Tensor products of modules for a vertex operator a

arXiv:hep-th/9401119v124Jan1994TENSORPRODUCTSOFMODULESFORAVERTEXOPERATORALGEBRAANDVERTEXTENSORCATEGORIESYI-ZHIHUANGANDJAMESLEPOWSKY1.IntroductionInthispaper,wepresentatheoryoftensorproductsofclassesofmodulesforavertexoperatoralgebra.Wefocusonmotivatingandexplainingnewstructuresandresultsinthistheory,ratherthanonproofs,whicharebeingpresentedinaseriesofpapersbeginningwith[HL4]and[HL5].Anannouncementhasalsoappeared[HL1].Thetheoryisbasedonboththeformal-calculusapproachtovertexoperatoralgebratheorydevelopedin[FLM2]and[FHL]andtheprecisegeometricinterpretationofthenotionofvertexoperatoralgebraestablishedin[H1].Recently,mathematicianshavebeenmoreandmoreattractedtoconformalfieldtheory,aphysicaltheorywhichplaysanimportantroleinbothcondensedmatterphysicsandstringtheory.Muchoftheresearchonconformalfieldtheoryhasbeencenteredontheconformalfieldtheoriesdeterminedbyholomorphicfieldsofweight1—thetheoriesassociatedwithcertainhighest-weightrepresentationsofaffineKac-Moodyalgebras.Manyimportantstructuresandconceptswhichhaveariseninre-centyearsarerelatedtothisspecialclassofconformalfieldtheories—Wess-Zumino-Novikov-Witten(WZNW)models,theKnizhnik-Zamolodchikovequationsandasso-ciatedmonodromy,quantumgroups,braidedtensorcategories,theJonespolynomialandgeneralizations,three-manifoldinvariants,Chern-Simonstheory,theVerlindeformula,etc.(seeforinstance[Wi1],[KZ],[J],[K1],[K2],[Ve],[TK],[MS],[Wi2],[TUY],[Dr1],[Dr2],[RT],[SV],[Va],[KL1]–[KL5],[Fi],[Fa]).Buttherearealsootherimportantmathematicalstructuresthatcanbestudiedasconformal-field-theoreticstructures,inparticular,highestweightrepresentationsoftheVirasoroalgebra,W-algebrasandtheirrepresentations,andmostparticularlyforus,themoonshinemod-uleV♮fortheFischer-GriessMonstersporadicfinitesimplegroup[Gr]constructedin[FLM1]and[FLM2].Fortheconformalfieldtheoriesassociatedwiththesestruc-tures,therearenononzeroholomorphicfieldsofweight1,andcorrespondingly,thespecialmethodsforstudyingthoseconformalfieldtheoriesassociatedwithaffineLiealgebrasdonotapply.Wemusthaveabroaderviewpointthanthatofthisspecialclassofconformalfieldtheories.1991MathematicsSubjectClassification.Primary08C99;Secondary18D10,32G15,81T40.12YI-ZHIHUANGANDJAMESLEPOWSKYThetheoryofvertexoperatoralgebras,developedin[B1],[FLM2]and[FHL](theresearchmonograph[FLM2]alsoincludesadetailedexpositionofthetheory,withexamples),providessuchaframework.Vertexoperatoralgebrasare“complexana-logues”ofbothLiealgebrasandcommutativeassociativealgebras.Allthestructuresmentionedintheprecedingparagraphcanbeandhavebeenilluminatedbytherepre-sentationtheoryofvertexoperatoralgebras.Oneimportantclassofvertexoperatoralgebras,generatedbyvectorsofweight1,isconstructedfromcertainhighestweightmodulesforaffineLiealgebras,andcertainotherhighestweightmodulescanbegiventhestructureofmodulesforthesevertexoperatoralgebras(seeforexample[FZ],[DL]).ThetheoriesofhighestweightrepresentationsoftheVirasoroalgebraandofrepresentationsofW-algebrascanalsobestudiedintermsoftherepresenta-tiontheoryofthecorrespondingvertexoperatoralgebras.ThemoonshinemoduleV♮[FLM1]isinfactanearlyexampleofvertexopera-toralgebra,exceptthatafullvertexoperatoralgebrastructureonV♮wasstatedtoexistonlylater,byBorcherdsintheannouncement[B1],inwhichthenotionof“vertexalgebra”wasintroduced.Intheseterms,thestructuredetailedintheannouncement[FLM1]wastheMonster-invariantgeneratingweight-twosubstruc-ture,equippedcorrespondinglywitha“cross-bracket”operation(ratherthantheLiebracketoperation),ofthevertexoperatoralgebrastructureonV♮.TheproofofthepropertiesoftheconstructionofV♮andoftheactionoftheMonsteronit[FLM1]wasgivenin[FLM2],alongwiththeconstructionofaMonster-invariantver-texoperatoralgebrastructureonit(statedtoexistin[B1]),andanaxiomaticstudyoftheconceptofvertexoperatoralgebrawaspresentedinthemonograph[FHL].Meanwhile,Belavin,PolyakovandZamolodchikov[BPZ]andotherphysicistsalsointroducedthebasicfeatures(onaphysicallevelofrigor)ofwhatmathematicianscametounderstandasvertexoperatoralgebras.Themostnaturalviewpointforusistheviewpointbasedonwhatwecallthe“Jacobiidentity”(see[FLM2],[FHL]andtheexposition[L1])forvertexoperatoralgebras.A“complexanalogue”oftheJacobiidentityforLiealgebras,thisidentityisthemaintoolintheformal-calculusapproachtothetheoryofvertexoperatorsalgebrasdevelopedin[FLM2]and[FHL]andusedinthepresentwork.Itcanbede-ducedfromthephysically-formulatedaxiomsin[BPZ]orfromtheaxiomsforvertexalgebrasin[B1].Thenotionofvertexoperatoralgebraof[FLM2]and[FHL]isinfactavariantofthenotionofvertexalgebraof[B1],equippedwiththeJacobiidentityasthemainaxiomandwithcertaingradingrestrictionsassumed.Itisthisvariantthatweneed.TheJacobiidentityexpressesaninfinitefamilyofgeneralizedcommu-tators(productsliketheZ-algebraproductsof[LW]orthecross-bracketsof[FLM1]mentionedabove)inclosedformviaageneratingfunctionbasedonformaldelta-functions.Inparticular,theJacobiidentityexhibitstheLie-algebra-likepropertiesofavertexoperatoralgebraratherthanitsassociative-algebra-likeproperties,whichareimplicitintheoperator-product-expansionformalisminthephysicsliterature,TENSORPROD