Quantum Field Theory and the Jones Polynomial.Edward Witten

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1、CommunicationsinCommun.Math.Phys.121,351-399(1989)MathematicalPhysics©Springer-Verlag1989QuantumFieldTheoryandtheJonesPolynomial*EdwardWitten**SchoolofNaturalSciences,InstituteforAdvancedStudy,OldenLane,Princeton,NJ08540,USAAbstract.Itisshownthat2+1dimensionalquantumYang-Millstheory,

2、withanactionconsistingpurelyoftheChern-Simonsterm,isexactlysolubleandgivesanaturalframeworkforunderstandingtheJonespolynomialofknottheoryinthreedimensionalterms.Inthisversion,theJonespolynomialcanbegeneralizedfromS3toarbitrarythreemanifolds,givinginvariantsofthreemanifoldsthatarecomp

3、utablefromasurgerypresentation.Theseresultsshedasurprisingnewlightonconformalfieldtheoryin1+1dimensions.InalectureattheHermannWeylSymposiumlastyear[1],MichaelAtiyahproposedtwoproblemsforquantumfieldtheorists.ThefirstproblemwastogiveaphysicalinterpretationtoDonaldsontheory.Thesecondpr

4、oblemwastofindanintrinsicallythreedimensionaldefinitionoftheJonespolynomialofknottheory.Thesetwoproblemsmightroughlybedescribedasfollows.Donaldsontheoryisakeytounderstandinggeometryinfourdimensions.Fouristhephysicaldimensionatleastmacroscopically,soonemaytakeaslightlibertyandsaythatD

5、onaldsontheoryisakeytounderstandingthegeometryofspace-time.Geometershavelongknownthat(viadeRhamtheory)theself-dualandanti-self-dualMaxwellequationsarerelatedtonaturaltopologicalinvariantsofafourmanifold,namelythesecondhomologygroupanditsintersectionform.Forasimplyconnectedfourmanifol

6、d,theseareessentiallytheonlyclassicalinvariants,buttheyleavemanybasicquestionsoutofreach.Donaldson'sgreatinsight[2]wastorealizethatmodulispacesofsolutionsoftheself-dualYang-Millsequationscanbepowerfultoolsforaddressingthesequestions.Donaldsontheoryhasalwaysbeenanintrinsicallyfourdime

7、nsionaltheory,andithasalwaysbeenclearthatitwasconnectedwithmathematicalphysicsatleastatthelevelofclassicalnonlinearequations.ThepuzzleaboutDonaldsontheorywaswhetherthistheorywastiedtomorecentralideasinphysics,whetheritcouldbeinterpretedintermsofquantumfieldtheory.Themostimportant*Ane

8、xpandedversi