The Hamiltonian geometry of the space of unitary connections with symplectic curvature

0.16 MB·English

by  Joel Fine

The Hamiltonian geometry of the space of unitary
1
connections with symplectic curvature
1
0
2
JoelFine

n
a
J

2 Abstract
1
LetL→M beaHermitianlinebundleoveracompactmanifold.Write

] S for thespaceofallunitary connections in L whosecurvatures define
G
symplecticformsonM andG forthegroupofunitarybundleisometries
S
of L, which acts on S by pull-back. The main observation of this note
.
h is that S carries a G-invariant symplectic structure, there is a moment
t mapfortheG-actionandthatthisembedsthecomponentsofS asG -
a 0
m coadjointorbits(whereG isthecomponentoftheidentity).Restrictingto
0
thesubgroupofG whichcoverstheidentityonM,weseethatprescribing
[
thevolumeformofasymplecticstructurecanbeseenasfindingazeroof

1
amomentmap. WhenM isaKählermanifold,thisgivesamoment-map
v
interpretationoftheCalabiconjecture. Wealsodescribesomedirections
0
2 forfutureresearchbaseduponthepictureoutlinedhere.
4
2
. 1 Introduction
1
0
1 Let L → M be a Hermitian line bundle over a compact 2n-dimensional man-
1
ifold. We assume throughout that c (L) contains symplectic forms. This note
: 1
v investigatesthespaceS ofallunitaryconnectionsA inL forwhichω = i F
i A 2π A
X isasymplecticformonM.ThegroupG ofunitarybundleisometries(notneces-
r sarilycoveringtheidentityonM)actsonS bypull-back.Themainobservation
a
ofthisnoteisthefollowing.
Theorem1.
• S carriesaG-invariantsymplecticform;
• Thereisanequivariantmoment-mapµ:S →Lie(G)∗fortheG-action;
• The map µ embeds each component of S as a coadjoint orbit of G , the
0
identitycomponentofG.
1
This is proved in §2.1. In §2.2 we show that the coadjoint orbit of A ∈ S
is integral if andonly if the Weinstein homomorphism π (Ham)→S1 is trivial
1
(whereHam=Ham(ω )isthegroupofHamiltoniandiffeomorphisms).
A
In§3weconsidertherestrictionofthemomentmapµfortheactionofthe
subgroup T = Map(M,S1) ⊂ G of bundle isometries covering the identity on
M. ItturnsoutthatthemomentmapsendsaconnectionA tothevolumeform
ωn/n!.Inthiswaytheproblemofprescribingthevolumeofasymplecticstruc-
A
turecanbeseenintermsofmomentmapgeometry.
Asweexplainin§3.1oneoutcomeofthisisthatwhenb (M)=0thespace
1
ofsymplecticformswithfixedvolumeformisnaturallyasymplecticmanifold.
When b (M) 6= 0 this space carries a torus-fibration with fibres of dimension
1
b (M)whosetotalspaceisnaturallyasymplecticmanifold.
1
In §3.2 we consider the problem of prescribing the volume formof a Käh-
ler metric. This is the renowned Calabi conjecture, now of course Yau’s theo-
rem [Yau78]. Using the picture outlined above we show how the Calabi con-
jecture canbephrasedasfindingazeroof themomentmapinside acomplex
group orbit. This puts the problem into the same framework as the Hitchin–
Kobayashi correspondence (concerning Hermitian–Einstein connections) and
theDonaldson–Tian–Yauconjecture(concerningKählermetricswithconstant
scalarcurvature).
Thefocusofthisnoteistoexplaintheabovegeometricpicture;noattempt
ismadehere,however,toexplorethepotentialapplications.Both§2and§3end
with abriefdiscussion ofsome ofthesepossible directions forfutureresearch
(somemorespeculativethanothers!).
Acknowledgements
Itisapleasuretoacknowledgetheinfluenceofconversationswiththefollowing
people: Frédéric Bourgeois, Baptiste Chantraine, Dmitri Panov, Simone Gutt,
JulienKellerandChrisWoodward.
ThisworkwaswrittenupwhilstIwasaguestattheSimonsCenterforGeom-
etryandPhysics,attheStateUniversityofNewYork,StonyBrook. Iamgrateful
for the hospitality and the stimulating research environment which they pro-
vided.
