–Eilenberg versus Hochschild cohomology . .

11 11 13 16

3

Dolbeault cohomology and the Kontsevich isomorphism 3.1 Complex manifolds . . . . . . . . . . . . . . . . . . 3.2 Atiyah and Todd classes . . . . . . . . . . . . . . . . 3.3 Hochschild cohomology of a complex manifold . . . 3.4 The Kontsevich isomorphism . . . . . . . . . . . . .

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19 19 21 22 24

Superspaces and Hochschild cohomology 4.1 Supermathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hochschild cohomology strikes back . . . . . . . . . . . . . . . . . .

25 25 28

5 The Duflo–Kontsevich isomorphism for Q-spaces 5.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Application: proof of the Duflo Theorem . . . . . . . . . . . . . . . . 5.3 Strategy of the proof . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 35 37

6

Configuration spaces and integral weights C . . . . . . . . . . . . . . . . . . . . . 6.1 The configuration spaces Cn;m C 6.2 Compactification of Cn and Cn;m à la Fulton–MacPherson . . . . . . 6.3 Directed graphs and integrals over configuration spaces . . . . . . . .

41 41 42 46

7 The map UQ and its properties 7.1 The quasi-isomorphism property . . . . . . . . . . . . . . . . . . . . 7.2 The cup product on poly-vector fields . . . . . . . . . . . . . . . . . 7.3 The cup product on poly-differential operators . . . . . . . . . . . . .

51 51 54 57

4

Lie algebra cohomology and the Duflo isomorphism 1.1 The original Duflo isomorphism . . . . . . . . . 1.2 Cohomology . . . . . . . . . . . . . . . . . . . 1.3 Chevalley–Eilenberg cohomology . . . . . . . . 1.4 The cohomological Duflo isomorphism . . . . .

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viii

Contents

8 The map HQ and the homotopy argument 8.1 The complete homotopy argument . . . . . . . . . . . . . . . . . . . 8.2 Contribution to W2 of boundary components in Y . . . . . . . . . . . 8.3 Twisting by a supercommutative DG algebra . . . . . . . . . . . . . .

61 61 63 68

9 The explicit form of UQ 9.1 Graphs contributing to UQ . . . . . . . . . . . . . . . . . . . . . . . 9.2 UQ as a contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The weight of an even wheel . . . . . . . . . . . . . . . . . . . . . .

71 71 73 76

10 Fedosov resolutions 10.1 Bundles of formal fiberwise geometric objects 10.2 Resolutions of algebras . . . . . . . . . . . . 10.3 Fedosov differential . . . . . . . . . . . . . . 10.4 Fedosov resolutions . . . . . . . . . . . . . . 10.5 Proof of Theorem 3.6 . . . . . . . . . . . . .

79 79 81 82 84 86

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Appendix Deformation-theoretical interpretation of Hochschild cohomology ˇ A.1 Cech cohomology: a (very) brief introduction . . . . . . . . . . . . . ˇ A.2 The link between Cech and Dolbeault cohomology: Dolbeault’s Theorem A.3 Twisted presheaves of algebras . . . . . . . . . . . . . . . . . . . . .

89 89 90 93

Bibliography

101

Index

105

1 Lie algebra cohomology and the Duflo isomorphism

Let g be a finite-dimensional Lie algebra over k. In this chapter we state the Duflo theorem and its cohomological extension. We take this opportunity to introduce standard notions of homological algebra and define the cohomology theory associated to Lie algebras, which is called Chevalley–Eilenberg cohomology. Preliminaries: tensor, symmetric, and universal enveloping algebras. For a kvector space V we define the tensor algebra T .V / of V as the vector space M T .V / ´ V ˝n .V ˝0 D k by convention/ n0

equipped with the product given by the concatenation. It is a graded algebra, whose subspace of homogeneous elements of degree n is T n .V / ´ V ˝n . The symmetric algebra of V , which we denote by S.V /, is the quotient of the tensor algebra T .V / by its two-sided ideal generated by v˝ww˝v

.v; w 2 V /:

Since the previous relations are homogeneous, then S.V / inherits a grading from the one on T .V /. Finally, if V D g, one can define the universal enveloping algebra U.g/ of g as the quotient of the tensor algebra T .g/ by its two-sided ideal generated by x ˝ y  y ˝ x  Œx; y

.x; y 2 V /;

where Œx; y denotes the Lie bracket between x and y. As the relations are not homogeneous, the universal enveloping algebra only inherits a filtration from the grading on the tensor algebra. Notation. Dealing with non-negatively graded vector spaces, we will use the symboly to denote the corresponding degree completions. Namely, if M is a graded k-vector space, then Y y ´ Mn M n0

is the set of formal series X n0

m.n/ ;

.m.n/ 2 M n /:

2

1 Lie algebra cohomology and the Duflo isomorphism

1.1 The original Duflo isomorphism The Poincaré–Birkhoff–Witt theorem. Recall the Poincaré–Birkhoff–Witt (PBW) theorem: the symmetrization map IPBW W S.g/ ! U.g/;

x1 : : : xn 7!

1 X x1 : : : xn ; nŠ 2Sn

is an isomorphism of filtered vector spaces, which further induces an isomorphism of the corresponding graded algebras S.g/ ! Gr.U.g//. Let us write  for the associative product on S.g/ defined as the pullback of the multiplication on U.g/ through IPBW . For any two homogeneous elements u; v 2 S.g/, u  v D uv C lower order terms. IPBW is obviously not an algebra isomorphism, unless g is abelian (since S.g/ is commutative while U.g/ is not). Remark 1.1. There are different proofs of the PBW Theorem: standard proofs may be found in [16], to which we refer for more details. More conceptual proofs, involving Koszul duality between quadratic algebras, may be found in [6], [37]. A proof of the PBW Theorem stemming from Kontsevich’s Deformation Quantization may be found in [39], [8]. Geometric meaning of the PBW theorem. We consider a connected, simply connected Lie group G with corresponding Lie algebra g. Then S.g/ can be viewed as the algebra of distributions on g supported at the origin 0 with (commutative) product given by the convolution with respect to the (abelian) additive group law on g. In the same way U.g/ can be viewed as the algebra of distributions on G supported at the origin e with product given by the convolution with respect to the group law on G. One sees that IPBW is nothing but the transport of distributions through the exponential map exp W g ! G (recall that it is a local diffeomorphism). The exponential map is obviously Ad -equivariant. In the next section we will translate this equivariance in algebraic terms. g-module structure on S.g/ and U.g/. On the one hand there is a g-action on S.g/ obtained from the adjoint action ad of g on itself, extended to S.g/ by Leibniz’ rule: for any x; y 2 g and n 2 N , adx .y n / D nŒx; yy n1 : On the other hand there is also an adjoint action of g on U.g/: for any x 2 g and u 2 U.g/, adx .u/ D xu  ux:

1.1 The original Duflo isomorphism

3

It is an easy exercise to verify that adx B IPBW D IPBW B adx for any x 2 g. Therefore IPBW restricts to an isomorphism (of vector spaces) from S.g/g to the center Z.U g/ D U.g/g of U g. Now we have commutative algebras on both sides. Nevertheless, IPBW is not yet an algebra isomorphism. Theorem 1.3 below is concerned with the failure of this map to preserve the product. Duflo element J . We define an element J 2 Sy.g / (the set of formal power series on g) as follows:   1  e adx J.x/ ´ det : adx It can be expressed as a formal power series with respect to ck ´ tr..ad/k /. Let us explain what this means. Recall that ad is the linear map g ! End.g/ defined by adx .y/ D Œx; y (x; y 2 g). Therefore ad 2 g ˝ End.g/ and thus .ad/k 2 T k .g / ˝ End.g/. Consequently tr..ad/k / 2 T k .g / and we regard it as an element of S k .g / through the projection T .g / ! S.g /. Notation. Here and below, for a vector space V we denote by End.V / the algebra of endomorphisms of V , and by V  the vector space of linear forms on V . Claim 1.2. ck is g-invariant. Here the g-module structure on S.g / is the coadjoint action on g extended by Leibniz’ rule. Proof. Let x; y 2 g. Then hy  ck ; x n i D hck ;

n P

x i Œy; xx ni1 i

iD1

D D

n P iD1 n P iD1

tr.adix adŒy;x adni1 / x / tr.adix Œady ; adx adni1 x

D  tr.Œady ; adnx / D 0: This proves the claim.



The Duflo isomorphism. Observe that an element  2 g acts on S.g/ as a derivation as follows: for any x 2 g,   x n D n.x/x n1 : By extension an element ./k 2 S k .g / acts as follows: ./k  x n D n : : : .n  k C 1/.x/k x nk :

4

1 Lie algebra cohomology and the Duflo isomorphism

y  / acts on S.g/.1 Moreover, one sees without difficulty that This way the algebra S.g  g g Sy.g / acts on S.g/ . We have: Theorem 1.3 (Duflo, [19]). IPBW BJ 1=2  defines an isomorphism of algebras S.g/g ! U.g/g . The proof we will give in these lectures is based on deformation theory and homological algebra, following the deep insight of M. Kontsevich [29] (see also [38]). Remark 1.4. c1 is a derivation of S.g/, thus exp.c1 / defines an algebra automorphism of S.g/. Therefore one can obviously replace J by the modified Duflo element 



e adx =2  e adx =2 JQ .x/ D det : adx Remark 1.5. It has been proved by Duflo that, for any finite-dimensional Lie algebra g, the trace of odd powers of the adjoint representation of g acts trivially on S.g/g . In [30], Theorem 8, Kontsevich states that such odd powers act as derivations on S.g/g , where now g may be a finite-dimensional graded Lie algebra (see Chapter 4). It is not known if, for a finite-dimensional graded Lie algebra g, the traces of odd powers of the adjoint representation act trivially on S.g/g : if not, they would provide a nontrivial incarnation of the action of the Grothendieck–Teichmüller group on deformation quantization.

1.2 Cohomology Our aim is to show that Theorem 1.3 is the degree 0 part of a more general statement. For this we need a few definitions. L Definition 1.6. 1. A DG vector space is a Z-graded vector space C  D n2Z C n equipped with a graded linear endomorphism d W C ! C of degree 1 (i.e., d.C n /  C nC1 ) such that d B d D 0. d is called the differential. 2. A DG (associative) algebra is a DG vector space .A ; d / equipped with an associative product which is graded (i.e., Ak  Al  AkCl ) and such that d is a graded derivation of degree 1: for homogeneous elements a; b 2 A d.a  b/ D d.a/  b C .1/jaj a  d.b/. 3. Let .A ; d / be a DG algebra. A DG A-module is a DG vector space .M  ; d / equipped with an A-module structure which is graded (i.e., Ak  M l  M mCl ) and such that d satisfies d.a  m/ D d.a/  m C .1/jaj a  d.m/ for homogeneous elements a 2 A, m 2 M . 1 This action can be regarded as the action of the algebra of differential operators with constant coefficients on g (of possibly infinite order) onto functions on g .

1.2 Cohomology

5

4. A morphism of DG vector spaces (resp. DG algebras, DG A-modules) is a degree-preserving linear map that intertwines the differentials (resp. the products, the module structures). DG vector spaces are also called cochain complexes (or simply complexes) and differentials are also known as coboundary operators. Recall that the cohomology of a cochain complex .C  ; d / is the graded vector space H  .C; d / defined by the quotient ker.d /=im.d /: H n .C; d / ´

fc 2 C n j d.c/ D 0g fn-cocyclesg D : n1 fb D d.a/ j a 2 C g fn-coboundariesg

Any morphism of cochain complexes induces a degree-preserving linear map at the level of cohomology. The cohomology of a DG algebra is a graded algebra. Example 1.7 (Differential-geometric induced DG algebraic structures). Let M be a differentiable manifold. Then the graded algebra of differential forms  .M / equipped with the de Rham differential d D ddR is a DG algebra. Recall that for any ! 2 n .M / and v0 ; : : : ; vn 2 X.M /, d.!/.u0 ; : : : ; un / ´

n X iD0

C

.1/i ui .!.u0 ; : : : ; ui ; : : : ; un // X

.1/iCj !.Œui ; uj ; u0 ; : : : ; ui ; : : : ; uj ; : : : ; un /:

0i<j n

In local coordinates .x 1 ; : : : ; x n /, the de Rham differential reads d D dx i @x@ i . The  corresponding cohomology is denoted by HdR .M /. For any C 1 map f W M ! N , one has a morphism of DG algebras given by the pullback of forms f  W  .N / !  .M /. Let E ! M be a vector bundle and recall that a connection r on M with values in E is given by a linear map r W .M; E/ ! 1 .M; E/ such that for any f 2 C 1 .M / and s 2 .M; E/ one has r.f s/ D d.f /s C f r.s/. Observe that it extends in a unique way to a degree 1 linear map r W  .M; E/ !  .M; E/ such that for any  2  .M / and s 2  .M; E/, r.s/ D d./s C .1/jj r.s/. Therefore if the connection is flat (which is basically equivalent to the requirement that r B r D 0) then  .M; E/ becomes a DG .M /-module. Conversely, any differential r that turns .M; E/ into a DG .M /-module defines a flat connection. Definition 1.8. A quasi-isomorphism is a morphism that induces an isomorphism at the level of cohomology. Example 1.9 (Poincaré Lemma). Let us regard R as a DG algebra concentrated in degree 0 and with d D 0. The inclusion i W .R; 0/ ,! . .Rn /; d/ is a quasi-isomorphism of DG algebras. The proof of this claim is quite instructive as it makes use of a standard method in homological algebra.

6

1 Lie algebra cohomology and the Duflo isomorphism

Proof. We construct a degree 1 graded linear map  W  .Rn / ! 1 .Rn / such that d B  C  B d D id  i B p;

(1.1)

where p W  .M / ! k takes the degree 0 part of a form and evaluates it at the origin: p.f .x; dx// D f .0; 0/ (here we write locally a form as a “function” of the “variables” x 1 ; : : : ; x n ; dx 1 ; : : : ; dx n ).2 Then it is obvious that any closed form lies in the image of i up to an exact one; in other words, we have proved that p admits a homotopy inverse. It is left as an exercise to check that  defined by .1/ D 0 and Z 1  dt j ker.p/ .f .x; dx// D x i @i f .t x; t dx/ t 0 



obeys the desired requirements.

