A topological interpretation of the KZ system

0.34 MB·English

by  Eduard Looijenga

A TOPOLOGICAL INTERPRETATION OF THE KZ SYSTEM
EDUARDLOOIJENGA
1
1
0
ABSTRACT. Weshowthat theKZsystemhasa topologicalinterpretation
2
in the sense that it may be understood as a variation of complex mixed

n Hodge structure whose successive pure weight quotients are polarized.
a ThisinasensecompletesandelucidatesworkofSchechtman-Varchenko
J
done intheearly1990’s. Acentralingredientisanewrealizationofthe

1 irreduciblehighestweightrepresentationsofaLiealgebraofKac-Moody
1
type, namely on an algebra of rational polydifferentials on a countable

product of Riemann spheres. We also obtain the kind of properties that
]
A in the sl(2) case are due to Ramadas and are then known to imply the
unitarityoftheWZWsystemingenuszero.
Q
.
h
t
a
m
INTRODUCTION
[
We construct an identification of a given KZ-space with a space con-

6
structed out of cohomology with supports of a rank one local system. The
v
3 latter space is topologically defined, but depends on the choice of n dis-
3 tinctpoints on the affine line. By letting these points move,we getover the
0 parameter space of such n-tuples a (Gauß-Manin) connection on the triv-
2
ial bundle with fiber the KZ-space and we show that this connection gets
.
3
identified with the KZ-connection. In a sense this completes earlier work
0
of Schechtman-Varchenko, who constructed a flat map from the KZ-system
0
1 to a Gauß-Manin system for the ordinary cohomology (with no supports).
: Theirmapfactorsthroughthemapconstructedhere,buttheywerenotable
v
i to say much about how nontrivial that map might be in any given case (a
X
priori it could be the zero map), whereas ours is always an isomorphism.
r
a We should perhaps emphasize that the topological interpretation tells us a
great deal about the monodromy of the KZ-system that may be hard to ob-
tain otherwise: the fundamental group of the base is a colored braid group
and we can see it act on the cohomology of a rank one local system over
a space that is naturally attached to n distinct points on the complex line.
This space is like apartial Deligne-Mumfordcompactification of the moduli
space of genus zero curves that are endowed with 1+n+m punctures for
some m of which the first 1+n ones are fixed and the others freely move.
We do not work this out here, but we feel that a closer examination of this
representationis not onlyfeasible, butalso desirable.
2010MathematicsSubjectClassification. Primary32G34;Secondary14D07.
Keywordsandphrases. KZsystem,polydifferentials,highestweightmodule.
1
2 EDUARDLOOIJENGA
Our approach differs from earlier work in an other respect by putting at
thecenterabialgebraofrationalpolydifferentialsonaniteratedself-product
of a Riemann sphere (these are like ordinary differential forms, exceptthat
thisalgebra,ratherthanobeyingaKoszulrule,isplainlycommutative). We
showthatifthenumberoffactorsiscountablyinfinite,thensuchanalgebra
receivesalltheirreduciblehighestweightrepresentationsofallLiealgebras
of Kac-Moody type, a fact that may have an interest in its own right. This
makesfor a smooth passage from the KZsystem to the Gauß-Manin system
and yields precious additional information. Thus the somewhat elusive KZ
systemisannexedbyalgebraicgeometry,whichinthepresentsettingmeans
thatthepowerfultoolsofmixedHodgetheorybecomeavailabletoferretout
anyfinerstructure. Thestrengthofthisapproachmanifestsitselfalsointhe
algebraic domain, witness the new vanishing property Theorem 2.14 and
the remarkableinvarianceproperty Theorem2.15.
Weexpectthisset-uptobeparticularlyusefulinestablishing whetherthe
WZW subsystem of the KZ-system has a flat unitary structure, as is con-
jectured by physicists and for which the evidence so far is limited to the
case sl(2) (due to Ramadas [7], see also our version [6]). We characterize
the WZW system in terms of the a simple vanishing property Corollary 4.2
that generalizes the one found by Ramadas. (Indeed, as is shown in [6]
this resulttogether with the vanishing theoremandthe invarianceproperty
mentionedabove suffice to obtain the unitarity for the sl(2)-case.)