2 Thespaceofconnectionswithsymplecticcurvature
Recall that L → M is a Hermitian line bundle over a compact 2n-dimensional
manifold. WewriteS forthespaceofallunitaryconnectionsA inL forwhich
2
ω = i F isasymplecticformonM.
A 2π A
2.1 Symplecticstructureandmomentmap
Webeginbydescribingasymplectic structureonS . ThesetS isopeninthe
spaceofallconnections(for,say,theC∞ topology). ThetangentspaceT S is
A
thespaceΩ1(M,iR)ofimaginary1-forms. Inordertoavoidfactorsof i inall
2π
ourformulae,wemultiplyby−2πi attheoutsetinidentifyingT S ∼=Ω1(M,R).
A
GivenA∈S ,wewriteω = i F fortheassociatedsymplecticform. Ourcon-
A 2π A
ventionsmeanthatfora ∈Ω1(M,R)correspondstoaninfinitesimalchangeof
da inω .
A
Definition2. Wedefinea2-formΩonS by
1
Ω (a,b)= a∧b∧ωn−1,
A (n−1)!Z A
X
fora,b ∈Ω1(M,R).
Proposition3. The2-formΩisasymplecticform.
Proof. Toprovenon-degeneracyonT S ,let J beanalmostcomplexstructure
A
onM compatiblewithω .Then,foranon-zero1-forma,
A
1
Ω (a,Ja)= |a|2ωn >0
A n!Z A
X
where|·|2istheRiemannianmetriccorrespondingto J andω .
A
NextwecheckΩisclosed. Forthisleta,b,c ∈Ω1(X,R),thoughtofasvector
fieldsonS .Then
dΩ(a,b,c)=a·Ω(b,c)+b·Ω(c,a)+c·Ω(a,b).
(TheformulaforgeneralvectorfieldsalsoincludestermswithLiebrackets,but
in our case these vanish since the vector fields a,b,c are linear on the affine
spaceofallconnectionsandsocommute.)Now,
1
a·Ω(b,c)= b∧c∧da∧ωn−2.
(n−2)!Z A
M
Hence
1
dΩ(a,b,c) = (da∧b∧c+db∧c∧a+dc∧a∧b)∧ωn−2,
(n−2)!Z A
M
1
= d a∧b∧c∧ωn−2 ,
(n−2)!Z A
M € Š
= 0.
3
WewriteG forthegroupofbundleisometriesofL,notnecessarilycovering
the identity on M. G acts by pull-back on S , preserving Ω. To describe the
moment map for this action, we first note that given a connection A in L and
η∈Lie(G),onecandefineafunctionA(η)∈C∞(M,R).Thinkingofηasavector
field on L, the connection A splits η into a vertical and a horizontal part. On
eachfibre,theverticalpartismultiplicationby i A(η).
2π
Alternatively, we can think of a connection A asanS1-invariant1-formon
the principal circle bundle P →M corresponding to L →M. Then η is anS1-
invariantvectorfieldonP andthefunctionA(η)givenbypairingthe1-formA
withthevectorfieldηisthefunctionweseek,pulledbackto P.(Again,normally
one considers connections on principal circle bundes as imaginary valued 1-
forms,butwemultiplyby−2πi throughoutanduseinsteadreal1-forms.) This
secondpointofview—via principalbundles—istheonewe normallyadoptin
thissection.
Proposition4. Themapµ:S →Lie(G)∗definedby
1
〈µ(A),η〉= A(η)ωn
n!Z A
M
isaG-equivariantmomentmapfortheactionofG onS .
Proof. Givenη∈Lie(G),leta ∈Ω1(M,R)bethevectorfieldonS correspond-
η
ingtotheinfinitesimalactionofη. Letb ∈Ω1(M,R)beanothervectorfieldon
S .Theidentitytobeprovedisb·〈µ,η〉=Ω(b,a ).
η
We begin with the left-hand-side. We use the description in terms of the
principal S1-bundle p:P → M given above, in which A is regarded as an S1-
invariant 1-formon P. The vector fieldb ∈ Ω1(M,R) on S corresponds to an
infinitesimalchangeofp∗b inA andhenceaninfinitesimalchangeofp∗b(η)=
b(p η)inA(η).Meanwhile,theinfinitesimalchangeinω isdb.Hence,
∗ A
1 1
b·〈µ,η〉= b(p η)ωn+ A(η)db∧ωn−1 .