Notice that we have proved at the same time that p W . .M /; d/ ! .k; 0/ is also a quasi-isomorphism. Moreover, one can check that  B  D 0. This allows us to decompose  .M / as ker./ ˚ im./, where  is defined to be the left-hand side of (1.1).  is often called the Laplacian and thus elements lying in its kernel are said to be harmonic.3

1.3 Chevalley–Eilenberg cohomology The Chevalley–Eilenberg complex. Let V be a g-module. The associated Chevalley– V  .g; V / is defined as follows: C n .g; V / D n .g/ ˝ V is the Eilenberg complex CV space of linear maps n .g/ ! V and the differential dC is defined on homogeneous elements by X .dC .l//.x0 ; : : : ; xn / ´ .1/iCj l.Œxi ; xj ; x0 ; : : : ; xi ; : : : ; xj ; : : : ; xn / 0i<j n

C

n X

.1/i xi  l.x0 ; : : : ; xi ; : : : ; xn /:

iD0

We prove below that dC B dC D 0. The corresponding cohomology is denoted by H  .g; V /. V Remark 1.10. Below we implicitly identify .g/ with antisymmetric elements in T .g/. Namely, we define the total antisymmetrization operator alt W T .g/ ! T .g/: 1 X alt.x1 ˝    ˝ xn / ´ .1/ x.1/ ˝    ˝ x.n/ : nŠ  2Sn

2

This comment will receive a precise explanation in Chapter 4, where we consider superspaces. This terminology is chosen by analogy with the Hodge–de Rham decomposition of  .M / when M is a Riemannian manifold. Namely, let  be the Hodge star operator and define  ´ ˙  d. Then  is precisely the usual Laplacian, and harmonic forms provide representatives of de Rham cohomology classes. 3

7

1.3 Chevalley–Eilenberg cohomology

V  It is a projection, and it factors through an isomorphism .g/V ! ker.alt V id/, that we also denote by alt. In particular this allows us to identify .g / with .g/ . Cup product. If V D A is equipped with an associative g-invariant product, meaning that, for any x 2 g and any a; b 2 A, x  .ab/ D .x  a/b C a.x  b/; then C  .g; A/ V naturally becomes a graded algebra with product [ defined as follows: for any ; 2 .g / and a; b 2 A, . ˝ a/ [ . ˝ b/ D  ^ ˝ ab: V V Another way to write the product is as follows: for l 2 m .g/ ˝ A, l 0 2 n .g/ ˝ A and x1 ; : : : ; xmCn 2 g, .l [ l 0 /.x1 ; : : : ; xmCn / X 1 D .1/ l.x.1/ ; : : : ; x.m/ /l 0 .x.mC1/ ; : : : ; x.mCn/ /: .m C n/Š  2SmCn

Remark 1.11. Observe that since l and l 0 are already antisymmetric then it is sufficient mŠnŠ to take .mCn/Š times the sum over .m; n/-shuffles (i.e., 2 SmCn such that .1/ <    < .m/ and .m C 1/ <    < .m C n/). Exercise 1.12. Check that [ is associative and satisfies dC .l [ l 0 / D dC .l/ [ l 0 C .1/jlj l [ dC .l 0 /:

(1.2)

The Chevalley–Eilenberg complex is a complex. In this section we prove that dC B dC D 0. Let us first prove it in the case when V D k is the trivial module. Let  2 g and x; y; z 2 g. Then ..dC B dC /.//.x; y; z/ D .dC .//.Œx; y; z/ C .dC .//.Œx; z; y/  .dC .//.Œy; z; x/ D .ŒŒx; y; z  ŒŒx; z; y C ŒŒy; z; x/ D 0: V Since .g / is generated as an algebra (with product [ D ^) by g , it follows from (1.2) that dC B dC D 0. V  V/ D .g / ˝ V is a Let V us come back to the general case. Observe that C  .g; V V    graded .g /-module: for any  2 .g / and ˝ v 2 .g / ˝ V ,   . ˝ v/ ´ . ^ / ˝ v:

8

1 Lie algebra cohomology and the Duflo isomorphism

Since C  .g; V / is generated by V as a graded verification is left as an exercise) that

V

.g /-module, and due to the fact (the

dC .  . ˝ v// D .dC .//  . ˝ v/ C .1/jj   dC . ˝ v/; it is sufficient to prove that .dC B dC /.v/ D 0 for any v 2 V . We do this now: if x; y 2 g then ..dC B dC /.v//.x; y/ D .dC .v//.Œx; y/ C x  .dC .v//.y/  y  .dC .v//.x/ D Œx; y  v C x  .y  v//  y  .x  v/ D 0: Interpretation of H 0 .g; V /, H 1 .g; V / and H 2 .g; V /. We will now give an algebraic interpretation of the low degree components of Chevalley–Eilenberg cohomology. • Obviously, the 0-th cohomology space H 0 .g; V / is equal to the space V g of g-invariant elements in V (i.e., those elements on which the action is zero). • 1-cocycles are linear maps l W g ! V such that l.Œx; y/ D x  l.y/  y  l.x/ for x; y 2 g. In other words 1-cocycles are g-derivations with values in V . 1coboundaries are those derivations lv (v 2 V ) of the form lv .x/ D x  v (x 2 g), which are called inner derivations. Thus H 1 .g; V / is the quotient of the space of derivations by inner derivations. V • 2-cocycles are linear maps ! W 2 g ! V such that !.Œx; y; z/ C !.Œz; x; y/ C !.Œy; z; x/  x  !.y; z/ C y  !.x; z/  z  !.y; z/ D 0

.x; y; x 2 g/:

This last condition is equivalent to the requirement that the space g ˚ V equipped with the bracket Œ.x; u/; .y; v/ D .Œx; y; x  v  y  u C !.x; y//

.x; y 2 g; v; w 2 V /

is a Lie algebra. Such objects are called extensions of g by V . 2-coboundaries ! D dC .l/ correspond exactly to those extensions that are trivial (i.e., such that the resulting Lie algebra structure on g ˚ V is isomorphic to the one given by !0 D 0; the isomorphism is given by .x; v/ 7! .x; l.x/ C v/).

1.4 The cohomological Duflo isomorphism  From the PBW isomorphism IPBW W S.g/  ! U.g/ of g-modules one obtains an   ! C  .g; U.g//. This is obviously isomorphism of cochain complexes C .g; S.g//  not a DG algebra morphism (even at the level of cohomology). The following result is an extension of the Duflo Theorem 1.3. It has been rigorously proved by M. Pevzner and C. Torossian in [35], after the deep insight of M. Kontsevich.

1.4 The cohomological Duflo isomorphism

9

Theorem 1.13. IPBW B J 1=2  induces an isomorphism of algebras at the level of cohomology  H  .g; S.g//  ! H  .g; U.g//: Again, one can obviously replace J by JQ .

2 Hochschild cohomology and spectral sequences

In this chapter we define a cohomology theory for associative algebras, which is called Hochschild cohomology, and explain the meaning of it. We also introduce the notion of a spectral sequence and use it to prove that, for a Lie algebra g, the Hochschild cohomology of U.g/ is the same as the Chevalley–Eilenberg cohomology of g.

2.1 Hochschild cohomology The Hochschild cochain complex. Let A be an associative algebra and M an Abimodule (i.e., a vector space equipped with two commuting A-actions, one on the left and the other on the right). The associated Hochschild complex C  .A; M / is defined as follows: C n .A; M / is the space of linear maps A˝n ! M and the differential dH is defined on homogeneous elements by the formula .dH .f //.a0 ; : : : ; an / D a0 f .a1 ; : : : ; an / C

n X

.1/i f .a0 ; : : : ; ai1 ai ; : : : ; an /

iD1

C.1/nC1 f .a0 ; : : : ; an1 /an : It is easy to prove that dH B dH D 0; the corresponding cohomology is denoted by H  .A; M /. If M D B is an algebra such that for any a 2 A and any b; b 0 2 B a.bb 0 / D .ab/b 0 and .bb 0 /a D b.b 0 a/ (e.g., B D A the algebra itself) then .C  .A; B/; dH / becomes a DG algebra; the product [ is defined on homogeneous elements by .f [ g/.a1 ; : : : ; amCn / D f .a1 ; : : : ; am /g.amC1 ; : : : ; amCn /: If M D A, then we write HH .A/ ´ H  .A; A/. Interpretation of H 0 .A ; M / and H 1 .A ; M /. We will now interpret the low-degree components of Hochschild cohomology. • Obviously, the 0-th cohomology space H 0 .A; M / is equal to the space M A of A-invariant elements in M (i.e., those elements on which the left and right actions coincide). In the case M D A is the algebra itself we then have H 0 .A; A/ D Z.A/.

12

2 Hochschild cohomology and spectral sequences

• 1-cocycles are linear maps l W A ! M such that l.ab/ D al.b/ C l.a/b for a; b 2 A, i.e., 1-cocycles are A-derivations with values in M . 1-coboundaries are those derivations lm (m 2 M ) of the form lm .a/ D ma  am (a 2 A), which are called inner derivations. Thus H 1 .A; M / is the quotient of the space of derivations by inner derivations. Interpretation of HH2 .A/ and HH3 .A/: deformation theory. Now let M D A be the algebra itself. • An infinitesimal deformation of A is an associative -linear product  on AŒ= 2 such that a  b D ab mod . This last condition means that for any a; b 2 A, a  b D ab C .a; b/, with W A ˝ A ! A. The associativity of  is then equivalent to a .b; c/ C .a; bc/ D .a; b/c C .ab; c/ which is exactly the 2-cocycle condition. Conversely, any 2-cocycle allows us to define an infinitesimal deformation of A. Two infinitesimal deformations  and 0 are equivalent if there is an isomorphism of kŒ= 2 -algebras .AŒ= 2 ; / ! .AŒ= 2 ; 0 / that is the identity mod . This last condition means that there exists l W A ! A such that the isomorphism maps a to a C l.a/. Being a morphism is then equivalent to .a; b/ C l.ab/ D 0 .a; b/ C al.b/ C l.a/b; which is equivalent to  0 D dH .l/. Therefore HH2 .A/ is the set of infinitesimal deformations of A up to equivalences. • An order n (n > 0) deformation of A is an associative -linear product  on AŒ= nC1 such that a  b D ab mod . This last condition means that the product is given by n X  i i .a; b/; a  b D ab C iD1

with i W A ˝ A ! A. Let us define ´ associativity is then equivalent to

Pn iD1

i  i 2 C 2 .A; AŒ/. The

dH . /.a; b; c/ D . .a; b/; c/  .a; .b; c// mod  nC1 : Proposition 2.1 (Gerstenhaber, [22]). If  is an order n deformation then the linear map nC1 W A˝3 ! A defined by

nC1 .a; b; c/ ´

n X

. i . nC1i .a; b/; c/  i .a; nC1i .b; c///

iD1

is a 3-cocycle, i.e., dH . nC1 / D 0.

2.2 Spectral sequences

13

Proof. Let us define .a; b; c/ ´ . .a; b/; c/  .a; .b; c// 2 AŒ. The associativity condition then reads dH . / D mod  nC1 and nC1 is precisely the coefficient of  nC1 in . Therefore it remains to prove that dH . / D 0 mod  nC2 . We leave it as an exercise to prove that dH . /.a; b; c; d / D .a; dH . /.b; c; d //  dH . /. .a; b/; c; d / C dH . /.a; .b; c/; d /  dH . /.a; b; .c; d // C .dH . /.a; b; c/; d /: Then it follows from the associativity condition that mod  nC2 the left-hand side is equal to

. .a; b/; c; d /  .a; .b; c/; d / C .a; b; .c; d //  . .a; b; c/; d / C .a; .b; c; d //: Finally, a straightforward computation shows that this last expression is identically zero.  Given an order n deformation one can ask if it is possible to extend it to an order n C 1 deformation. This means that we ask for a linear map nC1 W A ˝ A ! A such that nC1 nC1 X X i . nC1i .a; b/; c/ D i .a; nC1i .b; c//; iD0

iD0

which is equivalent to dH . nC1 / D nC1 . In other words, the only obstruction for extending deformations lies in HH3 .A/. This deformation-theoretical interpretation of Hochschild cohomology is due to M. Gerstenhaber [22].

2.2 Spectral sequences Spectral sequences are essential algebraic tools for working with cohomology. They were invented by J. Leray [32], [33]. Definition. A spectral sequence is a sequence .Er ; dr /r0 of bigraded spaces M Er D Erp;q .p;q/2Z2

together with differentials dr W Erp;q ! ErpCr;qrC1 ;

dr B dr D 0

14

2 Hochschild cohomology and spectral sequences

such that H.Er ; dr / D ErC1 (as bigraded spaces). One says that a spectral sequence converges (to E1 ) or stabilizes if for any .p; q/ p;q for all r  r.p; q/. We then define there exists r.p; q/ such that Erp;q D Er.p;q/ p;q p;q E1 ´ Er.p;q/ . This arises when drpCr;qrC1 D drp;q D 0 for r  r.p; q/. A convenient way to think about spectral sequences is to draw them:

Ep;qC1 O

EpC1;qC1

EpC2;qC1

p;q

d0

d

p;q

1 / E pC1;q Ep;q UUUU EpC2;q UUUU  p;q UUUU d2 UUUU UUUU UU* p;q1 pC1;q1 E E EpC2;q1

The spectral sequence of a filtered complex. A filtered complex is a decreasing sequence of complexes T i  C  D F 0 C       F p C   F pC1 C       F C D f0g: i2N

Here we have assumed that the filtration is compatible with differentials and separated (\p F p C n D f0g for any n 2 Z). Let us construct a spectral sequence associated to a filtered complex .F  C  ; d /. We first define F p C pCq E0p;q ´ Gr p .C pCq / D pC1 pCq F C p;q p;qC1 and d0 D d W E0 ! E0 . It is well defined as d is compatible with the filtration. We then define E1p;q ´ H pCq .Gr p .C pCq // D

fa 2 F p C pCq j d.a/ 2 F pC1 C pCqC1 g d.F p C pCq1 / C F pC1 C pCq

and d1 D d W E1p;q ! E1pC1;q . More generally, we define Erp;q ´

fa 2 F p C pCq j d.a/ 2 F pCr C pCqC1 g d.F prC1 C pCq1 / C F pC1 C pCq

15

2.2 Spectral sequences

and dr D d W Erp;q ! ErpCr;qrC1 . Here the denominator is implicitly understood as fdenominator as writteng \ fnumeratorg. Exercise 2.2. Prove that H.Er ; dr / D ErC1 . We now observe that the cohomology of a (not necessarily separated) filtered complex .F  C  ; d/ inherits a natural filtration, namely F p H q .C  / D .kerfd W C q ! C qC1 g \ F p C q /; where  is the natural surjective projection onto the cohomology. In more downto-earth terms, the filtration on the complex defines automatically a filtration on the subspace of cocycles; the surjective projection from cocycles onto cohomology induces naturally a filtration. We now have the following Proposition 2.3. If the spectral sequence .Er /r associated to a filtered complex .F  C  ; d / converges, then p;q D Gr p H pCq .C  /: E1

Proof. Let .p; q/ 2 Z2 . For r  max.r.p; q/; p C 1/, Erp;q D

fa 2 F p C pCq j d.a/ D 0g F p H pCq .C  / D pC1 pCq  D Gr p H pCq .C  /: pCq1 pC1 pCq d.C /CF C F H .C / 

This proves the proposition.

Example 2.4 (spectral sequences of a double complex). Assume that we are given a double complex .C ; ; d; d 0 /, i.e., a Z2 -graded vector space together with degree .1; 0/ and .0; 1/ linear maps d 0 and d 00 such that d 0 B d 0 D 0, d 00 B d 00 D 0 and  ; dtot / is defined as d 0 B d 00 C d 00 B d 0 D 0. Then the total complex .Ctot M n Ctot ´ C p;q ; dtot ´ d 0 C d 00 : pCqDn

There are two filtrations, and thus two spectral sequences, naturally associated to  ; dtot /: .Ctot M M k n k n ´ C p;q and F 00 Ctot ´ C p;q : F 0 Ctot pCqDn qk

pCqDn pk

Therefore the first terms of the corresponding spectral sequences are: E 0 1 D H q .C ;p ; d 0 / p;q E 00 1 D H q .C p; ; d 00 / p;q

with d1 D d 00 ; with d1 D d 0 :

In the case the d 0 -cohomology is concentrated in only one degree q, the spectral sequence stabilizes at E2 and the total cohomology is given by    D H q H q .C; d 0 /; d 00 : Htot

16

2 Hochschild cohomology and spectral sequences

Spectral sequences of algebras. A spectral sequence of algebras is a spectral sequence such that each Er is equipped with a bigraded associative product that turns .Er ; dr / into a DG algebra. Of course, we require that H.Er ; dr / D ErC1 as algebras. As in the previous section a filtered DG algebra .F  A ; d / gives rise to a spectral sequence of algebras .Er /r such that • E0p;q ´ Gr p .ApCq /, • E1p;q ´ H pCq .Gr p .ApCq //, p;q D Gr p H pCq .A /. • if it converges, then E1

2.3 Application: Chevalley–Eilenberg versus Hochschild cohomology Let M be a U.g/-bimodule. Then M is equipped with a g-module structure via x  m D xm  mx

for all x 2 g and for all m 2 M;

We want to prove the following. Theorem 2.5. (1) There is an isomorphism H  .g; M / Š H  .U.g/; M /. (2) If M D A is equipped with a U.g/-invariant associative product, then the previous isomorphism becomes an isomorphism of (graded ) algebras. We define a filtration on the Hochschild complex C  .U.g/; M /: F p C n .U.g/; M / is given by linear maps U.g/˝n ! M that vanish on M U.g/i1 ˝    ˝ U.g/in : i1 CCin 0 one defines the n-th scalar Atiyah class an .E/ as V an .E/ ´ tr.atnE / 2 H@Nn .M; n .T 0 / /: but we regard Observe that tr..R1;1 /n / lies in 0;n .M; ˝n .T 0 / /,N V V it as an element in n 0;n .M; .T 0 / / thanks to the natural projection .T 0 / ! .T 0 / . The Todd class of E is then tdE ´ det





atE : 1  e atE

One sees without difficulties that it can be expanded formally in terms of an .E/. Remark 3.3. We want to observe that there is an alternative definition of the Atiyah class: as has been noted before, a holomorphic vector bundle E over a complex manifold X always admits a connection compatible with the complex structure. On the other hand, we may say that a connection r on E is holomorphic, if it maps holomorphic sections to holomorphic sections: again, it is clear that, locally, holomorphic connections exist. Still, one cannot in general glue local holomorphic connections to a global one: the Atiyah class of E may be also viewed as the obstruction against the existence of a global holomorphic connection on E. For more details on the Atiyah class, we refer to [26]. We notice briefly that in the case of a Kähler manifold X and a holomorphic vector bundle E over X , the n-th Chern class of E coincides with the n-th scalar Atiyah class of E. We refer again to [26] for more details on this issue.

