Operator Product Expansion in Logarithmic Conforma

arXiv:hep-th/0107242v221Mar2002OperatorProductExpansioninLogarithmicConformalFieldTheoryMic~aelFlohr∗InstituteforTheoreticalPhysics,UniversityofHannoverAppelstraße2,D-30167Hannover,GermanyE-mail:[e-mail]20,2002AbstractInlogarithmicconformalfieldtheory,primaryfieldscometogetherwithlogarithmicpart-nerfieldsonwhichthestress-energytensoractsnon-diagonally.Exploitingthisfactandglobalconformalinvarianceoftwo-andthree-pointfunctions,operatorproductexpan-sionsoflogarithmicoperatorsinarbitraryranklogarithmicconformalfieldtheoryareinvestigated.Sincethepreciserelationshipbetweenlogarithmicoperatorsandtheirpri-marypartnersisnotyetsufficientlyunderstoodinallcases,thederivationofoperatorproductexpansionformulæisonlypossibleundercertainassumptions.Theeasiestcasesarestudiedinthispaper:firstly,whereoperatorproductexpansionsoftwoprimariesonlycontainprimaryfields,secondly,wheretheprimaryfieldsarepre-logarithmicoper-ators.Somecommentsongeneralizationtowardsmorerelaxedassumptionsaremade,inparticulartowardsthecasewherelogarithmicfieldsarenotquasi-primary.Weidentifyanalgebraicstructuregeneratedbythezeromodesofthefields,whichprovesusefulindeterminingsettingsinwhichourapproachcanbesuccessfullyapplied.∗ResearchsupportedbyEUTMRnetworkno.FMRX-CT96-0012andtheDFGStringnetwork(SPPno.1096),Fl259/2-1.I.IntroductionDuringthelastfewyears,so-calledlogarithmicconformalfieldtheory(LCFT)es-tablisheditselfasawell-definednewanimalinthezooofconformalfieldtheoriesintwodimensions.Toourknowledge,logarithmicsingularitiesincorrelationfunctionswerefirstnotedbyKnizhnikbackin1987[41].Althoughmanyfeaturessuchasloga-rithmicdivergencesofcorrelatorsandindecomposablerepresentationswereobservedinvariousplaces,mostnotablyin[74,73],ittooksixyears,untiltheconceptofacon-formalfieldtheorywithlogarithmicdivergentbehaviorwasintroducedbyGurarie[27].Fromthenone,therehasbeenaconsiderableamountofworkonanalyzingthegen-eralstructureofLCFTs,whichbynowhasgeneralizedalmostallofthebasicnotionsandtoolsof(rational)conformalfieldtheories,suchasnullvectors,characters,partitionfunctions,fusionrules,modularinvarianceetc.,tothelogarithmiccase,seeforexample[16,34,21,67,72,23,46,68,13,38,58,60,25]andreferencestherein.Besidesthebestunderstoodmainexampleofthelogarithmicc=−2theoryanditscp,1relatives,otherspecificmodelswereconsideredsuchasWZWmodels[1,45,64,65,22]andLCFTsrelatedtosupergroupsandsupersymmetry[73,11,39,37,55,2,71,51].Also,quiteanumberofapplicationshavealreadybeenpursued,andLCFTshaveemergedinmanydifferentareasbynow.Sometimes,longstandingpuzzlesinthedescrip-tionofcertaintheoreticalmodelscouldberesolved,e.g.theHaldane-RezayistateinthefractionalquantumHalleffect[28,7,70],multi-fractality[12],ortwo-dimensionalconfor-malturbulence[18,66,76].Otherapplicationsworthmentioningaregravitationaldress-ing[5],polymersandabeliansandpiles[74,33,8,57],the(fractional)quantumHalleffect[17,31,49],and–perhapsmostimportantly–disorder[9,43,56,29,10,69,30,3,4].Finally,thereareevenapplicationsinstringtheory[42],especiallyinD-branerecoil[14,44,15,59,52,6,53,26],AdS/CFTcorrespondence[24,40,35,47,63,48,75,62],aswellasinSeiberg-WittensolutionstosupersymmetricYang-Millstheories,e.g.[19],Last,butnotleast,arecentfocusofresearchonLCFTsisinitsboundaryconformalfieldtheoryaspects[60,50,54,32,36].However,thecomputationofcorrelationfunctionswithinanLCFTstillremainsdiffi-cult,andonlyinafewcases,four-pointfunctions(orevenhigher-pointfunctions)couldbeobtainedexplicitly.ThemainreasonforthisobstructionisthattherepresentationtheoryoftheVirasoroalgebraismuchmorecomplicatedintheLCFTcaseduetothefactthatthereexistindecomposablebutnon-irreduciblerepresentations(Jordancells).Thisfacthasmanywiderangingimplications.Firstofall,itisresponsiblefortheappearanceoflogarithmicsingularitiesincorrelationfunctions.Furthermore,itmakesitnecessarytogeneralizealmosteverynotionof(rational)conformalfieldtheory,e.g.characters,highest-weightmodules,nullvectorsetc.WenoteherethatindecomposablerepresentationsneednotoccurwithrespecttotheVirasoroalgebra,butmayoccurwithrespectto(partof)aextendedchiralsymmetryalgebrasuchascurrentalgebrasorW-algebras.Forthesakeofsimplicity,wewillconfineourselvesinthispapertothecasewhereJordancellsarewithrespecttotheVirasoroalgebra.Inparticular,whatwaslackingsofarisaconsistentgenericformofoperatorproductexpansions(OPEs)betweenarbitraryranklogarithmicfields.AlthoughsuchOPEscanbederivedfromco-productconsiderationsinthepurelyrepresentationtheoreticalframework[34,21],adirectapproachtryingtofixthegenericformfromglobalconformalcovarianceofthefieldsisclearlydesirable.ForthesimplecaseofaranktwoLCFT,whereJordancellsaretwo-dimensional,itwasknownsincesometime[27,9]thatthetwo-pointfunctions1ofaprimaryΨ(h;0)(z)anditsonlylogarithmicpartnerΨ(h;1)(z)arehΨ(h;0)(z)Ψ(h;0)(w)i=0,hΨ(h;0)(z)Ψ(h;1)(w)i=D(h,h;1)(z−w)−2h,(1.1)hΨ(h;0)(z)Ψ(h;1)(w)i=[D(h,h;2)−2D(h,h;1)log(z−w)](z−w)−2h.However,asweshallsee,eveninthissimplecaseOPEsturnouttobemorecomplicated,andoneneedsallpossiblethree-pointfunctionsaswell.Firstresultsinthisdirectioncanbefoundin[67,23,38,61].H