Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields

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by  Christine Jost

GLOBALIZING L -AUTOMORPHISMS OF THE
∞
SCHOUTEN ALGEBRA OF POLYVECTOR FIELDS
CHRISTINEJOST
Abstract. Recently, Willwacher showed that the Grothendieck-Teichmu¨ller
2 groupGRTactsbyL∞-automorphismsontheSchoutenalgebraofpolyvector
1 fieldsTpoly(Rd)onaffinespaceRd. Inthisarticle,weprovethatalargeclass
0 of L∞-automorphisms on Tpoly(Rd), including Willwacher’s, can be global-
2 ized. Thatis,givenanL∞-automorphismofTpoly(Rd)andageneralsmooth
manifold M with the choice of a torsion-free connection, we give an explicit
c
e construction of an L∞-automorphism of the Schouten algebra Tpoly(M) on
D the manifoldM, depending on the chosen connection. The method weuseis
theFedosovtrick.

4

]
A
1. Introduction
Q
The Grothendieck-Teichmu¨ller groupis a mysterious groupdefined by Drinfel’d
.
h
in his study of associators and deformation of Lie algebras [5]. In recent years,
t
a connections to objects studied in deformation quantization have become more and
m
more apparent.
[ In [14] Willwacher proves among other things that the (graded version of the)

Grothendieck-Teichmu¨ller group GRT acts on the Schouten algebra of polyvector
2
fieldsT (Rd)onaffinespace. Inthisarticle,weextendWillwacher’sresulttothe
v poly
2 SchoutenalgebraofpolyvectorfieldsTpoly(M)onageneralsmoothmanifoldM. In
9 fact,weprovethatanyL -automorphismofT (Rd)satisfyingcertainconditions
∞ poly
3
can be globalized. The method we use goes back to Dolgushev’s globalization [4]
1
of Kontsevich’s deformation quantization [8]. Dolgushev’s result in turn uses the
.
1 famous Fedosov trick [6]. Similar methods have also been used in [3] and [2]. The
0
main result is the following:
2
1 MainTheorem. LetF beanL -automorphism oftheSchoutenalgebraT (Rd)
∞ poly
:
v on affine space satisfying the following conditions:
Xi (1) F extends to an L∞-automorphism of Tpoly(Rdformal), the Schouten algebra on
the “fat point” Rd =Spec R[[x ,...,x ]].
r formal 1 d
a (2) The extension of F to T (Rd ) is invariant under linear change of coor-
poly formal
dinates of Rd .
formal
(3) F =id, and for n≥2, the n-ary part F of F vanishes on vector fields. That
1 n
means F (v ,...,v )=0 for vector fields v ,...,v .
n 1 n 1 n
(4) F vanishes if one of the inputs is a vector field that is linear in the standard
coordinates on Rd. That means F (γ ,...,Aixj ∂ ,...,γ ) = 0 for arbitrary
n 1 j ∂xi n
polyvector fields γ ,...,γ and a vector field Aixj ∂ .
1 n j ∂xi
Let M be a smooth manifold with a torsion-free connection. Then the construction
described in the proof yields an L -automorphism Fglob of the Schouten algebra
∞
T (M) on M, depending on F and the chosen connection.
poly
2000 Mathematics Subject Classification. 53D55,20B27,16E45.
Key words and phrases. deformation quantization, polyvector fields, Fedosov quantization,
Grothendieck-Teichmu¨ller group.
1
2 C.JOST
In[14]WillwacherconstructsaLiealgebraisomorphismfromtheGrothendieck-
Teichmu¨ller algebra grt to the zeroth cohomology H0(GC ) of Kontsevich’s graph
2
complex GC . The graph complex in turn acts on the Schouten algebra T (Rd)
2 poly
on affine space by L -automorphisms defined up to homotopy equivalence. Using
∞
our main theorem, we prove that these automorphisms can be used to construct
L -automorphismsof the Schouten algebraT (M) on any smooth manifold M.
∞ poly
The essential detail in Willwacher’s work is that any element of the Grothendieck-
Teichmu¨lleralgebracanbe representedby agraphallofwhoseverticesareatleast
trivalent. From this fact it follows that Willwacher’s automorphisms satisfy the
third and forth condition in our main theorem.
Corollary. TheGrothendieck-Teichmu¨ller groupGRTactsontheSchoutenalgebra
T (M) on a general smooth manifold M by L -automorphisms defined up to
poly ∞
homotopyequivalence. Theactiondependsonthechoiceofatorsion-freeconnection
on M.
Note that both the main theorem and its corollary depend on the choice of a
connection on M. The ideas in Appendix C of [1] might be used to prove that
the homotopy class of the L -automorphismdoes not depend on the choice of the
∞
connection.