3.3 Hochschild cohomology of a complex manifold Hochschild cohomology of a differentiable manifold. Let M be a differentiable   manifold. We introduce the differential graded algebras Tpoly M and Dpoly M of polyvector field and poly-differential V operators on M .   First of all Tpoly M ´ .M; TM/ with product ^ and differential d D 0. The algebra of differential operators is the subalgebra of End.C 1 .M // gener ated by functions and vector fields. Then we define the DG algebra Dpoly M as the  1 1 DG subalgebra of .C .C .M /; C .M //; [; dH / whose elements are cochains being differential operators in each argument (i.e., if we fix all the other arguments then it is a differential operator in the remaining one). The following result, due to J. Vey [42] (see also [29]), computes the cohomology  of Dpoly M . It is an analogue for smooth functions of the original Hochschild–Kostant– Rosenberg theorem [25] for regular affine algebras.

3.3 Hochschild cohomology of a complex manifold

23

Theorem 3.4. The degree 0 graded map   IHKR W .Tpoly M; 0/ ! .Dpoly M; dH /; P 1 "./ v1 ^    ^ vn ! 7 .f1 ˝    ˝ fn 7! nŠ v.1/ .f1 / : : : v.n/ .fn //;  2Sn .1/

is a quasi-isomorphism of complexes that induces an isomorphism of (graded ) algebras at the level of cohomology. Here we have used the signature ", which is a group morphism Sn ! f˙1g that is defined by "..i; i C 1// D 1 on transpositions. Proof. First of all it is easy to check that it is a morphism of complexes (i.e., images of IHKR are cocycles). Then one can see that everything is C 1 .M /-linear: the products ^ and [, the  is nothing but the differential dH and the map IHKR . Moreover, one can see that Dpoly 1 Hochschild complex of the algebra JM of 1-jets of functions on M with values in C 1 .M /.1 1 As an algebra JM can be identified (non-canonically) with global sections of the y bundle of algebras S.T  M /, and  with the projection on degree 0 elements. Therefore the statement follows immediately if one applies Lemma 2.6 fiberwise to V D Tm M (m 2 M ).  Hochschild cohomology of a complex manifold. Let us now return to the case of a complex manifold M . V 0 0 First, for any vector bundle E over M we define Tpoly .M; E/ ´ .M; E ˝ T /. Then we define @-differential operators as endomorphisms of C 1 .M / generated by functions and type .1; 0/ vector fields, and for any vector bundle E we define E-valued @-differential operators as linear maps C 1 .M / ! .M; E/ obtained by composing @-differential operators with a section of E or T 0 ˝ E (sections of T 0 ˝ E are E-valued type .1; 0/ vector fields). 0 The complex Dpoly .M; E/ of E-valued @-poly-differential operators is defined as the subcomplex of .C  .C 1 .M /; .M; E//; dH / consisting of cochains that are @differential operators in each argument. We have the following obvious analogue of Theorem 3.4: Theorem 3.5. The degree 0 graded map 0 0 IHKR W .Tpoly .M; E/; 0/ ! .Dpoly .M; E/; dH /; P 1  .v1 ^    ^ vn / ˝ s 7! .f1 ˝    ˝ fn 7! nŠ 2Sn .1/ v.1/ .f1 / : : : v.n/ .fn /s/; 1

1 1 Recall that JM ´ HomC 1 .M / .Dpoly M; C 1 .M // with product given by

j1  j2 .P / ´ .j1 ˝ j2 /..P //

1 1 .j1 ; j2 2 JM ; P 2 Dpoly M /;

2 where .P / 2 Dpoly M is defined by .P /.f; g/ ´ P .fg/. The module structure on C 1 .M / is 1 1 ! C 1 .M / obtained as the transpose of C 1 .M / ,! Dpoly M. given by the projection W JM

24

3 Dolbeault cohomology and the Kontsevich isomorphism

is a quasi-isomorphism of complexes. V 0 Now observe that T is a holomorphic bundle of graded algebras with product being ^. Namely, T 0 has an obvious holomorphic structure: for any v 2 .M; T 0 / and any f 2 C 1 .M /, N N N //; [email protected]//.f / ´ @.v.f //  [email protected] V 0 and it extends uniquely to a holomorphic structure on T , which is a derivation with 0 respect to the product ^: for any v; w 2 .M; Tpoly /, N N ^ w/ D @.v/ N @.v ^ w C .1/jvj v ^ @.w/: V 0 V 0 Therefore @N turns 0; .M; T / D Tpoly .M; .T 00 / / into a DG algebra. One also has an action of @N on @-differential operators defined in the same way: for any f 2 C 1 .M /, N //.f / D @.P N .f //  P [email protected] N //: [email protected] It canVbe extended uniquely to a degree 1 derivation of the graded algebra  0 Dpoly .M; .T 00 / /, with product given by .P [ Q/.f1 ; : : : ; fmCn / D .1/mjQj P .f1 ; : : : ; fm / ^ Q.fmC1 ; : : : ; fmCn /; where j  j refers to the exterior degree.

3.4 The Kontsevich isomorphism Theorem 3.6. The map IHKR B td1=2 T 0  induces an isomorphism of (graded ) algebras V

H@N .

V 0  N T 0 /  ! H.. T 00 / ˝ Dpoly ; dH C @/

at the level of cohomology. This result has been stated by M. Kontsevich in [29] (see also [10]) and proved in a more general context in [9]. V 0 Remark 3.7. Since a1 .T 0 / is V a derivation of H@N . T 0 /, it follows that e a1 .T / is an 0 algebra automorphism of H@N . T /. Therefore, as for the usual Duflo isomorphism (see Remark 1.4), one can replace the Todd class of T 0 by the modified Todd class z T 0 ´ det td





atT 0 : =2 at 0 e T  e atT 0 =2

4 Superspaces and Hochschild cohomology

In this chapter we provide a short introduction to supermathematics and deduce from it a definition of the Hochschild cohomology for DG associative algebras. V Moreover we prove that the Hochschild cohomology of the Chevalley algebra . .g/ ; dC / of a finite-dimensional Lie algebra g is isomorphic to the Hochschild cohomology of its universal enveloping algebra U.g/.

4.1 Supermathematics Definition 4.1. A super vector space (simply, a superspace) is a Z=2Z-graded vector space V D V0 ˚ V1 . In addition to the usual well-known operations on G-graded vector spaces (direct sum ˚, tensor product ˝, spaces of linear maps Hom.; /, and duality ./ ) one has a parity reversion operation …: .…V /0 D V1 and .…V /1 D V0 . In the sequel V is always a finite-dimensional super vector space. Supertrace and Berezinian. For any endomorphism X of V (also referred as a su x10 permatrix on V ) one can define its supertrace str as follows: if we write X D xx00 , 01 x11 meaning that X D x00 C x10 C x01 C x11 with xij 2 Hom.Vi ; Vj /, then str.X / ´ tr.x00 /  tr.x11 /: On invertible endomorphisms we also have the Berezinian Ber (or superdeterminant) which is uniquely determined by the two defining properties Ber.AB/ D Ber.A/ Ber.B/ and

Ber.e X / D e str.X/ :

A very nice, short and complete introduction to the Berezinian can be found in [15], to which we refer for more details. Symmetric and exterior algebras of a super vector space. The (graded) symmetric algebra S.V / of V is the quotient of the tensor algebra T .V / of V by its two-sided ideal generated by v ˝ w  .1/jvjjwj w ˝ v; where v and w are homogeneous elements of degree jvj and jwj in V , respectively. It has two different (Z-)gradings:

26

4 Superspaces and Hochschild cohomology

• The first one (by the symmetric degree) is obtained by assigning degree 1 to elements of V . Its degree n homogeneous piece, denoted by S n .V /, is the quotient of the space V ˝n by the action of the symmetric group Sn by superpermutations: .i; i+1/  .v1 ˝    ˝ vn / ´ .1/jvi jjvi C1 j v1 ˝ : : : vi ˝ viC1    ˝ vn : • The second one (the internal grading) is obtained by assigning degree i 2 f0; 1g to elements of Vi . Its degree n homogeneous piece is denoted by S.V /n , and we write jxj for the internal degree of a homogeneous element x 2 S.V /. Example 4.2. (a) If V D V0 is purely even, then S.V / D S.V0 / is the usual symmetric algebra of V0 , S n .V / D S n .V0 / and S.V / is concentrated in degree 0 for the internal grading. V (b) If V D V1 is V purely odd, then S.V / D .V1 / is the exterior algebra of V1 . Moreover, S n .V / D n .V1 / D S.V /n . V The (graded) exterior algebra .V / of V is the quotient of the tensor algebra T .V / of V by its two-sided ideal generated by v ˝ w C .1/jvjjwj w ˝ v: It has two different (Z-)gradings: • The first one (by the exterior degree) is obtained by assigning V degree 1 to elements of V . Its degree n homogeneous piece is, denoted by n .V /, is the quotient of the space of V ˝n by the action of the symmetric group Sn by signed superpermutations: .i; i C 1/  .v1 ˝    ˝ vn / ´ .1/jvi jjvi C1 j v1 ˝ : : : vi ˝ viC1    ˝ vn : • The second one (the internal grading) is obtained by assigning degree V i 2 nf0; 1g .V / , and to elements of V1i . Its degree n homogeneous piece is denoted by V we write jxj for the internal degree of a homogeneous element x 2 .V /. In other words, n X jv1 ^    ^ vn j D n  jvi j: iD1

V V Example 4.3. (a) V If V D V0 V is purely even, then .V / D .V0 / is the usual exterior V algebra of V0 and n .V / D n .V0 / D .V /n . V .V / D S.V1 / is the symmetric algebra of V1 . (b) If V VD V1 is purely odd, then V n .V / D S n .V1 / and .V / is concentrated in degree 0 for the internal Moreover, grading.

27

4.1 Supermathematics

Observe that one has an isomorphism of bigraded vector spaces V  S.…V /  ! .V /; Pn

v1 : : : vn 7! .1/

j D1 .j 1/jvj j

v1 ^    ^ vn :

(4.1)

Note that it remains true without the sign on the right. The motivation for this quite mysterious sign modification we make here is explained in the next paragraph. Graded-commutative algebras Definition 4.4. A graded algebra A is graded-commutative if for any homogeneous elements a; b one has a  b D .1/jajjbj b  a. Example 4.5. (a) The symmetric algebra S.V / of a super vector space is gradedcommutative with respect to its internal grading. (b) The graded algebra  .M / of differentiable forms on a smooth manifold M is graded-commutative. The exterior algebra of a super vector space, with product ^ and the internal grading, is not a graded-commutative algebra in general: for vi 2 Vi (i D 0; 1) one has v0 ^ v1 D v1 ^ v0 : One way to correct this drawback is to define a new product  on V V v 2 k .V / and w 2 l .V /, then

V

.V / as follows: let

v  w ´ .1/k.jwjCl/ v ^ w: In this situation one can check (this is an exercise) that the map (4.1) defines a graded algebra isomorphism V .S.…V /;  / ! . .V /; /: Graded Lie algebras Definition 4.6. A graded Lie algebra is a Z-graded vector space g equipped with a degree 0 graded linear map Œ ;  W g ˝ g ! g that is graded-skew-symmetric, i.e., Œx; y D .1/jxjjyj Œy; x; and satisfies the graded Jacobi identity Œx; Œy; z D ŒŒx; y; z C .1/jxjjyj Œy; Œx; z: Examples 4.7. (a) Let A be a graded associative algebra. Then A equipped with the super-commutator Œa; b D ab  .1/jajjbj ba

28

4 Superspaces and Hochschild cohomology

is a graded Lie algebra. (b) Let A be a graded associative algebra and consider the space Der.A/ of super derivations of A: a degree k graded linear map d W A ! A is a super derivation if d.ab/ D d.a/b C .1/kjaj ad.b/: Der.A/ is stable under the super-commutator inside the graded associative algebra End.A/ of (degree non-preserving) linear maps A ! A (with product the composition). The previous example motivates the following definition: Definition 4.8. Let g be a graded Lie algebra. 1. A graded g-module is a graded vector space V with a degree 0 graded linear map g ˝ V ! V such that x  .y  v/  .1/jxjjyj y  .x  v/ D Œx; y  v: In other words it is a morphism g ! End.V / of graded Lie algebras. 2. If V D A is a graded associative algebra, then one says that g acts on A by derivations if this morphism takes values in Der.A/. In this case A is called a g-module algebra.

4.2 Hochschild cohomology strikes back Hochschild cohomology of a graded algebra. Let A be a graded associative algebra. Its (shifted) Hochschild (cochain) complex C  .A; A/ is defined as the sum of spaces of (not necessarily graded) linear maps A˝.1/ ! A. Let us denote by j  j the degree of those linear maps; the grading on C  .A; A/ is given by the total degree, denoted by k  k. For any f W A˝m ! A, kf k D jf j C m  1. The differential dH is given by .dH .f //.a1 ; : : : ; amC1 / D .1/kf k.ja1 j1/ a1 f .a2 ; : : : ; amC1 / m Pi 1 P C .1/i1C j D1 jaj j f .a1 ; : : : ; ai aiC1 ; : : : ; amC1 / iD1

C f .a1 ; : : : ; am /amC1 :

(4.2)

Again, it is easy to prove that dH B dH D 0. As in Section 2.1 .C  .A; A/; dH / is a DG algebra with product [ defined by .f [ g/.a1 ; : : : ; amCn / ´ .1/jgj.ja1 jCCjam j/ f .a1 ; : : : ; am /g.amC1 ; : : : ; amCn /: Hochschild cohomology of a DG algebra. Let A be a graded associative algebra. We now prove that C  .A; A/ is naturally a Der.A/-module.

