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1、TWO-DIMENSIONALTOPOLOGICALQUANTUMFIELDTHEORIESANDFROBENIUSALGEBRASLOWELLABRAMSJohnsHopkinsUniversityDepartmentofMathematics3400CharlesStreetBaltimore,MD21218ABSTRACTWecharacterizeFrobeniusalgebrasAasalgebrashavingacomultiplicationwhichisamapofA-modules.Thischaracterizationallowsasimpledem
2、onstrationofthecompatibilityofFrobeniusalgebrastructurewithdirectsums.WethenclassifytheindecomposableFrobeniusalgebrasasbeingeitherannihilatoralgebras"
3、algebraswhosesocleisaprincipalideal
4、or eldextensions.Therelationshipbetweentwo-dimensionaltopologicalquantum eldtheoriesandFrobeniusalge
5、brasisthenformulatedasanequivalenceofcategories.TheproofhingesonournewcharacterizationofFrobeniusalgebras.Theseresultstogetherprovideaclassi cationoftheindecomposabletwo-dimensionaltopologicalquantum eldtheories.Keywords:topologicalquantum eldtheory,frobeniusalgebra,two-dimensionalcobordi
6、sm,categorytheory1.IntroductionTopologicalQuantumFieldTheories(TQFT's)were rstdescribedaxiomati-callybyAtiyahin[1].Sincethen,muchworkhasbeendonetounderstandthealgebraicstructuresarisinginthethreeandfour-dimensionalcases(see[2]andthereferencescitedthere.)Inthetwo-dimensionalcase,thealgebra
7、icstructureoflattice eldtheoriesarewelldiscussedin[3],butthecaseofatwo-dimensionaltheorynothavingdistinguishedzero-cells,orcorners,"hasnotbeencompletelyunderstood.Ofcourse,thesetwotheoriesarenotthesame;themostimmediatelyapparentdi erencebetweenthelatticeandregularcasesisthelackofcommuta-
8、tivityintheformer.Aclassi cationofthetwo-dimensionalcaseintermsofthespectrumofaspeci clinearoperatorhasbeeno eredin[4],butactuallydealswitharestrictedcase,aswillbediscussedbelow.Interestspeci callyinthetwodimensionalcasegoesbacktosuchsourcesasSegal'spresentationin[5]oftwo-dimensionalconfo
9、rmal eldtheoriesandWitten'sworkin[6]relatingthesametoresultsinhigherdimensions.In[7]Voronovpresentsafolktheorem"assertingthatatwo-dimensionalTQFTisequivalenttoaFrobeniusalgebra"(FA),andsketchesaproof.(See[8]foraphysicist'saccount.)Nevertheless,therehasbeendiculty