We give a short outline of the content of this article: In Section 2, we construct
thebasicobjectsusedfortheglobalization,verticalpolyvectorfieldsanddifferential
forms with values in them. In Section 3, we construct a non-trivial section of the
vectorbundlewhosesectionsaredifferentialformswithvaluesinverticalpolyvector
fields. TheconstructiondependsonthechoiceofaconnectiononM. Thisso-called
Fedosov trick yields a resolution of T (M) as a Lie algebra. In Section 4, we
poly
prove the main theorem constructing an L -automorphism of T (M) from an
∞ poly
automorphism of T (Rd) using the Fedosov resolution. In the last section, we
poly
firstrecallWillwacher’sactionofGRT onT (Rd)andfinallyprovethe corollary,
poly
i.e., that this action can be globalized using our main theorem.
We use the Einstein summation convention. The degree of an element f of a
graded vector space is denoted by |f|.
2. Vertical polyvector fields
In this section, we presentthe basic constructionsneeded in this article,namely
vertical polyvector fields, differential forms with values in them and the vertical
Schouten bracket. The aim is to construct a large vector bundle on a smooth
manifoldM such thatthe fiber of eachpoint is isomorphic asa vectorspace to the
Schouten algebra T (Rd ) of the “fat point” Rd , the formal completion
poly formal formal
of affine space Rd at the origin. The reason we have to use Rd instead of Rd
formal
is to ensure convergence of certain recurrence equations later. We will often work
with local trivializations of the different vector bundles we consider.
Let M be a d-dimensional smooth real manifold. We start by recalling the
definition of the usual Schouten algebra T (M) of polyvector fields on M. The
poly
d q
Schouten algebra is defined as the exterior algebra T of the tangent
q=0 M
bundle T of M. The usual Lie bracket on vector fields extends by the graded
M
L V
Leibniz rule to an odd graded Lie bracket on the exterior algebra, called Schouten
bracket. On a local chart U ⊂ M, the underlying graded R-vector space of the
Schouten algebra T (U) is isomorphic to C∞(U)[ϕ ,...,ϕ ], writing ϕ for ∂
poly 1 d i ∂xi
and with the grading |ϕ |=1. The Schouten bracket [−,−]S is on this local chart
i
given by the formula
∂f ∂g ∂g ∂f
[f,g]S =(−1)|f| +(−1)|f||g|+|g|
∂xi∂ϕ ∂xi∂ϕ
i i
GLOBALIZING L∞-AUTOMORPHISMS OF THE SCHOUTEN ALGEBRA 3
for f,g ∈T (U).
poly
We continue with the definition of vertical polyvector fields. Let TM be the
total space of the tangent bundle of M. The projection π : TM →M induces the
differential dπ : T → π∗T . We are interested in the kernel of dπ, a vector
TM M
bundle over TM. To describe it in a more detailed way, we observe that a local
chart of TM is isomorphic to an open subset of R2d. It is possible to choose the
local coordinate system on TM in such a way that the projection π to M is given
by xi 7→xi and yi 7→0. We obtain a local description of sections of ker dπ,
∂
Γ(U,ker dπ)={fi(x,y) ,fi(x,y) smooth}.
∂yi
Global sections of ker dπ are what is usually called verticalvector fields. However,
we will need a slightly different definition allowing formal power series in the ys.
Define the C∞(U)[[y1,...,yd]]-module of vertical vector fields on U ⊂M by
∂
Tvert(U)={ fi(x)yI ,fi(x) smooth}.
I ∂yi I
IX∈Nd
One checks that these glue together to a C∞(M)-module Tvert(M). Besides the
C∞(M)-module structure, Tvert(M) also is a C∞(M)[[y1,...,yd]]-module, where
C∞(M)[[y1,...,yd]] is glued together from the rings C∞(U)[[y1,...,yd]]. The
C∞(M)-module of vertical polyvector fields is then defined as the exterior alge-
bra of Tvert(M) over C∞(M)[[y1,...,yd]], i.e.,
Tvert(M)= d q Tvert(M).
poly q=0 C∞(M)[[y1,...,yd]]
Furthermore, differential formsLwith Vvalues in vertical polyvector fields are defined
by
Ω(M,Tpvoelryt(M))=Tpvoelryt(M)⊗C∞(M)ΩM,
where ΩM denotes as usual the de Rham algebra of differential forms on M. We
workonalocalchartU ofTM,denotingcoordinatefunctionsbyx1,...,xd,y1,...,yd
as before. Writing ∂ =ψ and dxi =ηi, differential forms with values in vertical
∂yi i
polyvector fields are then elements of
C∞(U)[[y1,...,yd]][ψ ,...,ψ ,η1,...,ηd].