While this paper may help to resolve one mystery, it creates another and
thatisthelackofaconceptualexplanationfortheappearanceofpolydiffer-
entials. This is not the first place where these do occur in this context: they
alsodoinapaperofStoyanovski-Feigin[9]andinonebyBeilinson-Drinfeld
[1] (both are mainly concerned with the WZW system) and our feeling is
that especiallythe first ofthese mightbe linkedto the presentpaper.
Although this article owes much to earlier work done in this area, the
chosenapproachalmostdemandsittobeself-contained. Forthisreasonwe
havegivenacompleteproofthatthevectorbundlemapbetweenthevector
bundlesunderlyingthe KZsystem to the Gauß-Manin system isflat,despite
thefactthatthepatternofthisproofisquitesimilarto(andindeed,inspired
by) the one ofSchechtman-Varchenko[8].
Here is brief review of the separate sections. The first section is prepara-
tory in nature: we introduce here the basic algebraof polydifferentials that
isatthecenterofthispaperandderivesomeofitsproperties. Thisisusedin
Section 2 to show that this algebra contains the highest weight representa-
tionsofaLie algebradefinedbyageneralizedCartan matrix. We alsomake
the transition of its weight spaces to spaces of twisted logarithmic forms.
In Section 3 we use the Casimir operator in order to define a Gauß-Manin
connectionandweshowitscompatibility withtheKZconnection: the main
result in this direction, Theorem 3.6, as well as the version that yields a
genuine topological interpretation of the KZ system, Theorem 3.7, in terms
POLYDIFFERENTIALSANDTHEKZSYSTEM 3
of cohomology with supports are stated here. As the underlying computa-
tions are somewhat delicate, we decided to write these out in full. We state
their core as an operator formula, but in order not to interrupt the flow of
arguments, we relegated the proof to an appendix. Our motivation was a
desire to better understand the WZW-systems in genus zero as subsystems
oftheKZ-systemsandSection4testifiestothat. Wecharacterizethissystem
in terms of our polydifferentials and Conjecture 4.3 links them to a square
integrability property. If true, then this should realize a WZW-system as a
flat subbundle of polarized variation of Hodge structure which is of pure
bidegree (m,0) for some m (so that this part is not really varying after all!)
andwouldgivetheflatunitarystructureonthissystem thatphysicists since
longconjectureto exist.
Conventions. For any finite sequence I = (i ,i ,...,i ) taken from an
N N−1 1
(index) set I, we denote by |I| := N its length, by {I} := {i ,...,i } ⊂ I
N 1
the set of its terms and by I∗ := (i ,i ,...,i ) the opposite sequence. If
1 2 N
(f ) is an indexed collectionof elementsof an algebrawith unit, then f
i i∈I I
denotesthenoncommutativemonomialf ...f (understoodtobe1ifI =
iN i1
∅). Similarly,if (f ) is an indexed collectionof elementsof a Lie algebra,
i i∈I
then [f ] stands for the iterated Lie bracket ad(f )ad(f )···ad(f )(f )
I iN ki−1 i2 i1
(read f when I = (i) and zero when I = ∅). But if for a subset X ⊂ I,
i
(c ) is afinite collectionelementsof an abelian group(usually complex
x x∈X
numbers or elements of a vector space), then c is sometimes used as an
X
abbreviation for c by c . If t is a coordinate on a Riemann surface
x∈X x X i
C (i∈ I),sothat(t ) isasetofcoordinatesontheproduct C ,then
i P i i∈I i∈I i
anyfinitesubsetX ⊂ I definesalocuswhereallt with x ∈ X areequalto
x
Q
eachother; the resulting coordinate on that locusis the denoted by t (the
X
remaining coordinates are (t ) ). Strictly speaking these conventions
i i∈I−X
clash,butin practice confusionis unlikely.
The permutation groupof aset X is denoted by S(X).
1. SOME OPERATORS IN AN ALGEBRA OF POLYDIFFERENTIALS
We introduce an algebraof logarithmic polydifferentials which looks like
ashuffle algebraandidentify in it certain operatorsof interest.