4.2 Hochschild cohomology strikes back

29

For any d 2 Der.A/ and any f 2 C  .A; A/ one defines d.f /.a1 ; : : : ; am / D d.f .a1 ; : : : ; am // m Pi 1 P  .1/kd kkf k .1/kd k.i1C j D1 jaj j/ f .a1 ; : : : ; dai ; : : : ; am /: iD1

In other words, d is defined as the unique degree jd j derivation for the cup product that is given by the super-commutator on linear maps A ! A. Moreover, one can easily check that d B dH C dH B d D 0. Therefore if .A ; d / is a DG algebra then its Hochschild complex is C  .A; A/ together with dH C d as a differential. It is again a DG algebra, and we denote its cohomology by HH .A; d /. Remark 4.9 (Deformation theoretic interpretation). In the spirit of the discussion in Section 2.1, one can prove that HH2 .A; d / is the set of equivalence classes of infinitesimal deformations of A as an A1 -algebra (an algebraic structure introduced by J. Stasheff in [40]) and that the obstruction to extending such deformations order by order lies in HH3 .A; d /. More generally, if .M; dM / is a DG-bimodule over .A; dA /, then the (shifted) Hochschild complex C  .A; M / of A with values in M consists of linear maps A˝n ! M (n  0) and the differential is dH C d , with dH given by (4.2) and d.f /.a1 ; : : : ; am / D dM .f .a1 ; : : : ; am // m Pi 1 P  .1/kd kkf k .1/kd k.i1C j D1 jaj j/ f .a1 ; : : : ; dA ai ; : : : ; am /: iD1

Hochschild cohomology of the Chevalley–Eilenberg algebra. One has the following important result: Theorem 4.10. Let g be aV finite-dimensional Lie algebra. Then there is an isomorphism  of graded algebras HH . g ; dC /  ! HH .U.g//. Let us emphazise that this result is related to some general considerations about Koszul duality for quadratic algebras (see e.g., [36]). Observe that, although we are interested here mainly in finite-dimensional Lie algebras concentrated in degree 0, the above theorem holds true for any finite-dimensional graded Lie algebra g with due changes. V  Proof. Due to Theorem 2.5 it suffices to prove that HH . g ; dC /  ! H  .g; U.g//. Let us define a linear map V V V V V (4.3) C. g ; g / D g ˝ T . g/ ! g ˝ U.g/ D C.g; U.g//; V given by the projection p W T . g/  T .g/  U.g/.

30

4 Superspaces and Hochschild cohomology

The previous map defines a morphism of DG algebras V V .C. g ; g /; dH C dC / ! .C.g; U.g//; dC /: It can be checked directly that, using the previous identification for Hochschild chains and the restriction morphism to T .g /, the only Hochschild cochains f whose differential with respect to dH C morphism must VdC is not annihilated by the restriction V contain factors only in g or 2 .g/ (while the only factor .g / is left untouched). The quadratic factors, which without loss of generality may be written as x1 ^ x2 , are sent by the sum of the Hochschild and Chevalley–Eilenberg differentials to .x1 ^ x2 / ˝ 1 C 1 ˝ .x1 ^ x2 / C x1 ˝ x2  x2 ˝ x1  Œx1 ; x2 I such terms are annihilated precisely by the projection p. It is left as an exercise to check that the remaining terms contribute exactly to the Chevalley–Eilenberg differential. It remains to prove that it is a quasi-isomorphism, for which we use a spectral sequence argument. V Lemma 4.11. We equip k (endowed with the trivialV differential) with the . g ; dC /DG-bimodule structure by the g ! k (left and right actions  V given  projection  W   coincide). Then H . g ; dC /; k Š U.g/. V

Proof. We consider the filtration F p C n .. M

Vi1

g ; dC /; k/, consisting of linear forms on

.g / ˝    ˝

V ik

.g /

k0 i1 CCik Dkn

that vanish on the components for which n  k < p. Then we have L

E0p;q D Hom.

i1 CCiq Dp

V i1

.g / ˝    ˝

V iq

.g /; k/;

d0 D dH :

Applying a “super” version of Lemma 2.6 to V D ….g / one obtains that E1p;q D E1q;q D

Vq

.….g / / D S q .g/;

and it follows that the spectral sequence stabilizes at E1 . Consequently we obtain V Gr.H  .. g ; dC /; k// Š S.g/ D Gr.U.g//, and the isomorphism is given by the composed map V T . .g// ! T .g/ ! S.g/: 

This ends the proof of the lemma. V

Lemma 4.12. The map (4.3) is a quasi-isomorphism: HH .

g ; dC / Š H  .g; U.g//.

4.2 Hochschild cohomology strikes back

31

Proof. Let us consider the descending filtration onVthe Hochschild complex that is induced from the following descending filtration on g : M Vk V F n . g / ´ g : kn

Thus, the 0-th term of the associated spectral sequence (of algebras) is V  V   E0; D g ˝ C  .. g ; dC /; k/ with d0 D id ˝ .dH C dC /: V     Using Lemma 4.11, one obtains that E1; D E1;0 D g ˝ U.g/ with d1 D dC . Therefore the spectral sequence stabilizes at E2 and the result follows.  This ends the proof of Theorem 4.10.



5 The Duflo–Kontsevich isomorphism for Q-spaces

In this chapter we prove a general Duflo-type result for Q-spaces, i.e., superspaces equipped with a vector field of degree 1, which squares to 0. This result implies in particular the cohomological version of Duflo’s Theorem 1.13, and will be used in the sequel to prove Kontsevich’s Theorem 3.6. This approach makes more transparent the analogy between the adjoint action and the Atiyah class.

5.1 Statement of the result Let V be a superspace. Hochschild–Kostant–Rosenberg for superspaces. We introduce • OV ´ S.V  /, the graded super-commutative algebra of functions on V ; • XV ´ Der.OV / D S.V  / ˝ V , the graded Lie super-algebra of vector fields on V ; V • Tpoly V ´ S.V  ˚ …V / Š OV XV , the XV -module algebra of poly-vector fields on V . We now describe the gradings we will consider. The grading on OV is the internal one: elements in Vi have degree i . The grading on XV is the restriction of the natural grading on End.OV /: elements in Vi have degree i and elements in Vi have degree i . There are three different gradings on Tpoly V : V (i) the one given by the number of arguments: degree k elements lie in kOV XV . In other words elements in V  have degree 0 and elements in V have degree 1; (ii) the one induced by XV : elements in Vi have degree i and elements in Vi have degree i . It is denoted by j  j; (iii) the total (or internal) degree: it is the sum of the previous ones. Elements in Vi have degree i and elements in Vi have degree 1  i . It is denoted by k  k. Unless otherwise made precise, we always consider the total grading on Tpoly V in the sequel. We also have

34

5 The Duflo–Kontsevich isomorphism for Q-spaces

• the XV -module algebra DV of differential operators on V , which is the subalgebra of End.OV / generated by OV and XV ; • the XV -module algebra Dpoly V of poly-differential operators on V , which consists of multilinear maps OV ˝    ˝ OV ! OV being differential operators in each argument. The grading on DV is the restriction of the natural grading on End.OV /. As for Tpoly there are three different gradings on Dpoly : the one given by the number of arguments, the one induced by DV (denoted by j  j), and the one given by their sum (denoted by k  k). Dpoly is then a subcomplex of the Hochschild complex of the algebra OV introduced in the previous chapter since it is obviously preserved by the differential dH . An appropriate super-version of Lemma 2.6 gives the following result: Proposition 5.1. The natural inclusion IHKR W .Tpoly V; 0/ ,! .Dpoly V; dH / is a quasiisomorphism of complexes that induces an isomorphism of algebras in cohomology. Cohomological vector fields Definition 5.2. A cohomological vector field on V is a degree 1 vector field Q 2 XV that is integrable: ŒQ; Q D 2QBQ D 0. A superspace equipped with a cohomological vector field is called a Q-space. Let Q be a cohomological vector field on V : then, .Tpoly V; Q/ and .Dpoly V; dH C Q/ are DG algebras, where Q denotes its natural adjoint action on both Tpoly V and Dpoly V by graded commutators. By a spectral sequence argument one can show that IHKR still defines a quasi-isomorphism of complexes between them. Nevertheless it no longer preserves the product at the level of cohomology. In a way similar to Theorems 1.13 and 3.6, Theorem 5.3 below remedies this situation. Let us remind the reader that the graded algebra of differential forms on V is .V / ´ S.V  ˚ …V  / and that it is equipped with the following structures: • for any element x 2 V  we write dx for the corresponding element in …V  , and then we define a differential on .V /, the de Rham differential, given on generators by d.x/ D dx and d.dx/ D 0; • there is an action  of differential forms on poly-vector fields by contraction, where x 2 V  acts by left multiplication and dx acts by derivation in the following way: for any y 2 V  and v 2 …V one has dx .y/ D 0

and dx .v/ D hx; vi:

We then define the (super)matrix valued 1-form „ 2 1 .V / ˝ End.V Œ1/ with coefficients explicitly given by  j @Q @2 Qj j „i ´ d dx k ; D @x i @x k @x i

5.2 Application: proof of the Duflo Theorem

35

where fx i g, i D 1; : : : ; n, are coordinates on V associated to a linear basis of V . A direct computation shows that a change of basis of V produces a conjugation of the matrix-valued 1-form „ji by a constant matrix (naturally associated to the base change): thus, if we set   1  e „ j.„/ ´ Ber 2 .V /; „ then j.„/ does not depend on the choice of linear coordinates on V . Theorem 5.3. IHKR B j.„/1=2 W .Tpoly V; Q/ ! .Dpoly V; dH C Q/ defines a quasiisomorphism of complexes that induces an algebra isomorphism on cohomology. As for Theorems 1.3, 1.13 and 3.6 one can replace j.„/ by 



e „=2  e „=2 : jz.„/ ´ Ber „

5.2 Application: proof of the Duflo Theorem In this section we discuss an important application of Theorem 5.3, namely the “classical” Theorem of Duflo (see Theorems 1.3 and 1.13): before entering into the details of the proof, we need to establish a correspondence between the algebraic tools of Duflo’s Theorem and the differential-geometric objects of 5.3. We consider a finite-dimensional Lie algebra g, to which we associate the superspace V D …g. In this setting, we have the identification V  OV Š g ; i.e., the super-algebra of polynomial functions on V is identified with the graded vector space defining the Chevalley–Eilenberg graded algebra for g with values in the trivial g-module; we observe that the natural grading of the Chevalley–Eilenberg complex of g corresponds to the aforementioned grading of OV . The Chevalley–Eilenberg differential dC identifies, under the above isomorphism, with a vector field Q of degree 1 on V ; Q is cohomological since dC squares to 0. In order to make things more understandable, we make some explicit computations with respect to supercoordinates on V . For this purpose, a basis fei g of g determines a system of (purely odd) coordinates fx i g on V : the previous identification can be expressed in terms of these coordinates as x i1 : : : x ip 7! "i1 ^    ^ "ip ;

1 i1 <    < ip n;

f"i g being the dual basis of fei g. Hence, with respect to these odd coordinates, Q can be written as @ 1 Q D  cji k x j x k i ; 2 @x

36

5 The Duflo–Kontsevich isomorphism for Q-spaces

where cji k are the structure constants of g with respect to the basis fei g. It is clear that Q has degree 1 and total degree 2. Lemma 5.4. The DG algebra .Tpoly V; Q/ identifies naturally with the Chevalley– Eilenberg DG algebra .C  .g; S.g//; dC / associated to the g-module algebra S.g/. Proof. By the very definition of V , we have an isomorphism of graded algebras V  .g / ˝ S.g/: S.V  ˚ …V / Š More explicitly, in terms of the aforementioned supercoordinates, the previous isomorphism is given by x i1 : : : x ip @x j1 ^    ^ @x jq 7! "i1 ^    ^ "ip ˝ ej1 : : : ejq ; where the indices .i1 ; : : : ; ip / form a strictly increasing sequence. It remains to prove that the action of Q on Tpoly V coincides,Vunder the previous  isomorphism, with the Chevalley–Eilenberg differential dC on .g / ˝ S.g/. It suffices to prove the claim on generators,i.e., on the coordinate functions fx i g and on the derivations f@x i g: the action of Q on both of them is given by 1 Q  x i D Q.x i / D  cji k x j x k ; 2 Q  @x i D ŒQ; @x i  D cijk x j @x k : Under the above identification between Tpoly V and identifies with dC , thus the claim follows.

V

.g / ˝ S.g/, it is clear that Q 

Similar arguments and computations imply the following: Lemma 5.5. There is a V naturalVisomorphism from the DG algebra .Dpoly V; dH C Q/ to the DG algebra .C  . g ; g /; dH C dC /. Coupling these results with Lemma 4.12, we obtain the following commutative diagram of quasi-isomorphisms of complexes, all inducing algebra isomorphisms at the level of cohomology: .Tpoly V; Q/

IHKR B j.„/1=2

.C  .g; S.g//; dC /

/ .Dpoly V; dH C Q/ IPBW BJ 1=2 

V

.C  .

g ;

V

g /; dH C dC /

 / .C  .g; U g/; dC /.

Using the previously computed explicit expression for the cohomological vector field Q on V , one can easily prove the following:

5.3 Strategy of the proof

37

Lemma 5.6. Under the obvious identification V Œ1 Š g, the supermatrix-valued 1-form „, restricted to g (which we implicitly identify with the space of vector fields on V with constant coefficients) satisfies „ D ad: Proof. Namely, since 1 Q D  cji k x j x k @x i ; 2 we have i „ji D d.@x j Qi / D cji k dx k D ckj dx k ;

and the claim follows by a direct computation, when e.g. evaluating „ on ek D @x k .  Hence, Theorem 5.3, together with Lemmas 5.4, 5.5 and 5.6, imply Theorem 1.13; we observe that „ is in this case an even endomorphism of V , whence its Berezinian reduces to the standard determinant.

5.3 Strategy of the proof The proof of Theorem 5.3 occupies the next three chapters. In this section we explain the strategy we are going to adopt in Chapters 6, 7, 8 and 9. The homotopy argument. Our approach relies on a homotopy argument (in the context of deformation quantization, this argument is briefly sketched by Kontsevich in [29] and proved in detail by Manchon and Torossian in [34] in a particular case). Namely, we construct a quasi-isomorphism of complexes1 UQ W .Tpoly V; Q/ ! .Dpoly V; dH C Q/ and a degree 1 map HQ W Tpoly V ˝ Tpoly V ! Dpoly V satisfying the homotopy equation UQ .˛/ [ UQ .ˇ/  UQ .˛ ^ ˇ/ D .dH C Q/.HQ .˛; ˇ// C HQ .Q  ˛; ˇ/ C .1/k˛k HQ .˛; Q  ˇ/ for any poly-vector fields ˛; ˇ 2 Tpoly V . We sketch below the construction of UQ and HQ . 1

It is the first structure map of Kontsevich’s tangent L1 -quasi-isomorphism [29].

(5.1)

38

5 The Duflo–Kontsevich isomorphism for Q-spaces

Formulae for UQ and HQ , and the scheme of the proof. For any poly-vector fields ˛; ˇ 2 Tpoly V and functions f1 ; : : : ; fm we set UQ .˛/.f1 ; : : : ; fm / ´

X 1 nŠ n0

X

W B .˛; Q; : : : ; Q/.f1 ; : : : ; fm / „ ƒ‚ …

2GnC1;m

(5.2)

n times

and HQ .˛; ˇ/.f1 ; : : : ; fm / ´

X 1 nŠ n0

X

z B .˛; ˇ; Q; : : : ; Q/.f1 ; : : : ; fm /: W „ ƒ‚ …

2GnC2;m

n times

(5.3) The sets Gn;m consist of suitable directed graphs with two types of vertices, to which z and poly-differential operators B . we associate scalar (integral) weights W and W We define in the next paragraph the sets Gn;m and the associated poly-differential z are introduced in Chapter 6 and 8, respectively. operators B . The weights W and W In Chapter 7 (resp. 8) we prove that U.˛ ^ ˇ/ and U.˛/ [ U.ˇ/ (resp. the right-hand side of (5.1)) are given by a formula similar to (5.3) with new weights W0 and W1 (resp. W2 ) so that, in fine , the homotopy property (5.1) reduces to W0 D W1 C W2 : Polydifferential operators associated to a graph. Let us consider, for given positive integers n and m, the set Gn;m of directed graphs described as follows: 1. there are n vertices of the “first type”, labeled by 1; : : : ; n; N : : : ; m; 2. there are m ordered vertices of the “second type”, labeled by 1; x 3. the vertices of the second type have no outgoing edge; 4. there are no short loops (a short loop is a directed edge having the same source and target) and no multiple edges (a multiple edge is a set of edges of cardinality strictly bigger than 1 with common source and target). Let us define  D idV0  idV1 2 V  ˝ V , and let it act as a derivation on S.…V / ˝ S.V  / simply by contraction. In other words, using coordinates .x i /i on V and dual odd coordinates .i /i on …V  one has X i D .1/jx j @i ˝ @x i : i

This action naturally extends to S.V  ˚ …V / ˝ S.V  ˚ …V / (the action on additional variables is zero). For any finite set I and any pair .i; j / of distinct elements in I we denote by ij the endomorphism of S.V  ˚ …V /˝I given by  which acts by the identity on the k-th factor for any k ¤ i; j .