1 d
Settingthedegrees|yi|=0and|ψ |=|ηi|=1,weobtainagradingonΩ(M,Tvert(M)).
i poly
Denote by Ωr(M,TvertM)) the subspace of elements of degree r. Put in another
poly
way, we have
Ωr(M,TvertM))= Tvert,p⊗Γ( q(T∗M)),
poly p+q=r poly
where the elements of Tvert,p haveLdegree p with respVect to the ∂ . Define the
poly ∂yi
vertical Schouten bracket on Ω(M,Tvert(M)) locally by
poly
∂f ∂g ∂g ∂f
[f,g]vert =(−1)|f| +(−1)|f||g|+|g|
∂yi∂ψ ∂yi∂ψ
i i
for f,g ∈ Ω(U,Tvert(U)). One checks that the definition is independent on the
poly
choice of the local chart. As the Schouten bracket, the vertical Schouten bracket
hasdegree-1andprovidesΩ(M,Tvert(M))withthestructureofanoddgradedLie
poly
algebra.
We also introduce the sub-C∞(M)-module Tvert| (M) of vertical polyvector
poly y=0
fields which are constant with respect to the y. On a local chart U, we have that
Tvert| (U) is isomorphic as an C∞(U)-module to C∞(U)[ψ ,...,ψ ]. One sees
poly y=0 1 d
easily that Tvert| (M) and T (M) are isomorphic as C∞(M)-modules.
poly y=0 poly
4 C.JOST
We conclude this sectionwiththe firstlemma onverticalpolyvectorfield, show-
ing that Ω(M,Tvert(M)) is a resolution of T (M) as a C∞(M)-module. Define
poly poly
δ :Ωr(M,Tvert(M))→Ωr+1(M,Tvert(M))
poly poly
locally by δ =dxi ∂ . Furthermore, define a contracting homotopy
∂yi
δ∗ :Ωr(M,Tvert(M))→Ωr−1(M,Tvert(M))
poly poly
by
1 ∂
δ∗(f (x,ψ)yIηJ)= ya f(x,ψ)yIηJ,
I,J p+q ∂ηa
where I and J are multi-indices of total degree p and q, respectively. For f =
f(x,ψ)y0η0, we set δ∗(f)=0. Furthermore, define the projection
σ :Ω(M,Tvert(M))→Tvert| (M)⊂Ω(M,Tvert(M))
poly poly y=0 poly
locally by σ(ηi)=σ(yi)=0 and σ(ψ )=ψ .
i i
Lemma 1. For any f ∈Ω(M,Tvert(M)), it holds that
poly
(1) f =σf +δδ∗f +δ∗δf,
hence
(i) Hn(Ω(M,Tvert(M)),δ))=0 for n6=0,
poly
(ii) H0(Ω(M,Tvert(M)),δ)∼=T (M).
poly poly
Proof. Equation (1) is easy to check. The remainder of the lemma follows because
Tvert| (M) and T (M) are isomorphic as C∞(M)-modules. (cid:3)
poly y=0 poly
3. The Fedosov resolution
It is possible to see Ω(M,Tvert(M)) as sections of a vector bundle over the
poly
total space of a vector bundle whose sections form Tvert| (M). In the previous
poly y=0
lemma, we have showed that the zero section of that bundle yields a resolution of
Tpoly(M)∼=Tpvoelryt|y=0(M) asa C∞(M)-module. Now wewill constructa verynon-
trivial section that yields a resolutionof T (M) as a Lie algebra. It is called the
poly
Fedosovresolution,asit uses the Fedosovtrick[6]of usinga differentialdepending
onthechoiceofatorsion-freeconnection. BesidesFedosov’swork,wealsorelyvery
much on Dolgushev’s [4]. The first two lemmas are taken directly from [4] and are
included mostly for self-containedness.
We will work on a local chart of M from now on. All local formulas are inde-
pendent of the choice of the chart, if not said otherwise.
Choose a torsion-free connection on the manifold M and denote its Christoffel
symbols by Γk(x). Define a derivation∇ onΩ(M,Tvert(M)) as ∇=d+[Γ,−]vert,
ij poly
where d = dxi ∂ and Γ = −dxiΓk(x)yj ∂ . This derivation is not a differential.
∂xi ij ∂yk
However, it holds that
• ∇2 = [R,−]vert, where R = −1dxidxjRl (x)yk ∂ is given by the Rie-
2 kij ∂yl
mann curvature tensor of the connection, and
• δ∇+∇δ =0.
Lemma 2. There exists an element A in Ω(M,Tvert(M)) such that δ∗A=0 and
poly
D :=∇−δ+[A,−]vert
is a differential, where A has the form A= ∞ dxkAj (x)yi1...yip ∂ .
p=2 k,i1...ip ∂yj
This and the following lemma use the samePtechnique of proof, which we explain
in detail at first and use in a more sketchy way later. The main idea is to use the
contracting homotopy equation (1) to produce a recursive equation.