Polydifferentials. We consider polydifferentials on CI, where C is a non-
singular complex curve and I is a finite set. A polydifferential of degree d
lookslikead-formonCI,butshouldnotbeconfusedwithitasdifferentials
comingfrom differentfactors commuterather than anti-commute.
The definition is as follows. Let Ω stand for the O -module of differ-
C C
entials and denote by Ω• the sheaf of graded O -algebras O ⊕ Ω . For
C C C C
every X ⊂ I, we have a natural, S(X)-equivariant, identification of O -
CX
modules
Ω⊠X ∼= Ω|X| ⊗or(X)
C CX
4 EDUARDLOOIJENGA
where Ω⊠X denotes the exterior product ⊗ π∗Ω1 and or(X) stands for
C x∈X x C
the orientation module ∧|X|ZX. We define the sheaf of polydifferentials on
CI as the sheaf of graded O -algebras (Ω• )⊠I. So its degree N part is
CI C
π∗ Ω⊠X, where π : CI → CX is the evident projection. A to-
X⊂I,|X|=N X C X
talorderonI identifiesthegradedsheafofpolydifferentialswiththegraded
L
sheaf of holomorphic differential forms on CI, but there is no natural ana-
logue of the exterior derivative defined on (Ω• )⊠I. If ω ∈ (Ω• )⊠I, we will
C C
write ωX for its componentin π∗ Ω⊠X, where π : CI → CX is the evident
X C X
projection. So ω = ωX. We also write ω ∈ π∗ (Ω• )⊠X for the sum
X⊂I X X C
ωY.
Y⊂X P
WesaythatameromorphicpolydifferentialonC islogarithmiciflocallyit
P
canbewrittenasaproduct(df /f )...(df /f )oflogarithmicdifferentials
1 1 N N
,with f ,...,f ∈O .
1 N CI
Residue operators that involve a single factor (and a priori only those)
have a meaning for polydifferentials. Recall for any p ∈ C is defined a
residue operator Res as a linear form on the space meromorphic dif-
(t=p)
ferentials on C at p . This extends to polydifferentials: Res maps the
i (ti=p)
meromorphic polydifferentials on CI to meromorphic polydifferentials on
CI−{i}. But a residue along any other type of hypersurface must be treated
with care,asit requiresan orderingof the index set. For instance,if (i,j) is
an orderedpair, then the residue along the diagonal (t = t ) where the ith
i j
andjthcomponentareequalistobeunderstoodasviewingt astheresidue
i
variable and t as a parameter, in other words, we take the residue in the
j
t -directionanddothisatthepointwitht -coordinateequaltot . Theresult
i i j
is a polydifferential on the diagonal divisor (t = t ). In order to emphasize
i j
the asymmetric roles of i and j, we shall denote this by Res . This is
(ti→tj)
clearlynotthe same thing asRes .
(tj→ti)
Ashufflealgebra of polydifferentials. WenowtakeC tobetheprojective
line (which we will denote by P), we pick a point ∞ ∈ P and denote by A
theaffinelineP−{∞}. ItishelpfultofixanaffinecoordinateonA(which
depending on the context is denoted t or z), although this is inessential (in
practiceitwillonlyenterthroughdifferentialsoftheform(t(p )−t(p ))−1dt,
1 2
wherep ,p ∈ A are distinct, which is indeed independentof t).
1 1
We consider relative polydifferentials on (PI)A, i.e., on the projection
PI × A → A. The coordinate of the base A is denoted z and the affine
coordinate ofthe ith factor P in the productby t .
i
We first consider the graded C-vector space B• of the relative polydiffer-
I
entials(PI)A generatedby the expressionsof the form
dt dt ···dt
ζ :z 7→ ζ (z) := iN iN−1 i1 .
I I
(t −t )···(t −t )(t −z)
iN iN−1 i2 i1 i1
where I = (i ,i ,...,i ) runsover the finite sequencesin I (the degree
N N−1 1
ofsuchapolydifferentialisN). WestipulatethatforI = ∅weget1: ζ = 1.