39

5.3 Strategy of the proof

Let us then choose a graph  2 Gn;m , poly-vector fields 1 ; : : : ; n 2 Tpoly V D S.V  ˚ …V /, and functions f1 ; : : : ; fm 2 OV  S.V  ˚ …V /. We define  Q  B .1 ; : : : ; n /.f1 ; : : : ; fm / ´  .i;j /2E./ ij .1 ˝  ˝n ˝f1 ˝  ˝fm / ; (5.4) where E./ denotes the set of edges of the graph , W S.V  ˚ …V /˝.nCm/ ! S.V  ˚ …V / is the product, and  W S.V  ˚ …V /  S.V  / D OV is the projection onto 0-poly-vector fields, defined by setting i to 0. Remark 5.7. (a) If the number of outgoing edges of a first type vertex i differs from ji j then the right-hand side of (5.4) is obviously zero. (b) We could have allowed edges outgoing from a second type vertex, but in this case the right-hand side of (5.4) is obviously zero. (c) There is an ambiguity in the order of the product of endomorphisms ij . Since each ij has degree 1, there is a sign ambiguity in the right-hand side of (5.4). Forz , tunately the same ambiguity appears in the definition of the weights W and W ensuring that expressions (5.2) and (5.3) for UQ and HQ are well defined. Example 5.8. Consider three poly-vector fields 1 D 1ij k i j k , 2 D 2lp l p and 3 D 3qr q r , and functions f1 ; f2 2 OV . If  2 G3;2 is given as in Figure 1, then B .1 ; 2 ; 3 /.f1 ; f2 / D ˙ 1ij k .@i @q 2lp /.@j 3qr /.@l f1 /.@r @p @k f2 /:

1 11 00 00 11

00 211 00 11

1 0 03 1

1 0 0 1

f1

11 00 00 11

f2

Figure 1. An admissible graph  in G3;2 .

6 Configuration spaces and integral weights

The main goal of this chapter is to define the weights W appearing in the defining formula (5.2) for UQ . These weights are defined as integrals over suitable configuration spaces of points in the upper half-plane. We therefore introduce these configuration spaces, and also their compactifications à la Fulton–MacPherson, which ensure that the integral weights truly exist. Furthermore, the algebraic identities illustrated in Chapters 7 and 8 follow from factorization properties of these integrals, which in turn rely on Stokes’ Theorem: thus, we need to discuss the boundary stratification of the compactified configuration spaces.

C 6.1 The configuration spaces Cn;m

We denote by H C the complex upper half-plane, i.e., the set of all complex numbers, whose imaginary part is strictly bigger than 0; further, R denotes here the real line in the complex plane. Definition 6.1. For any two positive integers n, m, we denote by Conf C n;m the configuration space of n points in HC and m ordered points in R, i.e., the set of n C m-tuples .z1 ; : : : ; zn ; q1 ; : : : ; qm / 2 .HC /n  Rm satisfying zi ¤ zj if i ¤ j and q1 <    < qm . It is clear that Conf C n;m is a real manifold of dimension 2n C m. We consider further the semidirect product G2 ´ RC Ë R, where RC acts on R by rescalings: it is a Lie group of real dimension 2. The group G2 acts on Conf C n;m by translations and homotheties simultaneously on all components, by the explicit formula ..a; b/; .z1 ; : : : ; zn ; q1 ; : : : ; qm // 7! .az1 C b; : : : ; azn C b; aq1 C b; : : : ; aqm C b/ for any pair .a; b/ in G2 . It is easy to verify that G2 preserves Conf C n;m ; easy compuC tations also show that G2 acts freely on Conf n;m precisely when 2n C m  2. In this C case, we may take the quotient space Conf C n;m =G2 , which will be denoted by Cn;m : in fact, we will refer to it, rather than to Conf C n;m , as to the configuration space of n points in HC and m points in R. It is also a real manifold of dimension 2n C m  2.

42

6 Configuration spaces and integral weights

Remark 6.2. We will not be too much concerned about orientations of configuration C spaces; anyway, it is still useful to point out that Cn;m is an orientable manifold. In C fact, Conf n;m is an orientable manifold, as it possesses the natural volume form  ´ dx1 ^ dy1 ^    ^ dxn ^ dyn ^ dq1 ^    ^ dqm ; using real coordinates z D x C iy for a point in HC . It is then easy to prove that C the natural projection Conf C n;m ! Cn;m defines a locally-trivial principal G2 -bundle, if 2n C m  2 (it is not difficult to construct many different local sections, and we C invite the interested reader to try it): then we define an orientation on Cn;m by declaring any local trivialization to be orientation-preserving. Thus, the orientability of Conf C n;m C implies the orientability of Cn;m : we refer to [3] for explicit (local) computations C in terms of possible local trivializations of the principal of the volume form of Cn;m C C G2 -bundle Conf n;m ! Cn;m . We also need to introduce another kind of configuration space. Definition 6.3. For a positive integer n, we denote by Conf n the configuration space of n points in the complex plane, i.e., the set of all n-tuples of points in C, such that zi ¤ zj if i ¤ j . It is a complex manifold of complex dimension n, or also a real manifold of dimension 2n. We consider the semidirect product G3 D RC Ë C, which is a real Lie group of dimension 3; it acts on Conf n by the rule ..a; b/; .z1 ; : : : ; zn // 7! .az1 C b; : : : ; azn C b/: The action of G3 on Conf n is free, precisely when n  2: in this case, we define the (open) configuration space Cn of n points in the complex plane as the quotient space Conf n =G3 , and it can be proved that Cn is a real manifold of dimension 2n  3. Following the patterns of Remark 6.2, one can show that Cn is an orientable manifold.

C 6.2 Compactification of Cn and Cn;m à la Fulton–MacPherson

In order to clarify forthcoming computations in Chapter 8, we need certain integrals over C the configuration spaces Cn;m and Cn : these integrals are a priori not well defined, and we have to show that they truly exist. Later, we make use of Stokes’ Theorem on these integrals to deduce the relevant algebraic properties of UQ : therefore we will need the boundary contributions to the aforementioned integrals. Kontsevich [29] introduced C C for this purpose nice compactifications Cxn;m of Cn;m which solve, on the one hand, the C problem of the existence of such integrals (their integrands extend smoothly to Cxn;m , and so they can be understood as integrals of smooth forms over compact manifolds); C on the other hand, the boundary stratifications of Cxn;m and Cxn and their combinatorics yield the desired aforementioned algebraic properties.

C 6.2 Compactification of Cn and Cn;m à la Fulton–MacPherson

43

C Definition and examples. The main idea behind the construction of Cxn;m and Cxn is C that one wants to keep track not only of the fact that certain points in H , resp. in R, collapse together, or that certain points of HC and R collapse together to R, but one wants also to record, intuitively, the corresponding rate of convergence. Such compactifications were first thoroughly discussed by Fulton–MacPherson [21] in the algebro-geometric context: Kontsevich [29] adapted the methods of [21] for the conC figuration spaces of the type Cn;m and Cn . We introduce first the compactification Cxn of Cn , which will play an important rôle C . We consider the map also in the discussion of the boundary stratification of Cxn;m from Conf n to the product of n.n  1/ copies of the circle S 1 , and the product of n.n  1/.n  2/ copies of the 2-dimensional real projective space RP 2 ,

n

.z1 ; : : : ; zn / 7!

Y arg.zj  zi /  2

i¤j

Y



 jzi  zj j W jzi  zk j W jzj  zk j :

i ¤j;j ¤k i ¤k

The map n descends in an obvious way to Cn and defines an embedding of the latter into a compact manifold. Hence the following definition makes sense. Definition 6.4. The compactified configuration space Cxn of n points in the complex plane is defined as the closure of the image of Cn with respect to n in .S 1 /n.n1/  .RP 2 /n.n1/.n2/ . C . First of all, there is a natural Next, we consider the open configuration space Cn;m imbedding

C n;m

.z1 ; : : : ; zn ; q1 ; : : : ; qm / 7! .z1 ; : : : ; zn ; zS1 ; : : : ; zSn ; q1 ; : : : ; qm / of Conf C n;m into Conf 2nCm , which is obviously equivariant with respect to the acC 1 tion of G2 . Moreover, C n;m descends to an embedding Cn;m ! C2nCm . We may C C thus compose n;m with 2nCm in order to get a well-defined imbedding of Cn;m into 2 .2nCm/.2nCm1/.2nCm2/ 1 .2nCm/.2nCm1/ .S /  .RP / , which justifies the following definition. C of n points in HC and m Definition 6.5. The compactified configuration space Cxn;m ordered points in R is defined as the closure of the image with respect to the imbedding C 1 .2nCm/.2nCm1/ 2nCm B C  .RP 2 /.2nCm/.2nCm1/.2nCm2/ . n;m of Cn;m into .S /

We notice that there is an obvious action of Sn , the permutation group of n elements, C on Cn , resp. Cn;m , by permuting the points in the complex plane, resp. the n points in C . Thus, we may consider HC : the action of Sn extends to an action on Cxn and Cxn;m C more general configuration spaces CA and CA;B , where now A (resp. B) denotes a To see this, first remember that G3 D G2 Ë R and then  observe that any orbit of R (acting by C simultaneous imaginary translations) intersects C n;m Conf n;m in at most one point. 1

44

6 Configuration spaces and integral weights

finite (resp. finite ordered) subset of N; they also admit compactifications CxA and C CxA;B , which are defined similarly as in Definitions 6.4 and 6.5. C Another important property of the compactified configuration spaces CxA and CxA;B has to do with projections. Namely, for any non-empty subset A1  A (resp. pair A1  A, B1  B such that A1 t B1 ¤ ;) there is a natural projection .A;A1 / (resp. .A;A1 /;.B;B1 / ) from CA onto CA1 (resp. from CA;B onto CA1 ;B1 ) given by forgetting the points labelled by indices which are not in A1 (resp. not in A1 t B1 ). The projection .A;A1 / (resp. .A;A1 /;.B;B1 / ) extends to a well-defined projection between CxA and CxA1 (resp. CxA;B and CxA1 ;B1 ). Moreover, both projections preserve the boundary stratifications of all compactified configuration spaces involved. C inherit Finally, we observe that the compactified configuration spaces Cxn and Cxn;m C both orientation forms from Cn and Cn;m respectively; the boundary stratifications of both spaces, together with their inherited orientation forms, induce in a natural way orientation forms on all boundary strata. We ignore here the orientation choices of the C boundary strata of Cxn and Cxn;m , referring to [3] for all important details. C Examples 6.6. (i) The configuration space C0;m can be identified with the open .m2/simplex, consisting of m  2-tuples .q1 ; : : : ; qm2 / in Rm2 such that

0 < q1 <    < qm2 < 1: This is possible by means of the free action of the group G2 on Conf C 0;m , m  2, namely by fixing the first coordinate to 0 by translations and rescaling the last one to 1. C However, the compactified space Cx0;m , for m > 3, does not correspond to the closed simplex 4m2 : the strata of codimension 1 of 4m2 correspond to the collapse of C comprise only two consecutive coordinates, while the strata of codimension 1 of Cx0;m C x the collapse of a larger number of consecutive points. C0;m actually is the .m  2/-th Stasheff polytope [40]. C can be identified with an open interval: more (ii) The configuration space C1;1 precisely, by means of the action of G2 on Conf C 1;1 , we can fix the point q1 in R to 0 C and the modulus of the point z1 in HC to be 1. Hence, C1;1 Š S 1 \ HC Š .0; 1/. C The corresponding compactified configuration space Cx1;1 is simply the closed interval Œ0; 1: in terms of collapsing points, the two boundary strata correspond to the situation where the point z1 in HC approaches R on the left or on the right of the point q1 in R. (iii) The configuration space C2 can be identified with S 1 : by means of the action of the group G3 on Conf 2 , e.g., the first point can be fixed to 0 and its distance to the second point fixed to 1. Thus, Cx2 D C2 Š S 1 . C can be identified with HC n fig: by means of the (iv) The configuration space C2;0 action of G2 , we can fix, for instance, the first point p1 in HC to i. The corresponding C compactified configuration space Cx2;0 is often referred to as the Kontsevich eye: in fact, its graphical depiction resembles an eye. More precisely, the boundary stratification C consists of three boundary strata of codimension 1 and two boundary strata of of Cx2;0

C 6.2 Compactification of Cn and Cn;m à la Fulton–MacPherson

45

codimension 2. In terms of configuration spaces, the boundary strata of codimension C 1 are identified with Cx2 Š S 1 and Cx1;1 Š Œ0; 1, while the boundary strata of codiC C x mension 2 are both identified with C0;2 : the stratum Cx2 resp. Cx1;1 corresponds to the C C collapse of both points z1 and z2 in H to a single point in H resp. to the situation where one of the points z1 and z2 approaches R, while both strata of codimension 2 correspond to the situation where both z1 and z2 approach R, and the ordering on points on R yields two possible configurations. Pictorially, the boundary stratum Cx2 C corresponds to the pupil of the Kontsevich eye; the boundary strata Cx1;1 correspond to eyelids of the Kontsevich eye, and, finally, the codimension 2 strata correspond to the two interchapter points of the two eyelids.

Cx2 C Cx1;1

C Cx0;2

111111111111111111111 000000000000000000000 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111

C Cx0;2

C Cx1;1

Figure 2. A graphical representation of the Kontsevich eye.

For the sake of simplicity, from now on, points in HC resp. R are said to be of the first resp. second type. Description of a few boundary components. Now, for the main computations of Chapter 8, we need mostly only boundary strata of codimension 1 and, in Section 7.3, C : we list here the relevant boundary particular boundary strata of codimension 2 of Cxn;m strata of codimension 1 and of codimension 2, which are needed. For the boundary strata of codimension 1, we are concerned with two situations: i) For a subset A  f1; : : : ; ng, the points zi of the first type, i 2 A, collapse together to a single point of the first type; more precisely, we have the factorization C C @A Cxn;m Š CxA  CxnjAjC1;m I

here, 2 jAj n denotes the cardinality of the subset A. Intuitively, CxA C describes the configurations of distinct points of the first type in Cxn;m which collapse to a single point of the first type.

46

6 Configuration spaces and integral weights

ii) For a subset A  f1; : : : ; ng and an ordered subset B  f1; : : : ; mg of consecutive integers, the points of the first type zi , i 2 A, and the points of the second type qi in R collapse to a single point of the second type; more precisely, we have the factorization C C C @A;B Cxn;m Š CxA;B  CxnjAj;mjBjC1 : C describes the configurations of points of the first type and of Intuitively, CxA;B C the second type in Cxn;m , which collapse together to a single point of the second type.