GLOBALIZING L∞-AUTOMORPHISMS OF THE SCHOUTEN ALGEBRA 5
Proof. We have to show that D2 =0 or, equivalently, that
1
(2) R+∇A+ [A,A]vert =δA.
2
As we are looking for an A such that δ∗A=σA=0, we insert these identities into
the contracting homotopy equation (1) and get that A should suffice A = δ∗δA.
Inserting the left hand side of (2) for δA yields the following recursion formula for
A:
1
A=δ∗R+δ∗(∇A+ [A,A]vert).
2
The recursion converges because δ∗ increases the degree in the ys.
WestillhavetoprovethatAactuallysatisfiesC :=R+∇A+1[A,A]vert−δA=0.
2
ObservethatitfollowsfromtheBianchiidentitiesfortheRiemanncurvaturetensor
that δR=∇R=0. Using this, we get that δC =∇C+[A,C]vert. Furthermore,it
holds that σC = δ∗C = 0. Inserting these equations for δC, δ∗C and σC into the
contracting homotopy equation (1), we get the recursive equation
C =δ∗(∇C+[A,C]vert).
As δ∗ increases the degree in the ys, this equation has the unique solution C = 0,
which concludes the proof. (cid:3)
We are now ready to show that the differential D that we have just constructed
still yields a resolution of T (M) as a C∞(M)-module.
poly
Lemma 3. (i) Hn(Ω(M,Tvert(M)),D)=0 for n6=0
poly
(ii) H0(Ω(M,Tvert(M)),D)∼=T (M)
poly poly
Proof. We start by proving part (i). Let f be a cocycle of degree ≥ 1, i.e., f ∈
Ωr(M,Tvert(M)), r ≥ 1, Df = 0. We need to find a g ∈ Ωr−1(M,Tvert(M)) such
poly poly
that g lies the image of f, i.e., Dg = f. We restrict our search to such g that
satisfy σg = δ∗g = 0. For these g it holds that f = Dg = ∇g −δg +[A,g]vert.
Furthermore, by the contracting homotopy equation (1), it holds that g = δ∗δg.
Inserting, we get the recursive equation
g =−δ∗f +δ∗(∇g+[A,g]vert).
The convergence follows as usual from the fact that δ∗ increases the degree in the
ys.
Weshowthatinfacth:=Dg−f =0. Oneseesthatδ∗h=σh=0andDh=0.
Hence we get the same recursion equation as for C in the foregoing lemma,
h=δ∗(∇h+[A,h]vert),
which has the unique solution h=0.
Nowweprovepart(ii). Theaimisto findasuitable sectionτ :Tvert| (M)→
poly y=0
Ω(M,Tvert(M)). Let f ∈ Tvert| (M), find a unique f ∈ Ω0(M,Tvert(M)) such
poly 0 poly y=0 poly
that Df = 0 and σf = f . As f has degree 0 it follows that δ∗f = 0. Together
0
with Df =0 this yields the recursive equation
f =f +δ∗(∇f +[A,f]vert)
0
with convergence as usual.
We have to show that actually u:=Df =0. As σu=δ∗u=0 and Du=0,this
follows in the same way as for g in the first part of the proof. (cid:3)
AsfinalstepfortheFedosovresolution,weprovenowthattheFedosovresolution
of T (M) indeed is a resolution as Lie algebra.
poly
Lemma 4. The induced Lie algebra structure on T (M) induced by the vertical
poly
Schouten bracket on Ω(M,Tvert(M)) is isomorphic to the Schouten bracket.
poly
6 C.JOST
Proof. To simplify notation, we will identify T (M) and Tvert| (M) in this
poly poly y=0
proof. Especially, we will write σ for the composition
Ω(M,Tvert(M))−σ→Tvert(M)| −∼→T (M)
poly poly y=0 poly
and τ for the composition
Ω(M,Tvert(M))←τ−Tvert(M)| ←∼−T (M).
poly poly y=0 poly
We have to show that τ[f ,g ]S = [τf ,τg ]vert for f ,g ∈ T (M). By the
0 0 0 0 0 0 poly
definitionofτ intheproofofthepreviouslemma,thisisequivalenttoshowingthat
[σf,σg]S =σ[f,g]vert for f,g ∈Ω0(M,Tvert(M)) such that Df =Dg =0.
poly
From Df =0 it follows that
∂f ∂f ∂f
dxi =dxi −dxiΓk(x)ψ +dxi(terms containing y).