Z (cid:18)n! ∗ A (n−1)! A (cid:19)
M
Tocomputetheright-hand-sideofthemoment-mapidentity,stillthinking
ofA asa1-formonP,wehavethat
a = L (A)
η η
= (d◦ι +ι ◦d)A,
η η
= d(A(η))+ι ω .
p∗η A
4
(Wehaveimplicitlyidentifieda ∈Ω1(M,R)andp∗a ∈Ω1(P,R)inthefirsttwo
η η
lineshere.)Hence,evaluatedatthepointA∈S ,
1
Ω(b,a )= b∧ d(A(η))+ι ω ∧ωn−1.
η (n−1)!Z p∗η A A
M € Š
Next we use the following identity: on a 2n-dimensional manifold, given a 1-
formαanda2-formβ the(2n+1)-formα∧βn necessarilyvanishes;hence,for
anyvectorfieldv,
0=ι (α∧βn)=α(v)βn−nα∧ι β∧βn−1.
v v
Puttingα=b,β =ω andv =p η,thisgives
A ∗
1 1
Ω(b,a ) = b∧d(A(η))∧ωn−1+ b(p η)ωn ,
η (n−1)!Z (cid:18) A n ∗ A(cid:19)
M
1 1
= A(η)db∧ωn−1+ b(p η)ωn ,
Z (cid:18)(n−1)! A n! ∗ A(cid:19)
M
= b·〈µ,η〉.
Finally,G-equivariancefollowsimmediatelyfromthedefinitionofµ.
Weremarkthatthispicture ismotivatedbythewell-known observation of
AtiyahandBott[AB83]that“curvatureisamomentmap”.In[AB83],Atiyahand
Bottconsiderunitaryconnectionsinbundlesofarbitraryrank,butoverabase
withafixedsymplecitcform.
TocompletetheproofofTheorem1weshowthatthecomponentsofS are
identifiedviaµwithcoadjointorbits.
Lemma5. Themapµ:S →Lie(G)∗embedseachcomponentofS asacoadjoint
orbitofG .
0
Proof. Wemustshowtwothings:firstly,thatµisinjective;secondlythatG acts
0
transitivelyonthecomponentsofS .
Toproveinjectivityofµ,supposethatA6=A′.Thenwecanfindavectorfield
v onM suchthattheA′-horizontalliftηofv satisfiesA(η)>0,hence〈µ(A),η〉>
0.ButA′(η)=0andso〈µ(A′),η〉=0,henceµ(A)6=µ(A′).
NextweshowthatG actstransitivelyonthecomponentsofS . GivenA ∈
0
S , let ρ : Lie(G) → T S denote the infinitesimal action of G at A. We have
A A
alreadyseenthat
ρ (η)=a =d(A(η))+ι ω .
A η p∗η A
5
First we show that ρ is surjective. Given a ∈ Ω1(M,R), let v be the ω -dual
A A
vectorfieldandletηbetheA-horizontalliftofv toP.Thenρ (η)=a.
A
Now, givenapathA(t)inS ,letv(t)bethevectorfieldwhichisω -dual
A(t)
to dA(t)andletη(t)betheA(t)-horizontalliftofv(t)toP.Thetime-dependent
dt
vector field η(t) integrates up to a path g(t) in G with g(0) the identity. By
0
construction,g(t)·A(0)=A(t).
2.2 IntegralityandtheWeinsteinhomomorphism
We next turn to the question of whether or not the orbits of S are integral
coadjoint orbits. It turns out that the obstruction to this is a homomorphism
π (Ham )→S1,firstintroducedbyWeinstein[Wei89].
1 A
Webrieflyrecallthedefinitionofanintegralcoadjointorbit.Formoredetails
see,forexample,[Kir04]. GivenaLiegroupG withLiealgebrag,fix f ∈g∗. We
writeStab(f)⊂G forthestabiliserof f underthecoadjointactionandhforthe
Liealgebraofthestabiliser.Thelinearmapf :g→RrestrictstoaLiealgebraho-
momorphism f :h→R.TheorbitO of f iscalledintegralwhenthemaph→R
f
is (up to a factor of i) the derivative of a group homomorphism Stab(f) →S1.