As for the codimension 2 boundary strata, which will be of importance to us, we have the following situation: there exist disjoint subsets A1 , A2 of f1; : : : ; ng and disjoint ordered subsets B1 , B2 of f1; : : : ; mg of consecutive integers such that the corresponding boundary stratum of codimension 2 admits the factorization C : CxAC1 ;B1  CxAC2 ;B2  CxnjA 1 jjA2 j;mjB1 jjB2 jC2

Intuitively, CxAC1 ;B1 and CxAC2 ;B2 parametrize disjoint configurations of points of the first and of the second type, which collapse together to two distinct points of the first type. We will write later on such a boundary stratum a bit differently, namely, after reordering of the points after collapse, the third factor in the previous factorization can be written as CxAC3 ;B3 , for a subset A3 of f1; : : : ; ng of cardinality n  jA1 j  jA2 j, for an ordered subset B3 of f1; : : : ; mg of cardinality m  jB1 j  jB2 j C 2.

6.3 Directed graphs and integrals over configuration spaces The standard angle function. We introduce now the standard angle function.2 For this purpose we consider a pair of distinguished points .z; w/ in HC t R and we denote by '.z; w/ the normalized hyperbolic angle in HC t R between z and w; more explicitly, as shown in Figure 3, '.z; w/ D

z  w

1 arg : 2 zN  w

2 As observed by Kontsevich [29] one could in principle choose more general angle functions, starting from the abstract properties of the standard angle function.

6.3 Directed graphs and integrals over configuration spaces

47

'.z; w/ w

z

Figure 3. The hyperbolic angle function '.z; w/. C Observe that the assignment C2;0 3 .z; w/ 7! '.z; w/ 2 S 1 obviously extends to C 1 a smooth map from Cx2;0 to S , which enjoys the following properties (these properties play an important rôle in the computations of Chapters 7 and 8):

i) the restriction of ' to the boundary stratum Cx2 Š S 1 equals the standard angle coordinate on S 1 (possibly up to addition of a constant term); C ii) the restriction of ' to the boundary stratum Cx1;1 , corresponding to the upper eyelid of the Kontsevich eye (in other words, when z approaches R), vanishes.

We will refer to ' as the angle function. Integral weights associated to graphs. We consider, for given positive integers n and N : : : ; mg. m, directed graphs  with mCn vertices labelled by the set E./ D f1; : : : ; n; 1; x Here, “directed” means that each edge of  carries an orientation. Additionally, the graphs we consider are required to have no loop (a loop is an edge beginning and ending at the same vertex). To any edge e D .i; j / 2 E./ of such a directed graph , we associate the smooth map C 'e W Cn;m ! S 1 ; .z1 ; : : : ; zn ; z1N ; : : : ; zmx / 7! '.zi ; zj /; C to S 1 . which obviously extends to a smooth map from Cxn;m To any directed graph  without short loops and with set of edges E./ we associate a differential form V ! ´ d'e e2E./ C on the (compactified) configuration space Cxn;m .

Remark 6.7. We observe that, a priori, it is necessary to choose an ordering of the edges of  since ! is a product of 1-forms: two different orderings of the edges of  simply differ by a sign. This sign ambiguity precisely coincides (and thus cancels) with the one appearing in the definition of B , as it is pointed out in Remark 5.7.

48

6 Configuration spaces and integral weights

C We recall from Sections 6.1 and 6.2 that Cxn;m is orientable, and that the orientation C x of Cn;m specifies an orientation for any boundary stratum thereof.

Definition 6.8. The weight W of the directed graph  is Z ! : W ´

(6.1)

C xn;m C

Observe that the weight (6.1) indeed exists, because it is an integral of a smooth differential form over a smooth compact manifold (with corners). Vanishing lemmata. It follows immediately from the definition of W that it is nontrivial only if • the cardinality of E./ equals 2n C m  2 (i.e., ! is a top-degree form), •  has no multiple edges, • second type vertices do not have outgoing edges. In particular, W is non-trivial only if  2 Gn;m . For later purposes, we need a few non-trivial vanishing Lemmata concerning the above integral weights, which we use later on in Chapters 7, 8 and 9. Lemma 6.9. If  in Gn;m has a vertex v of the first type with exactly one incoming and one outgoing edge (see Figure 4), its integral weight W vanishes.

v

v

v

Figure 4. Three possible situations, where the bivalent vertex v appears.

We observe that the target of the outgoing edge may be of the first or of the second type, while the source of the incoming edge must be of the first type. Sketch of proof. Exemplarily, we consider the case where the vertices v1 , v2 connected to v are of the first type (v1 points to v and v points to v2 ); the corresponding points in C Cxn;m are denoted by z1 and z2 , respectively. Using Fubini’s Theorem, we isolate in the weight W the factor Z d'e1 ^ d'e2 : (6.2) HC nfz1 ;z2 g

6.3 Directed graphs and integrals over configuration spaces

49

The rest of the proof consists in showing that (6.2) vanishes. We observe that (6.2) is a function depending on .z1 ; z2 /. We first show that it is a constant function. Namely, (6.2) is the integral along the fiber of the integrand form with C xC  C2;0 , .z1 ; z; z2 / 7! .z1 ; z2 /: independence of respect to the natural projection Cx3;0 z1 and z2 follows by means of the generalized Stokes’ Theorem applied to the fibration C C  Cx2;0 (we observe that the initial, resp. final, manifold of the fibration, as Cx3;0 well as the fiber itself, is a smooth manifold with corners; for the precise statement of the generalized Stokes’ Theorem in this framework, we refer to [5], [11], where an extensive use of it is made), d. .d'e1 ^ d'e2 // D ˙ .d.d'.v1 ;v/ ^ d'.v;v2 / // ˙ @ .d'e1 ^ d'e2 /; where the second term on the right-hand side corresponds to the boundary contributions coming from fiber integration. Since the integrand is obviously closed, it remains to show the vanishing of the boundary contributions. It is clear that there are three boundary strata of codimension 1 of the fibers of , corresponding to i) the approach of z to z1 or z2 , or ii) the approach of z to R. The properties of the angle function imply that the contribution from ii) vanishes, while the two contributions from i) cancel together. Hence, we may choose, for instance, z1 D i and z2 D 2i: for this particular choice, the involution z 7! zN of HC n fi; 2ig reverses the orientation of the fibers, but preserves the integrand form, whence the claim follows.  Lemma 6.10. For a positive integer n  3, the integral over Cxn of the product of 2n  3 forms of the type d.arg.zi  zj //, i ¤ j , vanishes. Proof. The proof relies on an analytic argument, which involves a tricky computation with complex logarithms; for a complete proof we refer to [29] and [27] (see also [13], Appendix). 

7 The map UQ and its properties In this chapter we first discuss and then prove remarkable properties of the map UQ defined by the formula (5.2). Namely, we first prove that UQ is a quasi-isomorphism of complexes, and we then give, for any poly-vector fields ˛, ˇ, explicit formulae for UQ .˛ ^ ˇ/ and UQ .˛/ [ UQ .ˇ/ in terms of new weights associated to graphs. The proof follows closely the treatment of Manchon and Torossian [34], and strongly uses the remarkably rich combinatorics of the boundary stratification of the compactified configuration spaces introduced in the previous chapter.

7.1 The quasi-isomorphism property The present section is devoted to the proof of the following result. Proposition 7.1. The map UQ W Tpoly V ! Dpoly V defined by equation (5.2) is a quasi-isomorphism of complexes, i.e., for any poly-vector field ˛,   (7.1) UQ .Q  ˛/ D .dH C Q/ UQ .˛/ ; and UQ induces an isomorphism of graded vector spaces on cohomology. Sketch of the proof. We first sketch the proof of equation (7.1). The fact that it induces an isomorphism in cohomology then follows from a straightforward spectral sequence argument. Let  2 GnC1;mC1 be a graph with 2n C m edges, the first type vertex 1 having exactly n C m outgoing edges, and all other first type vertices having a single outgoing edge. We apply Stokes’ Theorem Z Z ! D d! D 0 xC @C nC1;mC1

C CnC1;mC1

and discuss the meaning of the following resulting identity: for any poly-vector field ˛ with n C m arguments, and any functions f1 ; : : : ; fmC1 , Z  X X ˙ ! B .˛; Q; : : : ; Q/.f1 ; : : : ; fmC1 / D 0: „ ƒ‚ … C C

2GnC1;mC1

n times

C , and the sign Here, C runs over all boundary strata of codimension 1 of CxnC1;mC1 C x depends on the induced orientation from CnC1;mC1 .

7 The map UQ and its properties

52

We now discuss the possible non-trivial contributions of each boundary stratum C . Using Fubini’s Theorem we find (up to signs coming from orientation choices) the factorization property Z Z Z ! D C

Cint

!int

Cout

!out ;

(7.2)

where int (resp. out ) is the subgraph of  whose edges are those with both source and target lying in the subset of collapsing points (resp. is the quotient graph of  by its subgraph int ). C Let us begin with the boundary components of the form C D @A CxnC1;mC1 (with jAj  2). It follows from Lemma 6.10 that there is no contribution if jAj  3. If jAj D 2 then int consists of a single edge and the first factor in the factorization on the right-hand side of (7.2) equals 1. There are two possibilities: • Either 1 … A and thus, taking the sum of the contributions of all graphs  leading to the same pair .int ; out /, one obtains something proportional to Wout Bout .˛; Q; : : : ; Q B Q; : : : ; Q/.f1 ; : : : ; fmC1 / D 0: „ƒ‚… D0

• Or 1 2 A and thus, again taking the sum of the contributions of all graphs  leading to the same pair .int ; out /, and adding up the terms coming from the same graphs  after reversing the unique arrow of int , one obtains Wout Bout .Q  ˛; Q; : : : ; Q/.f1 ; : : : ; fmC1 /: „ ƒ‚ …

(7.3)

n1 times

We then continue with the boundary components of the form C C C C D @A;B CxnC1;mC1 Š CA;B  CnjAj;mjBjC1 I

in this situation, we set C Cint D CA;B

and

C Cout D CnjAj;mjBjC1 :

Again there are two possibilities: • Either 1 … A. The integral associated to !int is non-trivial, only if its degree equals the dimension of Cint . The boundary conditions satisfied by Kontsevich’s angle form implies that no edge can depart from A (i.e., there is no edge connecting A to its complement), whence the non-triviality condition corresponds to jAj D 2jAj C jBj  2, i.e., jAj C jBj D 2. Therefore, the graph int can only be one of the following three graphs:

53

7.1 The quasi-isomorphism property C CxnC1jAj;mC2jBj

iN

C CxnC1jAj;mC2jBj

iN

i C1 C CxA;B

C CxA;B

C CxnC1jAj;mC2jBj

C CxA;B

Figure 5. The three possible boundary strata, when 1 … A.

Summing the contributions of all graphs leading to the same pair .int ; out /, one obtains Wout Bout .˛; Q; : : : ; Q/.f1 ; : : : ; fi fiC1 ; : : : ; fmC1 / (7.4) „ ƒ‚ … n times

for the first type of graphs, and Wout Bout .˛; Q; : : : ; Q/.f1 ; : : : ; Q  fi ; : : : ; fmC1 / „ ƒ‚ …

(7.5)

n1 times

for the second one. The third type of graph does not contribute since its weight vanishes by Lemma 6.9. • Or 1 2 A. In this case, we consider the differential form !out , whose integral over Cout is non-trivial, only if its degree equals the dimension of Cout . Since the special vertex 1 belongs to A, and again since no edge may depart from A, the non-triviality condition reads 2.n C 1  jAj/ C m  jBj D n C 1  jAj, i.e., jAjCjBj D nC1Cm. Additionally, since 0 jAj nC1 and 2 jBj mC1, out must be one of the three graphs in Figure 6. The corresponding contributions (after summing over graphs leading to the same decomposition) respectively are Wint .f1 Bint .˛; Q; : : : ; Q/.f2 ; : : : ; fmC1 / „ ƒ‚ … n times

˙ Bint .˛; Q; : : : ; Q/.f1 ; : : : ; fm /fmC1 / „ ƒ‚ …

(7.6)

n times

for the first one and Wint Q  .Bint .˛; Q; : : : ; Q/.f1 ; : : : ; fmC1 // „ ƒ‚ … n1 times

for the second one.

(7.7)

7 The map UQ and its properties

54 C CxnC1jAj;mC2jBj

C CxnC1jAj;mC2jBj C CxA;B

1N

C CxA;B

2N

m x

mC1

11 00 C CxnC1jAj;mC2jBj

C CxA;B

1 0

1 0 0 1

1 0 1 0 0 1

11 00

1N

11 00 11 00 1 0

1 0

11 00 00 11 1 0

11 00 1 0

1 0

mC1

Figure 6. The three possible boundary strata, when 1 2 A.

We now summarize all non-trivial contributions: (7.3) gives the left-hand side of equation (7.1), (7.6) together with (7.4) gives dH UQ .˛/, and (7.7) together with (7.5) gives Q  UQ .˛/. Therefore equation (7.1) is satisfied and it remains to prove that UQ induces an isomorphism at the level of cohomology. For this, we consider the mapping  cone CQ of UQ together with the decreasing filtration on it coming from the grading on Tpoly V and Dpoly V induced by the degree we have denoted by j  j in Chapter 5. The 0-th term of the corresponding spectral sequence is given by the mapping cone of the Hochschild–Kostant–Rosenberg map IHKR W .Tpoly V; 0/ ! .Dpoly V; dH /, and thus E1 D f0g (as IHKR is a quasi-isomorphism). This ends the proof of Proposition 7.1. 