∂yi ∂xi ij k∂ψ
j
Hence,usingthatf liesinΩ0(M,Tvert(M)),wehaveσ(∂f )=σ(∂f )−σ(Γk(x)ψ ∂f ).
poly ∂yi ∂xi ij k∂ψj
From the explicit formula for the vertical Schouten bracket, we obtain
σ[f,g]vert =[σf,σg]S −σS,
where
∂f ∂g ∂g ∂f
S =(−1)|f|Γk(x)ψ +(−1)|f||g|+|g|Γk(x)ψ ,
ij k∂ψ ∂ψ ij k∂ψ ∂ψ
j i j i
which is zero because Γk(x) is symmetric in the lower indices. This proves the
ij
claim. (cid:3)
4. Proof of the main theorem
Inthis section,we describe a constructionthat globalizesL -automorphismsof
∞
the Schouten algebra if they satisfy certain conditions. The construction depends
onthe choice of a torsion-freeconnectiononthe manifold M. The theoremand its
proof are analogous to Proposition 3 in Dolgushev’s article [4].
MainTheorem. LetF beanL -automorphism oftheSchoutenalgebraT (Rd)
∞ poly
on affine space satisfying the following conditions:
(1) F extends to an L -automorphism of T (Rd ), the Schouten algebra on
∞ poly formal
the “fat point” Rd =Spec R[[x ,...,x ]].
formal 1 d
(2) The extension of F to T (Rd ) is invariant under linear change of coor-
poly formal
dinates of Rd .
formal
(3) F =id, and for n≥2, the n-ary part F of F vanishes on vector fields. That
1 n
means F (v ,...,v )=0 for vector fields v ,...,v .
n 1 n 1 n
(4) F vanishes if one of the inputs is a vector field that is linear in the standard
coordinates on Rd. That means F (γ ,...,Aixj ∂ ,...,γ ) = 0 for arbitrary
n 1 j ∂xi n
polyvector fields γ ,...,γ and a vector field Aixj ∂ .
1 n j ∂xi
Let M be a smooth manifold and choose a torsion-free connection on M. Then
the construction below yields an L -automorphism Fglob of the Schouten algebra
∞
T (M) on M, depending on F and the chosen connection.
poly
TheconstructionoftheglobalL -morphismconsistsofthreesteps. Atfirst,we
∞
construct an L -automorphism of Ω(M,Tvert(M)). In the second step, we twist
∞ poly
this morphism with a suitable Maurer-Cartan element to yield an L -morphism
∞
of Ω(M,Tvert(M)) commuting with the differential D. In the third step, this will
poly
induce an L -automorphism on H0(Ω(M,Tvert(M)))∼=T (M).
∞ poly poly
GLOBALIZING L∞-AUTOMORPHISMS OF THE SCHOUTEN ALGEBRA 7
First step. Choose an open chart U of the manifold M such that Ω(U,Tvert(U))
poly
trivializes to
C∞(U)[[y1,...,yd]][ψ ,...,ψ ,η1,...,ηd],
1 d
asdescribedinsection2. Becauseofcondition(1),there is anextensionofF toan
L -automorphism Fformal of
∞
Tpoly(Rdformal)∼=R[[x1,...,xd]][ϕ1,...,ϕd]∼=R[[y1,...,yd]][ψ1,...,ψd].
Any element of Ω(U,Tvert(U)) can be written as a (possibly infinite) linear combi-
poly
nation
f(x1,...,xd,η1,...,ηd)yIψ ,
J
I,XJ∈Nd
where f ∈ C∞(U)[η1,...,ηd]. By C∞(U)[η1,...,ηd]-linear continuation, Fformal
induces an L -automorphism Fvert| of Ω(U,Tvert(U)). By condition (2), the
∞ U poly
constructionisindependentofthechoiceofU. HenceityieldsanL -automorphism
∞
Fvert of Ω(M,Tvert(M)) with the vertical Schouten bracket.
poly
Second step. By construction, the automorphism Fvert commutes with the dif-
ferential d=dxi ∂ . In general, however, it does not commute with
∂xi
∂
D =d+[Γ−dxi +A,−]vert.
∂yi
We denote Γ−dxi ∂ +A by B and obtain the compact description D = d+
∂yi
[B,−]vert.
Beforecontinuingwiththeproof,werecallsomewell-knownfactsaboutMaurer-
Cartan elements. Let
Φ:(g,[−,−]g,dg)→(g′,[−,−]g′,dg′)
be an L -morphism of dg-Lie algebras and π a Maurer-Cartan element of g.
∞
Then π′ = ∞ 1Φ (πi) is a Maurer-Cartan element of g′. It follows that g =
i=1 i! i π
(Fgu,r[t−h,e−rm]go,rdePg,+it[hπo,l−ds]gt)haantdΦg′π′==ex(gp′(,−[−π,′)−◦]gΦ′,d◦ge′x+p([ππ′),−is]ga′n) aLre-dmgo-Lrpiehiaslmgebfrroams.
