2024年5月9日发(作者：ssd硬盘寿命一般多久)

CONTACTDEGREEANDTHEINDEXOFFOURIER

INTEGRALOPERATORS

E

Date:WedMay2715:55:551998;Run:May27,1998

pticFourierintegraloperatoroforder0,associatedtoa

homogeneouscanonicaldiffeomorphism,onacompactmanifoldisFredholm

onL

2

.Theindexmaybeexpressedasthesumofaterm,whichwecall

thecontactdegree,associatedtothecanonicaldiffeomorphismandaterm,

computablebytheAtiyah-Singertheorem,

contactdegreeisshowntobedefinedforanyoriented-contactdiffeomorphism

ofacontactmanifoldandisthenreducedtotheindexofaDiracoperatoron

themappingtorus,

thiscase,ofanoperatoronafixedmanifold,theseresultsansweraquestion

ofWeinsteininamannerconsistentwithamoregeneralconjectureofAtiyah.

Introduction

LetXbeacompactcontactmanifold,withorientedcontactlinebundleL⊂

(X)bethecontactmappingclassgroupofX;thatisthegroupofcom-

ponentsofthespaceofcontactdifftructahomomorphism

whichwecallthecontactdegree

∗

(1)c-deg:M(X)−→Z.

Thisconstructionisbasedonthenotionofaquantizationofthecontactstructure

introducedbyBoutetdeMonvelandGuillemin[5].Inparticularifthehyperplane

bundle,W=L

◦

onX,isgivenanalmostcomplexstructurewhichispositivewith

respecttotheconformalsymplecticstructureandXisgivenacompatiblepartially

Hermitianmetricthenin[5]generalizedSzeg˝-

thoughtheanalysisin[5]isintermsoftheHermitecalculus,theseprojectionsalso

lieintheHeisenbergcalculusdiscussedbyBealsandGreiner[1]andTaylor[18],

andoriginallydescribedbyDynin[6].Extendinganearlierideaofthesecondau-

thor[8](fortheintegrable,thatisCR,case)weintroducebelowtherelativeindex

oftwosuchgeneralizedSzeg˝oprojectionsastheindexoftheFredholmoperator

whichistheircompositeactingbetweentheirranges:

(2)ind(S

0

,S

1

)=ind(S

1

S

0

:Ran(S

0

)−→Ran(S

1

)).

WeshowbelowinProposition2thatthisrelativeindexfaithfullylabelsthecom-

ponentsofthespaceofgeneralizedSzeg˝ionofthegroupof

contactdiffeomorphisms,byconjugation,onthespaceofgeneralizedSzeg˝oprojec-

tionsinducesthehomomorphism(1),

(3)c-deg(φ)=ind(S,S

φ

),S

φ

=(φ

∗

)

−1

Sφ

∗

.

1

2CHARLESEPSTEINANDRICHARDMELROSE

LetZ

φ

bethemappingtorusofφ,i.e.X×[−1,1]withtheendsidentifiedby

φ.ThecontactstructureonXgivesZ

φ

ð

φ

bethe

associatedDiracoperator.

orientedcontactdiffeomorphismofacontactmanifoldthe

contactdegreeisgivenbytheindexoftheDiracoperatorassociatedtotheSpin-C

structureonthemappingtorus,thatis

c-deg(φ)=ind(ð

φ

).

Thisisprovedbyfirstexhibitingthecontactdegreeasthespectralflowofa

ceofDiracoperators

associatedwithpartialHermitianstructuresonthecontactmanifoldiscontractible.

SincetheseDiracoperatorsareellipticandself-adjoint,thespectralflowalongany

curveconnectingonesuchDiracoperatortoitsφ-conjugateiswelldefined.

contactdiffeomorphism,c-deg(φ)isthespectralflowofthe

curveofDiracoperatorsonthecontactmanifoldassociatedwithanisotopyfrom

anyonepartialHermitianstructuretoitsφ-conjugate.

Theorem1followsfromthisbyasuspensionargument.

Thecontactdegreeisdirectlyrelatedtoalong-openquestionofWeinstein[20,19]

askingforageometricformulafortheindexofellipticFourierintegraloperators.

Forsuchoperatorsactingonafixedmanifoldthetheoremaboveprovidesananswer.

Namely,Zelditch(see[19],[21]and[22])hadobservedthatthepropertiesofthe

integraltransformationstudiedbyGuillemin[12]showtheequalityoftheindexof

theFourierintegraloperatorandtherelativeindexoftheSzeg˝oprojectiononS

∗

X

anditsφingtheseresultswededuce

compactmanifoldandX=S

∗

Yisitscospherebundlethen

forany(oriented)contactdiffeomorphism,φ,ofX,icaldiffeomorphism

ofT

∗

X0,

ind(F

φ

)=c-deg(φ)

whereF

φ

isaFourierintegraloperator(see[13])associatedtoφandwithellip-

ticsymbolcorrespondingtothepositivetrivializationoftheMaslovbundle,hence

FredholmonL

2

(Y).

Anexplicitcohomologicalformulaforthisindex,whichfollowsfromtheAtiyah-

Singerindextheorem,isdiscussedin§eopen,forthemoment,the

partofWeinstein’squestionconcerningtheindexofellipticFourierintegraloper-

atorsbetweendiffghlylikelythatmethodscloselyrelated

tothoseusedherecanbeappliedtothegeneralproblemandprovideananswer

atitiseasilyseenthatourpresent

formulaisconsistentwiththatconjecture(see[19]).

Thecentralresulthere,whichisthepassagefromProposition4,essentiallya

restatementof(3),toTheorem2,iscloselyrelatedtothehomotopyargumentfor

theToeplitzindexgivenbyBoutetdeMonvelin[4].IndeedhisToeplitzindex

theoremcanbeprovedinessentiallythesameway,andsomadeindependentof

theAtiyah-Singerindextheoremforpseudodiffovides

ananalyticalternativetotheusualK-theorypathfromthetheoremforDirac

operatorstothegeneralcaseofpseudodifferentialoperators.