on wittens 3-manifold invariants

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1、In1991IdistributedapreliminaryversionofsomenotesonWitten’srecentlydiscovered3-manifoldinvariants.Forvar-iousreasonsthepaperwasnevercompletedandpublished.Nevertheless,manypeoplehavetoldmethattheystillfindthe1991notestobeuseful.Forthisreason,Ihavepreparedthis

2、versionofthenoteswhichisdistributableinelectronicform.Ihavenotattemptedtocorrect,completeorimprovethe1991version.InfactIhavetakenpainstomakesurethatallthepagebreaksoccurinthesameplaces—thisversionisessentiallyidenticaltotheoriginalone.Thenumeroushand-drawn

3、figureswerescannedin.Unfortunately,Icouldnotfindtheoriginalfigures,soIhadtoscanfromphotocopies.Thepostalandemailaddressesonpage1arenolongercur-rent.Icanbereachedatkwalker@xmission.com.KevinWalkerMarch29,2001OnWitten’s3-manifoldInvariantsKevinWalker∗†March29,2

4、001[PRELIMINARYVERSION#2][Preintroduction.Somesectionsaremissing,andothersareinroughdraftform.ThedefinitionofaTQFTgiveninthisversiondiffersslightlyfromtheonegiveninthepreviousversion.(Therearenomoregluingcoefficients;see(2.3),(2.9)and(2.13).)][Introductionfore

5、xperts.Theinteresting(Ihope)partsofthispaperare:thedefinitionof“extended”2-and3-manifolds,whichallowsonetoresolvetheprojectiveambiguity;gluing3-manifolds“withcorners”;apreciseandrigorousversionofMooreandSeibergspolynomialequationsresult;aproofthatWitten-Jon

6、esTQFTsexistandsatisfyalloftheaxiomswhichtheyoughtto;aclarificationoftherelationshipbetweenWitten-JonesTQFTsandtheinvariantsofReshetikhin-Turaev;aproofthattheTuraev-ViroinvariantequaltothesquareofthenormofaW-JTQFT,forclosed3-manifolds.]Contents0Introduction

7、21Extended2-and3-manifolds72ThedefinitionofaTQFT14∗SupportedinpartbyanNSFfellowship.†Currentaddress:P.O.Box945/Moab,UT84532;email:kwalker@math.ucsd.edu.(Thisemailaddressmightnotbereliable.)13Basicdata214Miscellaneouscalculations305Reconstructingmodularfunct

8、orsfrombasicdata366Relationsforbasicdata427Constructingpartitionfunctionsfrommodularfunctors568Reviewof[RT2]729TQFTsfrommodularHopfalgebras8810Reducedtanglefunctors9811ModularreducedtanglefunctorsfromTQFTs103