π ∞
g to g′ , where exp(π)(T)= ∞ 1πi·T and exp(−π′) is defined analogously.
π π′ i=0 i!
Weapplythisconstructiontooursituation,whereFvert| isanL -automorphism
U ∞
P
of(Ω(U,Tvert(U)),[−,−]vert,d)andB isaMaurer-Cartanelement. Fromcondition
poly
(3),itfollowsthat ∞ 1(Fvert| ) (Bi)=B,asB isaverticalvectorfield. Hence
i=1 i! U i
itfollowsthat (Fvert| ) =exp(−B)◦Fvert| ◦exp(B) is anL -automorphismof
U B U ∞
P
(Ω(U,Tvert(U)),[−,−]vert,D), because D =d+[B,−]vert.
poly
WehaveworkedonthelocalchartU sofar,astheformulaforB isdependenton
the choiceoflocalcoordinates. It willbecome clear,however,thatthe definition of
(Fvert| ) isnot. We analyzehow B transformsunder changeofcoordinates. The
U B
termsdxi ∂ andAareinvariantunderchangeofcoordinates. Thetransformation
∂yi
ofΓismorecomplicatedduetothepresenceoftheChristoffelsymbols. Wecompute
that B transforms as B′ = B+dxiHk(x)yj ∂ for some Hk(x), where the exact
ij ∂yk ij
form of Hk(x) is not important.1 We take a closer look at the explicit formula
ij
for (Fvert| ) . It holds that (Fvert| ) = ∞ 1(Fvert| ) (Bi−). However,
U B U B,n i=1 i! U n+i
Fvert| is zero on any summand of the form dxiHk(x)yj ∂ by condition (4).
U P ij ∂yk
Hence the definition of (Fvert| ) is independent of the choice of U and thus glues
U B
to an L -automorphisms Fvert,D of Ω(M,Tvert(M)) commuting with D.
∞ poly
1Thisstepistakendirectlyfrom[4],seeEquation(58).
8 C.JOST
Third step. As Fvert,D constructed in the last step commutes with the differ-
ential D of Ω(M,Tvert(M)), it induces an L -automorphism on cohomology. As
poly ∞
H0(Ω(M,Tvert(M)))∼=T (M),ithencefollowsthatthisisanL -automorphism
poly poly ∞
Fglob on T (M) depending on F and the choice of a connection from which the
poly
differential D is constructed. This is the last step of the construction of the global
L -automorphism Fglob. As we have checked that each step of the construction
∞
works if the given automorphism F satisfies the conditions given in the theorem,
this also concludes the proof of the main theorem.
5. Proof of the corollary
We start by stating the theorem of Willwacher’s that we use.
Theorem 1. (Willwacher [14]) The Grothendieck-Teichmu¨ller group GRT acts on
the Schouten algebra T (Rd) on affine space by L -automorphisms.
poly ∞
Aswewillneedsomedetailsfromtheproofofthistheoremlater,weincludethe
proof here. The way we present it here is analogous to Section 3 in [12]. We start
by recalling the definitions and constructions needed for the proof: the Lie algebra
associatedto anoperad, Kontsevich’sgraphcomplex, and the Chevalley-Eilenberg
cochain complex.
Let P = {P(n)}n∈N be an operad with partial composition ◦i. Following [7],
define a Lie algebra structure on the graded vector space ∞ P(n) by
n=1
m n
Q
[µ,ν]= µ◦ ν−(−1)|µ||ν| ν◦ µ
i i
i=1 i=1
X X
for µ ∈ P(m) and ν ∈ P(n). This Lie bracket induces a Lie algebra structure on
the space ofS-coinvariants ∞ P(k) . As we workin characteristic0, the space
k=1 Sk
of S-coinvariantsand the space of S-invariants are isomorphic as vector spaces. By
thisisomorphism,theLiebrQacketon ∞ P(n)alsoinducesaLiealgebrastructure
n=1
on the space of S-invariants ∞ P(k)Sk.
We continue by defining thek=op1eraQd Gra and graph complex GC . Let gra be
2 n,k
Q
the set of all undirected graphs with n numbered vertices and k numbered edges.
The symmetric group S acts on gra by renumbering of the edges. Let sgn be
k n,k k
the sign representation of S . Then define
k
Gra(n)= Rhgra i⊗ sgn [k].
n,k Sk k
k≥0
M(cid:0) (cid:1)
The vector spaces Gra(n) admit an action of S by renumbering of the vertices.
n
Furthermore, there exists a partial composition product: For Γ in Gra(m) and Γ
1 2
in Gra(n) define Γ ◦ Γ in Gra(m+n−1)by inserting Γ for the i-thvertex of Γ
1 i 2 2 1
andthensummingupallpossiblereconnectionstoΓ ofedgespreviouslyconnected
2
to the i-th vertex of Γ . The edges are renumbered in a way that puts the edges
1
of Γ at the end while otherwise keeping the ordering of the edges. For details of
2
the definition, we refer to [14] and [12]. With the S-action and partialcomposition
just defined, Gra is an operad.