∅
Notice that ζ = 0 unlessthe sequenceI is without repetition. In this paper
I
POLYDIFFERENTIALSANDTHEKZSYSTEM 5
I shall be fixed (although we will later assume I to be countably infinite)
andso we often write B• instead of B•.
I
We also put
dt dt ···dt
ω = iN iN−1 i2 ,
I
(t −t )···(t −t )
iN iN−1 i2 i1
agreeing that when I = (i), ω = ω = 1 and that ω = 0. So when
(i) i ∅
j ∈ I − {I}, then ζ (t ) = ω and when I is nonempty and ends with i,
I j Ij
then ω ζ = ζ . It is clear that if X ⊂ I, then B is closed under ‘taking
I iJ IJ I
the X-component’.
ForasectionaofPA,theresiduealongaisamapfromthemeromorphic
differentials on PA to C(A). Given an x ∈ I, then this extends this in an
obvious manner as a map of degree −1 from to the meromorphic relative
polydifferentials on (PI)A to itself (strictly speaking it is the composite of
a residue and the pull-back along the projection that suppresses the xth
factor). For certain valuesof a,this preservesB:
Lemma-definition 1.1. Forx ∈ I,weput
E := −Res , E′ := Res .
x (tx=∞) x (tx=z)
Then for a sequence I in I without repetition, we have E (ζ ) = 0 resp.
x I
E′(ζ )= 0 unlessI is of the formxJ resp.Jx,in which casewegetζ .
x I J
If x,y ∈ I are distinct, then Res ζ is zero unless I is of the form
(tx→ty) I
I′′xyI′ orI′′yxI′, in whichcasewe getζ resp.−ζ .
I′′yI′ I′′yI′
The proof is straightforward. This helps us (among other things) to give
anintrinsiccharacterizationofB•. GivenasubsetX ⊂ I,denotebyD the
X
reduced effective divisor on PX × P that is the sum of the loci (t = ∞),
i
(t = z) andthe diagonals(t = t ) (i,j ∈X distinct).
i i j
Corollary 1.2. Thepolydifferentialsζ arelinearlyindependentoverC[z]and
I
(hence) constitute a C-basis of B•. Moreover, we have a decomposition BN =
⊕ BN and when X ⊂ I is finite, then the natural map of C[z]-
X⊂I,|X|=N X
modules
C[z]⊗ B|X| → H0(PX ×A,Ω|X| (logD ))⊗or(X)
C X PX×A/A X
is an isomorphism that maps B|X| onto the space of sections that vanish at
z = ∞ andrestricts foranyz ∈ A to anisomorphism
B|X| → H0(PX,Ω|X|(logD (z)))⊗or(X).
X PX X
Proof. The first assertion follows from the observation that for a sequence
I = (i ,...,i ), the iterated residue E′ ···E′ takes on ζ the value δ
N 1 iN i1 J I,J
when |J|≤ |I|.
It is clear that B• = ⊕ B|X| and it is easily checked that BN is a sub-
I X⊂I X X
space of H0(PX ×P, Ω⊠X (D −(z = ∞))). To see that this inclusion
P P X
is an equality, we observe that if an element ω of the last space is annihi-
(cid:0) (cid:1)
latedbyallthe iteratedresidueoperatorsofthe typeabove,thenω mustbe
6 EDUARDLOOIJENGA
independent of z. As ω is zero for z = ∞, it follows that ω = 0. The last
assertion is provedsimilarly. (cid:3)
HereisanotherinterestingconsequenceofLemma1.1. Givenasequence
I = (i ,...,i )in I without repetition of lengthN ≥ 2, then put
N 1
Res := Res ···Res .
I (ti2→ti1) (tiN→tiN−1)
We regard this operator as taking values in the polydifferentials on the di-
agonallocusdefinedbyt = t = ··· = t . We recallthatt denotesthe
i1 i2 iN {I}
restriction of anyt to this locus.
ik
Lemma 1.3. Let I = (i ,...,i ) be a sequencewithout repetition in I. Then
N 1
the followingidentitieshold fora ∈ {z,∞} in B:
[[[···[E ,E ],···],E ],E ]= −Res Res ,
iN iN−1 i2 i1 (t{I}=∞) I
[[[···[E′ ,E′ ],···],E′ ],E′ ]= Res Res .