This condition implies the existence of a line bundle L → O which carries a
f
connection whose curvatureis thesymplectic formon O ; moreover thesym-
f
plecticactionofG onO liftstoaconnection-preservingactiononL.
f
Accordingly, wenextinvestigatethestabiliserStab ⊂G ofapointA ∈S .
A 0
Foranalternativeexpositionofthefollowing,seeWeinstein’sarticle[Wei89].
Westartfromtheashortexactsequence
1→Map (M,S1)→G →Diff (M)→1
0 0 0
(wherethesubscripts0denotetheidentitycomponents.)
Lemma 6 (Weinstein [Wei89]). Restricting this sequence to Stab ⊂ G gives a
A 0
shortexactsequence
1→S1→Stab →Ham →1 (1)
A A
whereS1⊂Map (M,S1)aretheconstantgaugetransformations.
0
Proof. FirstnotethattherestrictionofthemapG →Diff(M)toStab certainly
0 A
takesvaluesinω -symplectomorphisms. ToverifythattheimageliesinHam ,
A A
recall the formula for the infinitesimal action ρ (η) of η ∈ Lie(G) at A given
A
above. From this it follows that η ∈ Lie(Stab ) if andonly if p η is a Hamilto-
A ∗
nianvectorfieldwithHamiltonian−A(η).
Next we check that the map π: Stab → Ham is surjective. Given a ω -
A A A
Hamiltonian vector field v on M with Hamiltonian h we write v♭ for the A-
horizontalliftofv.Thenthevectorfieldη=v♭−h ∂ onPisS1-invariant,hence
∂θ
6
inLie(G)andρ (η)=0. Soη∈Lie(Stab )andπ (η)=v,meaningπ issurjec-
A A ∗ ∗
tive.Integratingthisshowsthatπ: Stab →Ham issurjective.
A A
ThekernelofπisStab ∩Map (M,S1). Given f :M →S1,thecorresponding
A 0
change in A is fd(f−1). Hence kerπ=S1 is the constants, andtheshortexact
sequenceforG restrictstoStab asclaimed.
0 A
GivenA∈S themomentmapatA restrictstogiveaLiealgebrahomomor-
phism
µ(A): Lie(Stab )→R
A
Thekernelofthismapisanideal I ⊂Lie(Stab ); moreover, theinclusionS1 ⊂
A
Stab determinesacopyofR⊂Lie(Stab )whichismappedisomorphicallyonto
A A
R byµ. It follows thatthederivative of Stab →Ham(A) identifies I ∼=HVect
A A
andsothereisasplitting
Lie(Stab )∼=R⊕HVect (2)
A A
intoadirectsumofideals.
Usingleft-multiplicationwecanviewthesplitting(2)asdefiningaconnec-
tion on the principle S1-bundle Stab → Ham . Because the horizontal sub-
A A
space(theHVect summand)isaLiesub-algebraofLie(Stab ),thisconnection
A A
isflat.ItsholonomyistheWeinsteinhomomorphism,
w:π (Ham )→S1.
1 A
Proposition7. GivenA∈S ,thecorrespondingcoadjointorbitofG isintegralif
0
andonlyiftheWeinsteinhomomorphismw:π (Ham )→S1istrivial.
1 A
Proof. Thecoadjoint orbitofA isintegralprecisely whenthekerneloftheho-
momorphism µ(A): Lie(Stab ) → R integrates up to a subgroup of Stab . In
A A
our case this kernel defines the horizontal space of the flat connection whose
holonomyisw. Sotheorbitisintegralifandonlyifparalleltransportidentifies
alltheS1-fibresofStab →Ham . Thishappenspreciselywhentheholonomy
A A
istrival.
Ontheonehand,thereareexamplesofsymplecticmanifoldsforwhichthe
Weinsteinhomomorphismistrivial. Indeed,forasurfaceofgenusatleastone,
the Hamiltonian group iseven contractible. Onthe otherhand, therearealso
plentyofmanfioldsforwhichtheWeinsteinhomomorphismisnon-trivial;the
simplest beingS2. Tosee this, restrict theshortexactsequence (1)tothesub-
groupSO(3)⊂Hamtoobtainthesequence
1→S1→U(2)→SO(3)→1.