The S-invariants of Gra can be seen as graphs with “unidentifiable”, i.e., no
longer numbered, vertices. As explained above, the vector space of S-invariants
S
obtains the structure of a Lie algebra, as does the shifted space Gra{−2} , where
Gra{−2}(n) = Gra(n)[2−2n]. Willwacher denotes this Lie algebra Gra{−2}S of
graphs with unidentifiable vertices by fGC . The subset of graphs whose vertices
2
are at least trivalent forms a sub-Lie algebra denoted by GC . One checks that
2
the graph ∈fGC satisfies the Maurer-Cartanequation [, ]=0. Hence it equips
2
fGC with the differential [,−]. Furthermore, as provedin e.g. [14], GC is closed
2 2
GLOBALIZING L∞-AUTOMORPHISMS OF THE SCHOUTEN ALGEBRA 9
with respect to the differential. We have hence constructed a differential graded
Lie algebra (GC ,[−,−],[,−]). Willwacher’s main result in [14] is that the zeroth
2
cohomology of this complex is isomorphic as a Lie algebra to the Grothendieck-
Teichmu¨ller algebra: H0(GC )∼=grt.
2
Let now g be a graded vector space. We define CE(g), the Chevalley-Eilenberg
cochain complex of g. Its cochains are given by
CE(g)= Hom(Symng[1],g[1])∼= Hom( ng,g)[1−n]∼=CoDer(Sym g[1]),
n n
Y Y V
where Sym denotes the symmetric algebra and CoDer the space of coderivations.
TheChevalley-EilenbergcochainsformagradedLiealgebrawiththegradedNijenhuis-
Richardson bracket [−,−] . It is defined by all possible insertions of homomor-
NR
phismsintoeachother,see[13]and[9]. Observethata1-cochainπ ∈Hom0(g∧g,g)
g
in the Chevalley-Eilenberg complex is a Lie algebra structure on g if and only if it
satisfiesthe equation[π ,π ] . Furthermore,anyLie bracketπ ong yieldsadif-
g g NR g
ferentiald =[π ,−] onCE(g). Thiscochaincomplexwiththenewdifferential
πg g NR
d is called the Chevalley-Eilenberg cochain complex of the Lie algebra (g,π ).
πg g
We continue by stating some well-known facts that are treated in more detail
in e.g. [11], [15] or the book [10]. Given a zero-cocycle γ, i.e., γ ∈ CE0(g) =
CoDer0(Sym g[1]) and d (γ)=0, its exponential exp(γ)= ∞ 1γn is an L -
πg n=1 n! ∞
automorphismofg. Callzero-cocyclesγ,γ′ gauge-equivalentifγ−γ′ isacobound-
ary. Awell-knowntheoremstatesthatifγ andγ′ aregauge-ePquivalentthenexp(γ)
and exp(γ′) are homotopy-equivalent. Hence the zero-cohomologyH0(CE(g),d )
πg
of the Chevalley-Eilenberg cochain complex of the Lie algebra (g,π ) acts on g by
g
L -automorphisms modulo homotopy equivalence.
∞
Proof of Theorem 1. We start with an outline of the proof. In [14], Willwacher
showsthatthe Grothendieck-Teichmu¨lleralgebragrtisisomorphicasaLie algebra
to the zeroth cohomology H0(GC ) of the Kontsevich graph complex GC . Fur-
2 2
thermore, as we have stated above, H0CE(T (Rd)[1]) acts on T (Rd)[1] by
poly poly
L -automorphisms modulo homotopy equivalence. Hence it suffices to prove that
∞
there is a morphism of Lie algebras H0(GC ) → H0(CE(T (Rd)[1]), because it
2 poly
willfollowthatgrt∼=H0(GC )actsonT (Rd)[1]byL -automorphismsmodulo
2 poly ∞
homotopy equivalence. Observe that we have to shift T (Rd) by 1 to turn the
poly
odd Lie algebra into a usual graded Lie algebra.
The plan is to construct a Lie algebra morphism
GC →CE(T (Rd)[1])
2 poly
commuting with the differentials [,−] of GC and [π ,−] of CE(T (Rd)[1]),
2 S NR poly
where π denotes the shifted Schouten bracket on T (Rd)[1]. Recall that GC
S poly 2
is a subalgebra of the Lie algebra fGC of S-invariants of the operad Gra. We will
2
identify CE(T (Rd)[1]) with the Lie algebra of S-invariants of another operad,
poly
EndTpoly(Rd)[2]. An operad morphism between the two operads will induce a Lie
algebra morphism between fGC and CE(T (Rd)[1]), which restricts to GC .