iN iN−1 i2 i1 (t{I}=z) I
Proof. The last clause of Lemma 1.1 shows that if Res ζ 6= 0, then some
I K
permutation σ(I) of I must appear as an uninterrupted subsequence of K:
K = K′′JK′, and have the following property: i is adjacent to i and
N N−1
removingi makes i adjacent to i and so on. In other words, there
N N−1 N−2
is a proper subsequence I′ < I such that if I′′ is the residual subsequence
(obtained by removing the terms in I′), then σ(I) = (I′)∗I′′; let us call
this a back-forward permutation of I and denote by back(σ) the length of
I′. If we also require that Res Res (ζ ) 6= 0, then K′ = ∅ and its
(t{I}=z) I K
value is then by Lemma 1.1 equal to (−1)back(σ)ζ . If on the other hand
K′′
−Res Res (ζ ) is nonzero, then K′′ = ∅ and its value is then by
(t{I}=∞) I K
Lemma1.1 equalto (−1)back(σ)ζ .
K′
Expanding the iterated bracket of residue operators in a straightforward
fashion yields
[[[···[E′ ,E′ ],···],E′ ],E′ ]=
iN iN−1 i2 i1
= (−1)back(σ)E′ E′ ···E′ ,
σ(iN) σ(iN−1) σ(i1)
σ
X
wherethesumisoverallback-forwardpermutationsσ ofI. Itsvalueonζ
K
is nonzero only if K is of the form K′′σ(I) and is then equal to ζ . If we
K′′
replace E′ by E and use the antisymmetry of the bracket: then we find
ik ik
expansion aboveis also equal to
[[[···[E ,E ],···],E ],E ]=
iN iN−1 i2 i1
= (−1)back(σ)E E ···E .
σ(i1) σ(i2) (tσ(iN)
σ
X
Its value on is ζ is nonzero only if K is of the form σ(I)K′ and is then
K
equalto (−1)back(σ)ζ . (cid:3)
K′
POLYDIFFERENTIALSANDTHEKZSYSTEM 7
A subset X ⊂ I induces via pull-back an injective map from rational
polydifferentialsonCX torationalpolydifferentialsonCI. WenowallowI
to be infinite, but countableand define the space of polydifferentials on CI
as the injective limit of the polydifferentials on CX, where X runs over the
finitesubsets ofI. We denotebyBˆm the spaceof(possibly infinite) sumsof
these relative polydifferentials ofdegreem:
Bˆm = Bm
X
X⊂I,|X|=m
Y
and put Bˆ• := ⊕∞ Bˆm, but often omit the subscript I as that has been
m=0
fixed.
If I and J are sequences such that their juxtaposition IJ is without rep-
etition, then we denote by I ⋆ J the collection of shuffles of I and J, i.e.,
the set of sequences that are permutations of IJ and in which I and J
appear as subsequences. This shuffle product is associative: we have that
(I ⋆J)⋆K = I ⋆(J ⋆K)if IJK is without repetition.
Lemma1.4(Shufflerules). ThegradedvectorspaceBˆisclosedunderproduct
and is as a C-algebra isomorphic to a (completed) shuffle algebra: if I and J
are finite sequences in I, then ζ ζ = ζ (we get zero unless IJ is
I J K∈I⋆J K
withoutrepetition). Togetherwith the coproduct
P
δ : Bˆ→ Bˆ⊗Bˆ, δ(ζ ) = ζ ⊗ζ ,
I I′′ I′
I=I′′I′
X
thismakesBˆacommutativebialgebraoverC(withtheprojectiononthedegree
zerosummandas counit).
More generally,if I and J are finite nonemptysequencesin I,k ∈ {J} and
J is written J J ,then ω ζ = ζ .
<k ≥k Ik J K∈I⋆J<k KJ≥k
Proof. An induction argument showPs that it is enough to verify these state-
ments insofar they do not regard the Hopf property when I is a singleton.
That case easily followsfrom repeateduse from the simple identity
1 1 w−v
− = . (cid:3)
v−u w−u (v−u)(w−u)
The proof that δ is a coproduct that is compatible with the shuffle product
is straightforward.
