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1、D-branesandK-homologyBeiJiaThesissubmittedtotheFacultyoftheVirginiaPolytechnicInstituteandStateUniversityinpartialful llmentoftherequirementsforthedegreeofMasterofScienceinMathematicsPeterE.Haskell,ChairWilliamJ.FloydPeterA.LinnellEricR.SharpeApril19,2013Blacksburg,VirginiaKeywor
2、ds:K-homology,K-theory,D-brane,StringtheoryCopyright2013,BeiJiaD-branesandK-homologyBeiJia(ABSTRACT)InthisthesisthecloserelationshipbetweenthetopologicalK-homologygroupofthespace-timemanifoldXofstringtheoryandD-branesinstringtheoryisexamined.AnelementoftheK-homologygroupisgivenby
3、anequivalenceclassofK-cycles[M;E; ],whereMisaclosedspincmanifold,EisacomplexvectorbundleoverM,and :M!Xisacontinuousmap.ItisproposedthataK-cycle[M;E; ]representsaD-branecon gurationwrappingthesubspace (M).Asaconsequence,theK-homologyelementde nedby[M;E; ]repre-sentsaclassofD-brane
4、con gurationsthathavethesamephysicalcharge.Furthermore,theK-cyclerepresentationofD-branesresemblesthemodernwayofcharacterizingfundamentalstrings,inwhichthestringsarerepresentedastwo-dimensionalsurfaceswithmapsintothespacetimemanifold.Thisclassi cationofD-branesalsosuggeststheposs
5、ibilityofphysicallyinterpretingD-braneswrappingsingularsubspacesofspacetime,enlargingtheknowntypesofsingularitiesthatstringtheorycancopewith.AcknowledgmentsIwouldliketothankmyadvisor,ProfessorHaskell,forallthehelphegavemeduringmyMaster'sstudy.Ihavelearnedatremendousamountofmathem
6、aticsfromhim.HeintroducedthetopicsofK-theoryandK-homologytome,aswellasmanyotherrelevantmaterials.Heevenprovidedmanyofthetechnicaldetailsthatareusedinthisthesis.Iamespeciallythankfulthathelistenedtomanyofmyproblems,bothacademicandpersonal,withgreatpatience,ando eredmanyvaluableadv
7、ice.IconsidermyselfluckytohaveProfessorHaskellasmyadvisor.Iwouldalsoliketothankmycommitteemembers:ProfessorFloyd,ProfessorLinnell,andProfessorSharpe,fortheirhelpandsupport.Especially,IwouldliketothankProfessorSharpe,whoismyPh.D.advisorinphysics,forhissupportofmeinpursuingthisMast
8、er'sdegreeinmathematicssimultaneouslywit