7.2 The cup product on poly-vector fields In the present section we consider the cup product between any two poly-vector fields ˛ and ˇ: we want to express the result of applying (5.2) on the cup product ˛ ^ ˇ in C terms of integral weights over a suitable submanifold Z0  CxnC2;m , that we define now. C We recall from Section 6.2 that the compactified configuration space Cx2;0 can be depicted as the Kontsevich eye. We choose a general point x in the boundary stratum C C2 Š S 1  Cx2;0 . Furthermore, for any two positive integers n and m we consider

55

7.2 The cup product on poly-vector fields

C C the projection F ´ f1;2g;; from CxnC2;m onto Cx2;0 , using the same notation as in C given by the preimage Section 6.2. Then we denote by Z0 the submanifold of CxnC2;m with respect to F of the point x; accordingly, to a graph  2 GnC2;m we associate a new weight W0 given by Z

W0 ´

Z0

! ;

using the same notation of Section 6.3. Proposition 7.2. For any two poly-vector fields ˛ and ˇ on V , the following identity holds true: UQ .˛ ^ ˇ/ D

X 1 nŠ n0

X 2GnC2;m

W0 B .˛; ˇ; Q; : : : ; Q/: „ ƒ‚ …

(7.8)

n times

Notice in particular that the right-hand side of (7.8) does not depend on the choice of x 2 C2 (it is a consequence of the generalized Stokes’ Theorem). Proof. The proof relies on the following key lemma: Lemma 7.3. The integral weight W0 vanishes, for any graph  2 GnC2;m , unless  contains no edge connecting the vertices of the first type 1 and 2, in which case W0 D Wz ; z is the graph in GnC1;m obtained from  by collapsing the vertices 1 and 2. where  Proof. First of all, Z0 intersects non-trivially only boundary strata of the form CxA  C , where A is a subset of f1; : : : ; n C 2g and containing 1 and 2. Using CxnjAjC3;m Fubini’s Theorem, we obtain Z Z Z W0 D ! D !int !out : (7.9) xC Z0 \@A C nC2;m

xA Z0 \C

xC C njAjC3;m

The points corresponding to the vertices 1 and 2 of the first type are fixed by assumption. For dimensional reasons, the only (possibly) non-trivial contributions to the first factor in the factorization on the right-hand side of (7.9) occur only if the degree of the integrand !int equals 2jAj  4. The corresponding integral vanishes if jAj  3 by the arguments in the proof of Kontsevich’s Lemma 6.10, for which we refer to [29]: suffice it to mention that, in the proof in [29], Kontsevich reduces the case of the integral over Cxn of a product of 2n  3 forms to the case of the integral over a manifold of the form Z0 \ Cxn (i.e., he fixes two vertices) and then he extracts from the integrand the 1-form corresponding to the edge joining the two fixed points (i.e., there is no edge between the two fixed vertices). Then he shows that the latter integral vanishes by complicated

56

7 The map UQ and its properties

analytical arguments (tricks with logarithms and distributions): anyway, the very same arguments imply that the first factor in the factorization (7.9) vanishes. Hence, we are left with the case jAj D 2, i.e., A D f1; 2g: therefore, we obtain, again using Fubini’s Theorem, Z Z Z 0 !out : ! D !int W D xC Z0 \@f1;2g C nC2;m

x2 Z0 \C

xC C nC1;m

z 2 GnC1;m in the claim of the lemma. On It is clear that out is exactly the graph  the other hand, by properties of the angle function, when restricted to the boundary stratum Cx2 , we have Z !int D 1; x2 Z0 \C

observing that int consists of two vertices of the first type, with no edge connecting them. For jAj D 2, if there is an edge connecting the two vertices, then the corresponding contribution vanishes, as it contains the derivative of a constant angle (1 and 2 are infinitely near to each other with respect to a fixed direction). Thus, we have proved the claim.  z in GnC1;m and with ˛, ˇ and Q as before, the polyWe consider now, for a graph  differential operator Bz .˛ ^ ˇ; Q; : : : ; Q/, where there are n cohomological vector fields Q. By the very construction of B and by the definition of ^, we have X B .˛; ˇ; Q; : : : ; Q/; Bz .˛ ^ ˇ; Q; : : : ; Q/ D z GnC2;m 37!

z where the sum is over all possible graphs  in GnC2;m , which are obtained from  by separating the vertices 1 and 2 of the first type without inserting any edge between them; it is clear that contraction of the vertices 1 and 2 of a graph  as before gives the z This collapsing process is symbolized by writing  7! . z initial graph . We finally compute X X X Wz B .˛; ˇ; Q; : : : ; Q/ Wz Bz .˛ ^ ˇ; Q; : : : ; Q/ D z nC1;n 2G

z nC1;m GnC2;m 37! z 2G

D

X

X

W0 B .˛; ˇ; Q; : : : ; Q/

z nC1;m GnC2;m 37! z 2G

D

X

W0 B .˛; ˇ; Q; : : : ; Q/:

2GnC2;m

The second and third equalities follow from Lemma 7.3. This ends the proof of Proposition 7.2 

57

7.3 The cup product on poly-differential operators

7.3 The cup product on poly-differential operators Applying UQ on poly-differential operators ˛ and ˇ, we may then take their cup product in the Hochschild complex of poly-differential operators. We want to show that, in analogy with Proposition 7.2, this product can be expressed in terms of integral C weights over a suitable submanifold Z1 of CxnC2;m , that we define now. C C Let y be the unique point sitting in the copy of C0;2 inside @Cx2;0 in which the vertex 1 stays on the left of the vertex 2. Then for any two positive integers n and m, using C the same notation as in the previous section, we define Z1 ´ F 1 .y/  CxnC2;m and Z 1 W ´ ! : Z1

Proposition 7.4. Under the same assumptions of Proposition 7.2, the following identity holds true: X 1 X UQ .˛/ [ UQ .ˇ/ D W1 B .˛; ˇ; Q; : : : ; Q/: (7.10) „ ƒ‚ … nŠ n0

2GnC2;m

n times

Proof. By the very definition of the cup product in Hochschild cohomology, we obtain UQ .˛/ [ UQ .ˇ/ D

X k;l0

1 kŠlŠ

X

W1 W2 B1 .˛; Q; : : : ; Q/ „ ƒ‚ …

1 2GkC1;m 1 2 2GlC1;m 2

k times

 B2 .˛; Q; : : : ; Q/ „ ƒ‚ … D

X k;l0

1 kŠlŠ

X

(7.11)

l times

W1 t2 B1 t2 .˛; ˇ; Q; : : : ; Q/; „ ƒ‚ …

1 2GkC1;m 1 2 2GlC1;m 2

kCl times

where, for any two graphs 1 2 GkC1;m1 and 2 2 GlC1;m2 , we have denoted by 1 t 2 their disjoint union: it is again a graph in GkClC2;m1 Cm2 . The vertices of 1 t 2 are re-numbered starting from the numberings of the vertices of 1 and 2 to guarantee the last equality in the previous chain of identities: namely, denoting by an index i D 1; 2 the graph to which belongs a given vertex labelled by i , the new numbering of the vertices of 1 t 2 is f11 ; 12 ; 21 ; 31 ; : : : ; .k C 1/1 ; 22 ; 32 ; : : : ; .l C 1/2 g: Lemma 7.5. The integral weight W1 vanishes for any graph  in GnC2;m , unless  D 1 t 2 , with i 2 Gki ;mi , i D 1; 2, in which case W1 D W1 W2 :

7 The map UQ and its properties

58

Proof. It follows from its very definition that Z1 intersects non-trivially only those C C boundary strata @T CxnC2;m of CxnC2;m of codimension 2 which, according to Section 6.2, possess the factorization CxAC1 ;B1  CxAC2 ;B2  CxAC3 ;B3 ; where vertex 1 and vertex 2 lie in CxAC1 ;B1 and CxAC2 ;B2 , respectively; finally, the positive integers ni ´ jAi j and mi ´ jBi j obviously satisfy n1 C n2 C n3 D n C 2

m1 C m2 C .m3  2/ D m:

and

1 2 , resp. int , resp. out , the subgraph of For a graph  2 GnC2;m , we denote by int , whose vertices are labelled by A1 t B1 , resp. A2 t B2 , resp. by contracting the 1 2 and int to two distinct vertices of the second type. subgraphs int Using Fubini’s Theorem once again, we get Z Z Z Z ! D ! 1 ! 2 !out : xC Z1 \@T C nC2;m

xC C A ;B 1

int

1

xC C A ;B 2

int

2

xC C A ;B 3

3

1 2 By the properties of the angle function, there cannot be vertices of int or int , from which departs an external edge, i.e., an edge whose target lies in the set of vertices of out : otherwise, there would be an edge in out , whose source is of the second type. Hence, since the poly-vector fields ˛ and ˇ are respectively associated to vertices in A1 t B1 and A2 t B2 , then only copies of Q can be associated to the vertices of out . Therefore the vertices of out have all exactly one outgoing edge, and consequently out can be only the trivial graph with no vertex of the first type and exactly two vertices 1 2 of the second type. In other words,  is the disjoint union int t int . Summarizing all these arguments, we get Z Z Z W D ! D ! 1 ! 2 D W 1 W 2 : xC Z1 \@T C nC2;m

xC C A ;B 1

int

1

xC C A ;B 2

int

int

int

2

For any other graph , it follows from the previous arguments that W D 0.



Combining Lemma 7.5 with (7.11), we finally obtain X 1 X UQ .˛/ [ UQ .ˇ/ D W1 t2 B1 t2 .˛; ˇ; Q; : : : ; Q/ „ ƒ‚ … kŠ lŠ  2G k;l0

D

X 1 nŠ n0 1 nŠ

1 kC1;m1 2 2GlC1;m 2

X

kCl times

W1 B .˛; ˇ; Q; : : : ; Q/: „ ƒ‚ …

2GnC2;m

n times

1 , kŠlŠ

The combinatorial factor appears, instead of as a consequence of the fact that the sum is over graphs which split into a disjoint union of two subgraphs, and we have to take care of the possible equivalent graphs splitting into the same disjoint union. 

7.3 The cup product on poly-differential operators

59

Remark 7.6. We could have chosen y to be the unique point sitting in the other copy C C of C0;2 inside @Cx2;0 , i.e., the one in which vertex 2 is on the left of vertex 1. In this case Proposition 7.4 remains true if one replaces the left-hand side of (7.10) by ˙UQ .ˇ/ [ UQ .˛/. Since [ is known to be commutative at the level of cohomology, C the choice of the copy of C0;2 is not really important.

8 The map HQ and the homotopy argument z appearing in the defining formula (5.3) for In this chapter we define the weights W HQ and prove that, together with UQ , it satisfies the homotopy equation (5.1). We continue to follow closely the treatment of Manchon and Torossian [34]. To evaluate certain integral weights, we again need the explicit description of boundary strata of C codimension 1 of Cxn;m , for whose discussion we refer to the end of Section 6.2.

8.1 The complete homotopy argument We have proved in Sections 7.2 and 7.3, that the expressions UQ .˛ ^ ˇ/ and UQ .˛/ [ UQ .ˇ/ can be rewritten in terms of the integral weights over Z0 D F 1 .x/ and Z1 D C C C F 1 .y/, where we recall that F ´ f1;2g;; W CxnC2;m  Cx2;0 , and x 2 C2  @Cx2;0 C C and y 2 C0;2  @Cx2;0 are arbitrary. C It is thus natural to consider a continuous path ` W Œ0I 1 ! Cx2;0 such that x ´ C C `.0/ 2 C2 , y ´ `.1/ 2 C0;2 , and `.t / 2 C2;0 for any t 2 .0; 1/. We therefore define C Z ´ F 1 .`..0; 1///  CxnC2;m :

x is the preimage of `.Œ0; 1/ under the projection F . Then the boundary Its closure Z x of Z splits into the disjoint union x D Z0 t Z1 t .Z \ @Cx C @Z nC2;m /:

(8.1)

The third boundary component will be denoted by Y . Cx2 11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111

C Cx0;2

the curve `

Figure 7. The curve ` on the Kontsevich eye.

62

8 The map HQ and the homotopy argument

C C Since, by assumption, `..0; 1// lies in the interior C2;0  Cx2;0 , then it follows that Y intersects only the following five types of boundary strata of codimension 1 of C CxnC2;m :

(i) there is a subset A1 of f1; : : : ; n C 2g, containing 1, but not 2, such that the points of the first type labelled by A1 collapse together to a single point of the first type; (ii) there is a subset A2 of f1; : : : ; n C 2g, containing 2, but not 1, such that the points of the first type labelled by A2 collapse together to a single point of the first type; (iii) there is a subset A of f1; : : : ; n C 2g, containing neither 1 nor 2, such that the points of the first type labelled by A collapse together to a single point of the first type; (iv) there is a subset A of f1; : : : ; n C 2g, containing neither 1 nor 2, and an ordered subset B of f1; : : : ; mg of consecutive integers such that the points labelled by A (of the first type) and by B (of the second type) collapse together to a single point of the second type; (v) there is a subset A of f1; : : : ; n C 2g, containing both 1 and 2, and an ordered subset B of f1; : : : ; mg of consecutive integers such that the points labelled by A (of the first type) and by B (of the second type) collapse together to a single point of the second type. Remark 8.1. We observe that there is no intersection with a boundary stratum for which there is a subset A of f1; : : : ; n C 2g containing f1; 2g and such that the points labelled by A collapse together to a single point of the first type. This is because such a boundary stratum (by the arguments of Proposition 7.2) intersects non-trivially Z0 , and Y , Z0 and Z1 are pairwise disjoint. For a graph  2 GnC2;m we define new weights Z Z z D W2 D ! and W ! ; Z

Y

with the same notation as in Definition 6.8 of Section 6.3. Stokes’ Theorem implies that Z Z ! D d! D 0: x @Z

x Z

Using the orientation choices for Z, for which we refer to [34], together with (8.1), the previous identity implies the relation W0 D W1 C W2 : Using Proposition 7.2, Proposition 7.4 and the above identity involving Stokes’ Theorem, we obtain that the left-hand side of the homotopy equation (5.1) equals X 1 X  W2 B .˛; ˇ; Q; : : : ; Q/: „ ƒ‚ … nŠ n0 2GnC2;m

n times

8.2 Contribution to W2 of boundary components in Y

63

Hence, to prove that HQ , given by (5.3), satisfies (5.1) together with UQ , it remains to show that for fixed n and m, the following identity holds true: X W2 B .˛; ˇ; Q; : : : ; Q/ „ ƒ‚ … 2GnC2;m n times X z dH.B .˛; ˇ; Q; : : : ; Q// D W „ ƒ‚ … 2GnC2;m n times X z Q  .B .˛; ˇ; Q; : : : ; Q// n W (8.2) „ ƒ‚ … 2GnC1;m

C

X

2GnC1;m k˛k

C .1/

n1 times

z .B .Q  ˛; ˇ; Q; : : : ; Q// W „ ƒ‚ … X

2GnC1;m

n1 times

z .B .˛; Q  ˇ; Q; : : : ; Q// : W „ ƒ‚ … n1 times

In Section 8.2 we will sketch the proof of equation (8.2). For a more detailed treatment of signs appearing in the forthcoming arguments, we refer to [34]. Summarizing, the sum of (8.3) and (8.4) from Section 8.2.1, and of (8.6) from Section 8.2.2, we get the term in (8.2) involving the Hochschild differential of (5.2). The sum of (8.5) from Section 8.2.1 and of (8.7) from Section 8.2.2 yields the term with the action of the cohomological vector field Q on Dpoly V . In Section 8.2.3 one obtains the vanishing of terms which contain the action of Q on itself. Finally, (8.8) and (8.9) from Section 8.2.4 yield the remaining terms in (8.2). Thus, we have proved (5.1).

8.2 Contribution to W2 of boundary components in Y The discussion is analogous to the one sketched in the proof of Proposition 7.1. 8.2.1 Boundary strata of type (v). We consider a boundary stratum C of Y of type (v): there exists a subset A of f1; : : : ; nC2g and an ordered subset B of f1; : : : ; mg of consecutive integers, such that C C  CxnjAjC2;mjBjC1 /: C D Z \ .CxA;B

Accordingly, by means of Fubini’s Theorem, the integral weight of a graph  2 GnC2;m , restricted to C , can be rewritten as Z Z Z W jC D ! D !int !out : x @C Z

xC Z\C A;B

xC C njAjC2;mjBjC1

8 The map HQ and the homotopy argument

64

Here we have used the same improper notation as in the proof of Lemma 7.3, and, as usual, int (resp. out ) denotes the subgraph of  whose vertices are labelled by A t B (resp. the subgraph obtained by contracting int to a single point of the second type). The poly-vector fields ˛ and ˇ have been attached on the vertices labelled by 1 and 2, which belong to A: hence, only copies of the cohomological vector field Q can be attached on the first type vertices of out . In other words, first type vertices of out have a single outgoing edge. Then, for the same combinatorial reason as in the proof of Proposition 7.1, out can be of the following forms. C CxnC2jAj;mC1jBj

C CxnC2jAj;mC1jBj C Z \ CxA;B

1N

C Z \ CxA;B

2N

m x

mC1

11 00 00 11 C CnC2jAj;mC1jBj

C Z \ CxA;B

1 0 0 1

11 00

1 0 1 0

1 0 11 00

11 00

1N

1 0 1 0

1 0

1 0 11 00

11 00 11 00 11 00

11 00

m x

Figure 8. The three possible boundary strata of type (v).

In all three cases, the integral weight corresponding to out is normalized, up to some signs coming from orientation choices (which we will again ignore, as before). The directed subgraph int belongs obviously to GnC2;m1 , resp. GnC1;m , since in case (i), jAj D n C 2 and jBj D m  1, whereas, in case (ii), jAj D n C 1 and jBj D m. Case (i), furthermore, includes two subcases, namely, since jBj D m  1, and since B N : : : ; m  1g consists only of consecutive integers, it follows immediately that B D f1; N : : : ; mg. or B D f2; x From the point of view of poly-differential operators, the graph out corresponds, in both subcases of (i), to the multiplication operator, whereas, in case (ii), it corresponds to the action of the cohomological vector field Q, placed on the vertex of the first type, on a function on V , placed on the vertex of the second type.