7
∼
TheflatconnectioncorrespondstotheLiealgebraisomorphismu(2)=su(2)⊕
iR;itsholonomyisnon-trivialandgivesthestandardisomorphism
U(2)∼=SU(2)× S1.
±1
SimiliarremarksapplytoCPn withtheFubini–Studymetricand,moregenerally
to certain toric varieties. See the recent survey article of McDuff [McD10] for
moreonthissubject.
2.3 Furtherquestions
Given a subgroup H ⊂ Diff (M), the preimage under G → Diff (M) is a sub-
0 0 0
groupH′⊆G whichinheritsaHamiltonianactiononS .Themoment-mapµ′
0
fortheactionofH′issimplytheprojectionofµunderLie(G )∗→Lie(H′)∗.One
0
mightlookforzerosofµ′inthehopethattheygivesymplecticstructureswhich
respectinsomewaytheadditionalgeometryimposedinpassingfromDiff (M)
0
toH.
We explore this idea in the next section in its most extreme form, when
H =1isthetrivialgroup. Thisleadstotheproblemofprescribingthevolume
formof a symplectic structure. In a forthcomingpaper[Fin11]we exploit this
sameideaforcertainmanifoldsM andsubgroupsH. Themanifoldsinques-
tionareS2-bundlesoverfour-manifoldsandinthiswaywegiveamoment-map
interpretationoftheanti-self-dualEinsteinequationsforaRiemannianmetric
onafour-manifold.Besidesthesetwosituations,however,therearemanyother
possibilitiesonecouldstudyanditwouldbeinterestingtoseemoreexamples.
Weclosethissectionwithaspeculativeremark.Theabovepictureassociates
toeachisotopyclassofsymplecticformsinc (L)acertaincoadjointorbitofG .
1 0
Ontheonehand,distinguishingisotopyclassesofsymplecticformsisacentral
problem in symplectic topology; on the other hand, distinguishing coadjoint
orbitsisacentralprobleminthetheoryofinfintedimensionalLiegroups. One
might hope that Theorem 1 opens up the path for a transferof ideas between
thesetwoasyetpoorlyunderstoodquestions.
Animportantapproachtothestudyofcoadjointorbitsisthecelebrated“or-
bitmethod”(seeforexamplethetextofKirillov[Kir04]). ForthegroupG ,per-
0
hapsthefirstcasetoconsiderwouldbeasurfaceofgenusatleastone. There,
thecorrespondingcoadjointorbitisintegral.Moreover,aswewillseeinthefol-
lowingsection,itcomeswithanaturalisotropicfibrationwhoseinfinite-dimen-
sional fibresfailtobecoisotropic byafinitedimensional discrepancy (see Re-
mark12). Thuswe haveinplacemore-or-lesstheinitialdatarequiredbygeo-
metricquantisation. Thisstillleaves,ofcourse,theprincipaldifficultyofwhat
shouldplaytherôleofthe“square-integrablesections”oftheprequantumline
8
bundle, since thebase is infinitedimensional. Exactlyhowtoquantise such a
coadjointorbitis,inmyopinionatleast,aninterestinganddifficultquestion.
3 Prescribingthevolumeformofasymplecticstructure
Given a Hamiltonian action of a groupG with a moment map µ takingvalues
in g∗, the action of a sub-group H ⊂G has moment map given by composing
µ with the projection g∗ →h∗. In thissection we applythisobservation tothe
action ofthesubgroup T ⊂G ofbundleisometriesof L →M which cover the
identity.
3.1 Purelysymplecticcase
Ofcourse,T =Map(M,S1)andsoLie(T)=C∞(X,R).(Onenormallyusesimag-
inaryvaluedfunctionsherebutagainwehavemultipliedby−2πi throughout.)
Byintegratingagainsttop-degreeforms,wecanidentifyΩ2n(M,R)withasub-
set ofLie(T)∗. Withthisunderstood, we havethefollowing result, whichis an
immediatecorollaryofProposition4.
Proposition 8. There is an equivariant moment map ν:S → Lie(T)∗ for the
actionofT onS givenbyν(A)=ωn/n!