So B can be identified with a shuffle algebra over C[t]/(t2) on a set of
generatorsindexed by I. This makes us wonder whether there is a relation
with iterated integrals. From Corollary1.2 andLemma1.4we deduce:
Corollary 1.5. If I is a countably infinite set, then BˆS(I) is a polynomial
algebrawithprimitive generatorζ := ζ andwe have
i∈I i
ζN P dtx
= .
N! t −z
x
{X⊂I,|X|=N}x∈X
X Y
8 EDUARDLOOIJENGA
Two-parameter identities. The following two lemmas assert identities in
an algebra of meromorphic polydifferentials on P depending on two com-
plex variables. They will be needed later (beginning with the construction
of the Gauß-Manin connection in Lemma 3.9), but it is convenient to state
andprove them now.
Recallthat for asequenceI, I∗ denotes the opposite sequence.
Lemma 1.6. Let I be a nonempty sequence in I. If b(I) denotes its first ele-
ment,then
z−w
t −wζI(z) = (−1)|I2|ζI1(z)·ζI2∗(w),
b(I)
I=XI2I1
orequivalently,the operator (−1)|J|ζ (w)E (with the sum takenover all
J J J
finite sequencesJ in I)sendsζ to z−w ζ .
PI tb(I)−w I
More generally,if iappearsin I so that wecan write I = I′′iI′, then
z−w
t −wζI(z) = ζI(z)− (−1)|I2|ζI1(z)·ωI′′iζI2∗i(w),
i
I′X=I2I1
andif j ∈ I,j 6= i, then
z−t
t −tj ζI(z) = ζI(z)− (−1)|I2|ζI1(z)·ωI′′iωI2∗ij.
i j
I′X=I2I1
Proof. Thethirdidentityisnotreallydifferentfromthesecond(taket = w)
j
and the second follows from the first applied to iI′ (and multiply it with
ω ). Soweconcentrateonthefirstidentity andproveitwithinductionon
I′′i
|I|. ForI = ∅thereisnothingtoshowandforI = (i),thelemmastatesthat
z−w
ζ (z) = ζ (z)−ζ (w),
i i i
t −w
i
whichisasimpleconsequenceoftheidentity z−w = 1− ti−z. Nowassume
ti−w ti−w
I has length > 1 and write I = iI′ and I′ = jJ′. So ζ (z) = ζ ζ (z) =
I ij jJ
dti ζ (z). Since
ti−tj I′
z−w z−w 1 1
= − ,
(t −w)(t −t ) t −w t −t t −w
i i j j i j i
(cid:16) (cid:17)
we have
z−w z−w
ζ (z) = dt ζ (z)
I i I′
t −w (t −w)(t −t )
i i i j
z−w dt dt
i i
= − ζ (z)
I′
t −w t −t t −w
j i j i
(cid:16) (cid:17)
z−w z−w
= ζ (z)− ζ (z)·ζ (w),
I I′ i
t −w t −w
j j
which after invoking the induction hypothesis becomes
POLYDIFFERENTIALSANDTHEKZSYSTEM 9
ζI(z)− (−1)|J2|ζJ1(z)·ωJ2∗jζij(w)
J′=XJ2J1
− ζI′(z)− (−1)|J2|ζJ1(z)·ζJ2∗j(w) ζi(w) = ζI(z)+
(cid:16) J′=XJ2J1 (cid:17)
+ (−1)|J2|ζJ1(z) −ωJ2∗jζij(w)+ζJ2∗j(w)ζi(w) −ζI′(z)·ζi(w).
J′=XJ2J1 (cid:16) (cid:17)
According to the shuffle rules 1.4, we have −ωJ∗jζij(w) +ζJ∗j(w)ζi(w) =
2 2
ζJ2∗ji(w) = ζ(jJ2)∗i(w), so that we get
ζI(z)− (−1)|jJ2|ζJ1(z)ζ(jJ2)∗i(w)−ζI′(z)·ζi(w),
J′=XJ2J1
which indeedequalsζI(z)− I′=I2I1(−1)|I2|ζI1(z)ζI2∗i(w). (cid:3)
Remark 1.7. Notice that thePexpression ωI∗iζI′′i(w) in the right hand side
2
of the preceding lemma is, according to Lemma 1.4, equal to the sum
ζ (w), whereL runsover the shuffles ofI∗ andI′′.