8.2 Contribution to W2 of boundary components in Y

All these arguments yield the following expressions for the contributions left-hand side of (8.2) coming from boundary strata of type (v): X z f1 .B .˛; ˇ; Q; : : : ; Q/.f2 ; : : : ; fm //; ˙W „ ƒ‚ … 2GnC2;m1 n times X z .B .˛; ˇ; Q; : : : ; Q/.f2 ; : : : ; fm //fm ; ˙W „ ƒ‚ … 2GnC2;m1 n times X z Q  .B .˛; ˇ; Q; : : : ; Q/.f1 ; : : : ; fm //: ˙W „ ƒ‚ … 2GnC1;m

65 to the (8.3) (8.4) (8.5)

n1 times

Remark 8.2. The actual contribution of (8.5) has to be multiplied by n since there are precisely n distinct subgraphs int of  2 GnC2;m that coincide in GnC1;m . 8.2.2 Boundary strata of type (iv). We consider now a boundary stratum C of Y of the fourth type: in this case, there exists a subset A of f1; : : : ; n C 2g, containing neither the vertex labelled by 1 nor by 2, and an ordered subset B of f1; : : : ; mg of consecutive elements, such that C C C D Z \ .CxA;B  CxnjAjC2;mjBjC1 /:

Once more, Fubini’s Theorem implies the factorization Z Z Z W jC D ! D !int xC C A;B

C

xC Z\C njAjC2;mjBjC1

!out :

The vertices labelled by 1 and 2, to which we have attached the poly-vector vector fields ˛ and ˇ, are vertices of the graph out : hence, every first type vertex of int has exactly one outgoing edge. Again, as in the proof of Proposition 7.1 and thanks to the vanishing Lemma 6.9, int can be only one of the following three forms. C Z \ CxnC2jAj;mC1jBj

iN

C Z \ CxnC2jAj;mC1jBj

iN

i C1 C CxA;B

C CxA;B

C Z \ CxnC2jAj;mC1jBj

C CxA;B

Figure 9. The three possible boundary strata of type (iv).

8 The map HQ and the homotopy argument

66

In the first resp. second case, out is a graph in GnC2;m1 resp. in GnC1;m . In the first case A D ; and B D fiN ; i C 1g (since points of the second type are ordered), for N : : : ; m, iN D 1; x while in the second case A D fig and B D fjNg for i D 1; : : : ; n C 2 N N and j D 1; : : : ; m. x The third contribution vanishes once again in view of Lemma 6.9. Up to signs arising from orientation choices, which we have ignored so far, both integrals corresponding to (i) and (ii) are normalized. The graph int corresponds, in terms of the poly-differential operators B , to the product of two functions on V , which have been attached to the vertices labelled by iN and i C 1, in case i /; on the other hand, in case (ii), the graph int corresponds to the situation where the cohomological vector field Q acts as a derivation on a function on V , which has been put on the vertex jN. Using all previous arguments, we obtain the following two expressions for the contributions to the left-hand side of (8.2) coming from boundary strata of type (iv): m1 X

X

iD1 2GnC2;m1 m X

z B .˛; ˇ; Q; : : : ; Q/.f1 ; : : : ; fi fiC1 ; : : : ; fm /; ˙W „ ƒ‚ …

(8.6)

z B .˛; ˇ; Q; : : : ; Q/.f1 ; : : : ; Q  fi ; : : : ; fm / ˙W „ ƒ‚ …

(8.7)

X

iD1 2GnC1;m

n times

n1 times

for any collection ff1 ; : : : ; fm g of m functions on V . 8.2.3 Boundary strata of type (iii). We examine a boundary stratum C of Y of the third type. Thus, there is a subset A of f1; : : : ; n C 2g, containing neither the vertex labelled by 1 nor by 2, such that C C D Z \ .CxA  CxnjAjC3;m /:

The contribution coming from C to the integral weight is, again by means of Fubini’s Theorem, Z Z Z W jC D ! D !int !out : C

xA C

xC Z\C njAjC3;m

Since the poly-vector fields ˛ and ˇ have been attached on the vertices labelled by 1 and 2, which do not belong to A, it follows that only copies of Q have been attached on the vertices of int . We focus in particular on the integral contributions coming from int : by Lemma 6.10, if jAj  3, such contributions vanish, whence we are left with only one possible directed subgraph int , namely int consists of exactly two vertices of the first type joined by exactly one edge. The corresponding weight is normalized, by the properties of the angle function. The graph out is easily verified to be in GnC1;m ; the poly-differential operator corresponding to int represents the adjoint action of Q on itself, by its very construction. Since Q is, by assumption, a cohomological vector field, it follows that such a contribution vanishes by the property ŒQ; Q D 12 QBQ D 0. It thus follows that boundary strata of type (iii) do not contribute to the left-hand side of (8.2).

8.2 Contribution to W2 of boundary components in Y

67

8.2.4 Boundary strata of type (i) and (ii). We consider a boundary stratum C of Y of type (i). By its very definition, for such a stratum C there exists a subset A1 of f1; : : : ; n C 2g, containing the vertex labelled by 1, but not the vertex labelled by 2, such that C C D Z \ .CxA1  CxnjAjC3;m /: By means of Fubini’s Theorem, we obtain the following factorization for the integral weight W , when restricted to C , Z Z Z ! D !int !out : W jC D C

xA C 1

xC Z\C njAjC3;m

We focus our attention on the integral contribution coming from int : as in Section 8.2.3, by means of Lemma 6.10, the only possible subgraph int yielding a non-trivial integral contribution is the graph consisting of two vertices of the first type joined by exactly one edge, in which case the contribution is normalized (up to some signs, coming from orientation choices, which we ignore, as we have done before). By assumption, one of the two vertices is labelled by 1 and the other one is not labelled by 2: there are hence two possible graphs, namely, (i) when the edge has, as target, the vertex labelled by 1, and (ii) when the edge has, as source, the vertex labelled by 1. Since the other vertex is not labelled by 2, in terms of poly-differential operators, we have two situations: a copy of Q acts, as a differential operator of order 1, on the components of the polyvector field ˛, in case (i), or one of the derivations of the poly-vector field ˛ acts, as a differential operator of order 1, on the components of Q, in case (ii). Finally, the graph out belongs obviously to GnC1;m . By the previous arguments, and by the very definition of the Lie XV -module structure on poly-vector fields, the contributions to the left-hand side of (8.2) coming from boundary strata of type (i) can be written as X 2GnC1;m

z B .Q  ˛; ˇ; Q; : : : ; Q/: ˙W „ ƒ‚ …

(8.8)

n1 times

As for boundary strata of Y of type (ii), we may repeat almost verbatim the previous arguments, the only difference in the final result being that the rôle played by the polyvector field ˛ will be now played by ˇ, hence the contributions to the left-hand side of (8.2) coming from boundary strata of type (ii) are exactly X 2GnC1;m

z B .˛; Q  ˇ; Q; : : : ; Q/: ˙W „ ƒ‚ …

(8.9)

n1 times

Remark 8.3. The actual contribution of (8.8) and (8.9) has to be multiplied by n. For instance, for (8.8) one sees that there are precisely n distinct subgraphs  2 GnC2;m that induces the same out 2 GnC1;m (they are given by A D f1; kg, k D 3; : : : ; n C 2).

68

8 The map HQ and the homotopy argument

8.3 Twisting by a supercommutative DG algebra We consider finally a supercommutative DG algebra .m; dm /: typically, instead of considering Tpoly V and Dpoly V , for a superspace V as before, we consider their twists with respect to m: m V ´ Tpoly V ˝ m and Tpoly

m Dpoly V ´ Dpoly V ˝ m:

Since m is supercommutative, the Lie bracket on XV determines a graded Lie algebra structure on Xm V ´ XV ˝ m: Œv ˝ ; w ˝  D .1/jwjj j Œv; w ˝ : Hence, for any choice of a supercommutative DG algebra .m; dm /, there are two graded m m Lie Xm V -modules Tpoly V and Dpoly V . Moreover the differential dm extends naturally m m V . It is easy to verify that the differential dm to a differential on Tpoly V and Dpoly m (super)commutes with the Hochschild differential dH on Dpoly V. m We now consider an m-valued vector field Q 2 XV of degree 1 which additionally satisfies the so-called Maurer–Cartan equation 1 dm Q C ŒQ; Q D dm Q C Q B Q D 0: 2 We observe that, if m D k (with k placed in degree 0) then Q is simply a cohomological vector field on V as in Definition 5.2. The Maurer–Cartan equation implies that dm CQ m is a linear operator of (total) degree 1 on Tpoly V , which additionally squares to 0; moreover, the product ^ on Tpoly V extends naturally to a supercommutative graded m V , and dm C Q is obviously a degree 1 derivation of associative product ^ on Tpoly this product. Therefore, m V; ^; dm C Q/ .Tpoly is a DG algebra. One obtains in exactly the same way a DG algebra m .Dpoly V; [; dH C dm C Q/:

Theorem 5.3 can be generalized to these DG algebras as follows. Theorem 8.4. For any degree 1 solution Q 2 Xm V of the Maurer–Cartan equation, the m-linear map UQ given by (5.2) defines a morphism of complexes UQ

m m V; ^; dm C Q/ ! .Dpoly V; [; dH C dm C Q/; .Tpoly

which induces an isomorphism of (graded ) algebras on the corresponding cohomologies.

8.3 Twisting by a supercommutative DG algebra

69

Proof. The proof follows along the same lines as the proof of Theorem 5.3, which can be repeated almost verbatim. The differences arise when discussing • the morphism property (7.1) for UQ , • the homotopy property (5.1) for UQ and HQ . In both cases one must replace .dH C Q/ where it appears in the equation by .dH C dm C Q/. For the homotopy property (5.1), the core of the proof lies in the discussion of the boundary strata for the configuration spaces appearing in (8.2): the relevant boundary strata in the present proof are those of Section 8.2.3. We can repeat the same arguments in the discussion of the corresponding integral weights: using the very same notation as in Section 8.2.3, the poly-differential operator corresponding to int is one half times the adjoint action of Q on itself, which, in this case, does not square to 0, but equals (up to sign) dm Q by the Maurer–Cartan equation. Using the graded Leibniz rule for dm , we get all homotopy terms which contain dm . The discussion of the remaining boundary strata remains unaltered. The very same argument also works for the morphism property (7.1). Nevertheless, we see in the next chapter that (7.1) can be obtained as a consequence of the explicit form of UQ , avoiding the discussion on possible contributions of the boundary components in the proof of Proposition 7.1. 

9 The explicit form of UQ In this chapter we derive an explicit expression for the quasi-isomorphism UQ (5.2), following closely [9], Chapter 8. Namely, we first argue about the possible shapes of the graphs  involved in the construction of UQ : by the way, this was already done, although not as precisely as in the present chapter, in the proof of Proposition 7.4.

9.1 Graphs contributing to UQ We now recall that, in (5.2), we need a poly-vector field ˛ on the superspace V and a cohomological vector field Q. We consider a graph  2 GnC1;m , appearing in (5.2): on one of its vertices of the first type, we attach ˛, while, on the remaining n vertices of the first type we attach copies of Q. Since Q is a vector field, from any edge, where Q has been put, departs exactly one edge. A simple dimensional argument implies that  has no 0-valent vertices; similarly,  does not contain vertices, with exactly one edge landing or departing from it. Additionally, Lemma 6.9 from Section 6.3 implies that  cannot contain vertices of the first type with exactly one ingoing and one outgoing edge. In summary, a vertex of the first type, where a copy of Q has been put, has exactly one outgoing edge and at least two incoming edges. One can prove inductively with respect to the number of vertices that such a vertex has exactly two incoming edges, one of which comes from another vertex of the first type, where Q has been put, while the other one comes from the vertex of the first type, where ˛ has been put. Thus, a general graph  2 GnC1;m , contributing (possibly) non-trivially to (5.2), is a wheeled tree, i.e., there is a chosen vertex c of the first type, and a partition of f1; : : : ; ng into k disjoint subsets, such that from c depart m edges, joining c to the m vertices of the second type of , and such that to c are attached, by means of outgoing directed edges, k wheels, the i -th wheel having exactly li vertices (of the first type) (see Figure 10). For a wheeled tree  in GnC1;m , associated to k wheels, whose length is li , i D P 1; : : : ; k, and kiD1 li D n, we denote by †li , i D 1; : : : ; k, resp. Am , the i -th wheel with li vertices, resp. the graph with exactly one vertex of the first type and m vertices of the second type, and m edges, whose directions and targets are obvious (see Figure 11).

9 The explicit form of UQ

72





1N

2N

m x

Figure 10. A general wheeled tree.

†li

Am  1N

2N

m x

Figure 11. The graph Am and a wheel †li .

Lemma 9.1. For any positive integer m  1, the identity WAm D

1 mŠ

holds true. C . A Sketch of proof. The configuration space corresponding to the graph Am is C1;m direct computation using the explicit form of Kontsevich’s angle function shows that the assignment C C1;m ! 4m ;

Œ.z1 ; q1 ; : : : ; qm / 7! .'.z; q1 /; : : : ; '.z; qm //;

C is an orientation-preserving diffeomorphism from C1;m to the closed m-simplex 4m . Thus, in particular, the weight we want to compute is exactly the volume of the closed m-simplex, whence the claim follows. 

9.2 UQ as a contraction

73

Lemma 9.2. If l is an odd integer, then W†l vanishes. Sketch of the proof. All vertices of the wheel †l are of the first type: the corresponding C configuration space is Cl;0 . The action of G2 permits us to fix the central vertex of the wheel to i (this corresponds to a local trivialization of the principal G2 -bundle C C Conf C ! Cl;0 ): in this setting, Cl;0 equals the (compactified) configuration space of l;0 l  1 points of the complex upper half-plane, which do not coincide with i. Then the C involution z 7! zN on the complex upper half-plane extends to an involution of Cl;0 , C which changes the sign of the integrand and preserves the orientation of Cl;0 , since l  1 is even. 

9.2 UQ as a contraction By Lemma 9.2 we are concerned only with wheeled trees whose wheels have an even number of vertices. In order to compute explicitly the weight of such a wheeled tree C  in GnC1;m , we use the action of G2 on CnC1;m to put the central vertex of  in i, similarly to what was done in Lemma 9.2. Denoting by Cx the compactification of C CnC1;m , where one point of the first type has been put in i, the weight of  can be rewritten as Z n

V

V

n m V d'gi ^ d'ei ^ d'fi ; W D x C

iD1

j D1

kD1

where the big wedge products are ordered according to the indices, i.e., n V iD1

d'gi D d'g1 ^    ^ d'gn

and so on. Further, the notation is as follows: gi , resp. ej , resp. fk , denotes the only edge outgoing from the i -th vertex of the first type (where the vertex labelled by i does not coincide with the central vertex c), resp. the edge connecting the central vertex c to the j -th vertex of the first type, resp. the edge connecting the central vertex c to the k-th vertex of the second type. At this point, we may use the fact that there is an action of the permutation group C Sn  SnC1 on CxnC1;m , where Sn contains all permutations which keep the point of the first type corresponding to the central vertex c of  fixed. We choose a permutation

in such a way that the weight of  takes the form Z n

V

V

n m V W D d'g.i / ^ d'e .i / ^ d'pi : x C

iD1

j D1

kD1

The permutation is chosen so that for each wheel †l of  the i -th vertex of †l has the only outgoing edge g .i/ and the two incoming edges g.i1/ (modulo the length

9 The explicit form of UQ

74

of the wheel) and e .i/ . After reordering of the differential forms, the weight of  can be finally rewritten as P Z l1

l1 lp lq V V 1p