A
Soprescribingthevolumeofasymplecticstructureinc (L)canbeseenas
1
findingazeroofamomentmap. Moreprecisely, since T isabelian,thecoad-
jointactionistrivialandsowecanequallyuseν−θ asamomentmapforany
θ ∈Lie(T)∗. Givenavolumeformθ ∈Ω2n(M,R)with[θ]= 1c (L)n,theequa-
n! 1
tionforA ∈S givenbyωn/n!=θ isthesameasfindingazeroofthemoment
A
mapν−θ.
Givensuchaθ,wenextturntothesymplecticreductionν−1(θ)/T .Bystan-
dardtheorythisisasymplecticmanifold(ofinfinitedimension). Todescribeit
wewriteX forthespaceofsymplecticformsω∈c (L)withωn/n!=θ.
θ 1
Proposition9. Ifb (M)=0thenX =ν−1(θ)/T andso,inparticular,thespace
1 θ
ofsymplecticformswithfixedvolumeformisnaturallyasymplecticmanifold.In
generalthereisasubmersionfromthesymplecticreductionν−1(θ)/T →X with
θ
fibresisomorphictoH1(M,R)/H1(M,Z). Therestrictionofthesymplecticstruc-
turetothesefibresisidentifiedwiththe2-formonH1(M,R)definedby(α,β)7→
1 α∧β∧c (L)n−1.
(n−1)! M 1
R
Proof. Webeginwiththefollowingstandardfact. Givenasymplecticformω∈
c (L),writeS ⊂S forthesetofunitaryconnectionsAforwhichω =ω.Then
1 ω A
S /T canbeidentifiedwithH1(M,R)/H1(M,Z).
ω
9
More precisely, givenA ∈S ,anyotherconnection A ∈S isoftheform
0 ω ω
A=A + i a foraclosed1-forma. Thereisthusasurjectionc:S →H1(M,R)
0 2π ω
givenbyc(A)=[a]. NowT =Map(M,S1)actsonH1(M,R),theactionof f ∈T
on H1(M,R) is by addition of 1 [fd(f−1)] ∈ H1(M,Z). With this action un-
2πi
derstood, c is T-equivariant. Since any element of H1(M,Z) can be written
in as 1 [fd(f−1)] for some f ∈ T, the map c descends to an identification
2πi
S /T →H1(M,R)/H1(M,Z).
ω
The group T is abelian, so its orbits in S are isotropic and hence the re-
∼
strictionofthesymplecticformΩonS toS descendstoa2-formonS /T =
ω ω
H1(M,R)/H1(M,Z). ItfollowsfromthedefinitionofΩthatthe2-formisidenti-
fiedwiththe2-formonH1(M,R)givenby
1
(α,β)7→ α∧β∧c (L)n−1.
1
(n−1)!Z
M
Theresultfollowsfromthesetwoobservationsappliedfibrewisetothemap
ν−1(θ)→X whichsendseachconnectionAtoitscurvatureω .
θ A
Remark 10. Whenb (M)=0, the symplectic structureon X can be seen di-
1 θ
rectly (and with no need for the condition that the fixed choice of symplectic
class be integral). The tangent space at a point ω ∈ X is the space of exact
θ
2-formsγsuchthatωn−1∧γ=0.WenowdefineaskewpairingΘonT X by
ω θ
1
Θ(γ,γ′)= a∧a′∧ωn−1
(n−1)!Z
M
wherea,a′are1-formswithda =γ,da′=γ′.Ifa˜isanother1-formwithda˜=γ,
then d(a −a˜) = 0 and so, since b (M) = 0, we can write a −a˜ = df for some
1
function f.Hence,
(a−a˜)∧a′∧ωn−1=− fda′∧ωn−1
Z Z
M M
whichvanishessinceda′∧ωn−1=γ′∧ωn−1=0.ItfollowsthatΘ(γ,γ′)doesnot
dependonthechoiceofa ora′.
Whenthefixedsymplecticclass[ω]=c (L)isintegral,Θispreciselythe2-
1
form which arises from the identification ν−1(θ)/T ∼= X . It follows from the
θ
generaltheorythatΘisclosedandnon-degenerate,somethingwhichonecan
verifydirectlyfromthedefinition.
Remark 11. Still under the assumption that b (M) = 0, note that the group
1
Diff(M,θ)ofvolume-preservingdiffeomorphismsactsonthesymplecticmani-
foldX . ThisactionisHamiltonianinthesensethattheinfinitesimalactionof
θ
10