2 poly 2
ThisLiealgebramorphismwillnotcommutewiththedifferentials,butthestandard
trick of twisting with a Maurer-Cartan element will remedy that. As said before,
the existence of such a morphism proves the theorem.
The endomorphism operad End of a graded vector space A is defined by
A
End (n) = Hom(A⊗n,A), where partial composition is the insertion of endomor-
A
phisms into each other. The Lie algebra of S-invariants of End is the Chevalley-
A
Eilenberg cochain complex CE(A[−1]) of A[−1] with the Nijenhuis-Richardson
bracket. HenceCE(Tpoly(Rd)[1])istheLiealgebraofS-invariantsofEndTpoly(Rd)[2].
10 C.JOST
We constructanoperadmapGra→EndTpoly(Rd)[2] bymappinga graphΓ tothe
endomorphism
d
(3) Φ (γ [2],...,γ [2])= µ◦ ∆ (γ ⊗...⊗γ ) [2],
Γ 1 n  (i,j) 1 n 
(i,j)Y∈E(Γ)Xk=1
 
where ∆ is defined by
(i,j)
∂ ∂ ∂ ∂
∆ = + .
(i,j) ∂xk ∂ψ(i) ∂ψ(j) ∂xk
(j) k k (i)
Here,µismultiplicationofpolyvectorfields, ∂ and ∂ aredifferentiationwith
∂xk(j) ∂ψk(j)
respect to xk or ψ , respectively, in the j-th argument, and is concatenation of
k
differentialoperators,wheretheorderfollowstheorderingoftheedges. Onechecks
Q
that the mapin fact respects operadic composition. Hence it induces a Lie algebra
morphism fGC →CE(T (Rd)[1]) which restricts to the subalgebra GC .
2 poly 2
Recall that CE(T (Rd)[1]) is equipped with the differential [π ,−] , where
poly S NR
π denotes the shifted Schouten bracket, and GC with the differential [,−]. The
S 2
just constructed Lie algebra morphism GC → CE(T (Rd)[1]) does not repect
2 poly
these differentials, only the zero differential. However, it maps the Maurer-Cartan
element to the Schouten bracket. Hence twisting the morphism with yields a
dg-Lie algebra morphism compatible with both differentials. This concludes the
proof. (cid:3)
Corollary. TheGrothendieck-Teichmu¨ller groupGRTactsontheSchoutenalgebra
T (M) on a general smooth manifold M by L -automorphisms defined up to
poly ∞
homotopyequivalence. Theactiondependsonthechoiceofatorsion-freeconnection
on M.
Proof. WeneedtocheckthattheL -automorphismsintheimageofthemorphism
∞
grt∼=H0(GC )→H0(CE(T (Rd)[1])→Aut(T (Rd)[1])/∼
2 poly poly
satisfy the four conditions of the main theorem. Recall that we construct a mor-
phism from the graph complex GC to the Chevalley-Eilenberg cochain complex
2
CE(T (Rd)[1]). TheimageinCE0(T (Rd)[1])ofadegreezeroelementΓ∈GC
poly poly 2
is a degree zero map Ψ : Symn(T (Rd)[2]) → T (Rd)[2]. We have to show
Γ poly poly
thatitsexponentialsatisfiesthe fourconditionsofthemaintheorem. Observethat
it suffices to show that Ψ satisfies the conditions. The important fact is that the
Γ
graphs in GC have the property that each vertex is at least trivalent.
2
Condition1. Theformula(3)worksforγ ,...,γ inbothT (Rd)andT (Rd ).
1 n poly poly formal
ItfollowsthatthewholeconstructionofΨ runsthroughforT (Rd )aswell.
Γ poly formal
Condition 2. Because ∆ is invariant under linear change of coordinates of
(i,j)
Rd , this also holds for Φ and Ψ .
formal Γ Γ
Condition 3. It suffices to show that Φ as defined in (3) vanishes on vector fields
Γ
for any graph Γ ∈ Gra. Denote by D(Γ) the set of all directed graphs that can be
obtainedbygivingadirectiontoeachedgeinΓ. Wecanthenrewritethedefinition
of Φ as
Γ
d
∂ ∂
Φ (γ [2],...,γ [2])= µ◦ (γ ⊗...⊗γ ) [2].
Γ 1 n  Γ′∈XD(Γ)(i,j)Y∈E(Γ′)kX=1 ∂xk(j) ∂ψk(i)! 1 n 
As the vertices of Γ are at least trivalent, for any directed Γ′ in D(Γ) there is a
vertex l that has at least two outgoing edges. This means that we differentiate γ
l