L Li 2
PThe followinglemmageneralizesthe precedingone.
Lemma 1.8. If I := I′′iI′ andJ := J′′jJ′ are sequencesas above,then
z−w
ζ (z)·ζ (w) = ζ (z)·ζ (w)
I J I J
t −t
i j
− (−1)|I2|ζI1(z)·ωI2∗iωI′′ijζJ(w)
I′X=I2I1
− (−1)|J2|ωJ2∗jωJ′′jiζI(z)·ζJ1(w).
J′=XJ2J1
In particular, itis a linearcombinationofterms ofthe formζ (z)·ζ (w).
K L
Proof. The secondidentity ofLemma1.6givesafter multiplicationbyζ (w)
J
z−t
t −tj ζI(z)·ζJ(w) = ζI(z)·ζJ(w)− (−1)|I2|ζI1(z)·ωI2∗i·ωI′′ijζJ(w).
i j
I′X=I2I1
Likewise we find
w−t
t −tiζI(z)·ζJ(w) = ζI(z)·ζJ(w)− (−1)|J2|ωJ2∗jωJ′′jiζI(z)·ζJ1(w).
j i
J′=XJ2J1
The lemmathen followsby adding these two identities andusing that
z−t w−t z−w
j + i = +1. (cid:3)
t −t t −t t −t
i j j i i j
10 EDUARDLOOIJENGA
TheΦ-operators. Inthespaceofrationalrelativepolydifferentialson(PI)A,
we regard dt (i ∈ I) not just as an element, but also as the multiplication
i
operatorinthisspace. Itsadjointactsintheithtensorfactoronlyandsends
dt to 1 and1 to zero,henceis the contraction operator ι .
i ∂/∂ti
Let now be given complex numbers (p ) and (c ) . We define
i i∈I i,j i,j∈I,i6=j
for i ∈ I an operator Φ in the space of relative meromorphic polydifferen-
i
tials by
p dt dt dt
i i i j
Φ := − c ι ,
i t −z i,jt −t ∂/∂tj
i i j
j6=i
X
Sofor afinite subsetX ⊂ I,we have
p c
i i,x
Φ (dt )= − dt dt ,
i X i X
t −z t −t
i i x
(cid:16) xX∈X (cid:17)
whereit is understood that the righthand side is zerowhen i ∈ X.
Lemma 1.9. This operatorpreservesB (hencealso the completionBˆ), for
Φ (ζ )= (p −c )ζ .
x I x x,I′ I′′xI′
I=I′′I′
X
(Observe that the right hand side vanishes if x ∈ I and that the term indexed
by(I′′,I′)= (I,∅) reducesto p ζ .) Furthermore, foranyo ∈I,
x Ix
[Φ ,ω ] = −c ω .
x Io x,I′ I′′xI′o
I=I′′I′
X
Proof. We have Φ (ζ ) = p ζ ζ − c ω ζ , and the latter is by
x I x x I i∈{I} x,i x,i I
Lemma1.4 equalto
P
p ζ − c ζ = (p −c )ζ .
x I′′xI′ x,i I′′xI′ x x,I′ I′′xI′
I=I′′I′ I=I′′I′i∈{I′} I=I′′I′
X X X X
The secondidentity follows fromthis. (cid:3)
Products and powers. Notice that
−c dt
i,x i
[Φ ,dt ] = dt .
i X X
t −t
i x
x∈X
X
We also observethat for asequenceI = (i ,...,i )in I
N 1
Φ (dt ) := Φ Φ ···Φ (dt )
I X iN iN−1 i1 X
p c c
i i,x i,j
= dt − − .dt
i X
t −z t −t t −t
i i x i j
I=YI′′iI′ (cid:16) xX∈X jX∈I′ (cid:17)
dt t −z t −z
i i i
= p − c − c .dt .
i i,x i,j X
t −z t −t t −t
i i x i j
I=YI′′iI′ (cid:16) xX∈X j∈X{I′} (cid:17)