Discrete connections on the triangulated manifolds

arXiv:math-ph/0303035v25Apr2004S.Novikov1DiscreteConnectionsontheTriangulatedManifoldsanddifferencelinearequationsAbstract:Followingtheauthorsworks[1,2,3],wedevelopatheoryofthediscreteanalogsofthedifferential-geometricalGLn-connectionsoverthetriangulatedn-manifolds.WestudyanonstandarddiscretizationbasedontheinterpretationofDGConnectionasthelinearfirstorder(”triangle”)differenceequationinthesimplicialcomplexesactingonthescalarfunctionsofvertices.Thistheoryappearedasaby-productofthenewtypeofdis-cretizationforthespecialCompletelyIntegrableSystems,suchasthefamous2DTodaLatticeandcorresponding2DstationarySchrodingeroperators.Anonstandarddiscretizationofthe2DComplexAnalysisbasedontheseideaswasdevelopedintherecentwork[4].Acompleteclassificationtheoryiscon-structedherefortheDiscreteDGConnectionsbasedonthemixtureoftheabelianandnonabelianfeatures.I.GeneralDefinitions:TheDiscreteDGConnections.LetMbean-dimensionalsimplicialcomplex.BytheDiscreteDifferentially-Geometrical(DG)Connectionwecallanysetofcoefficients06=bT:P∈k∗,k=R,C,assigninganonzeronumbertoeverypairconsistingofthen-simplexTanditsvertexP∈T.EveryDG-connectiondefinesafirstorderdifferenceTriangleOperatorQmappingthespaceofk-valuedfunctionsofverticesψPintothespaceoffunctionsofn-simplices:(Qψ)T=XP∈TbT:PψPSuchoperatorsplayedanimportantroleintheworks[1,2,3].FortheneedsofthetheoryofdiscreteDGConnectionsonlythelinearTriangle1LandauInstituteforTheoreticalPhysics,Kosyginastr.2,Moscow117940,Rus-siaandIPST,UniversityofMaryland-CollegePark,MD,20742-2431,USA,[e-mail];thefirstversionofthisworkwascompletedduringthestayinKoreanInstitutefortheAdvancedStudies(KIAS),Seoul,S.KoreainNovember2002;itwaspartiallysupportedbytheNSFGrantDMS-0072700.Theauthormadeseveralim-provementsinFebruary2004,inparticular,concerningthereconstructionofthediscreteconnectionforn≥3.1EquationisimportantQψ=0welldefineduptotheAbelianGaugeTransformationsQ→fTQg−1P,ψP→gPψPwheref6=0,g6=0.Beginningfromnowwedenoteverticesbythelettersi,j,l,....Thereforeforeveryn-simplexTwithverticesi,j∈(i0,...,in)onlytheratiosareessentialμTij=bT:i/bT:jwherebT:iarethecoefficientsofDGcon-nectionassociatedtotheverticesofthesimplexT.WeassumebeginningfromnowthattheDGConnectionisgivenbythesetofnonzeronumbersμTijforalln-simplicesandpairsoftheirvertices.ObviouslywehaveμTii=1,μTijμTji=1,μTijμTjkμTki=−1.LetT,T′beapairofn-simplicessuchthati,j∈TTT′.WedefineaGauge-InvariantCoefficientsμTijμT′ji=ρTT′ijLemma1ThewholesetofthegaugeinvariantcoefficientsρTT′ijcanbere-coveredfromtheMinimalSubsetsuchthatTandT′aretheclosestneigh-bors,i.e.theintersectionsTTT′arethe(n−1)-dimensionalfaces.Thereare”trivial”setsAandBofrelationsonthesequantities:A.Foreverytriangle[ijl]⊂TTT′wehaveρTT′ijρTT′jlρTT′li=1B.ForeveryclosedpathT0T1...TNT0inthePoincaredualcellsubdivisionofthetriangulatedmanifoldMwheren-simplicesTdefinethevertices,allpairsTpTTp+1definetheedges(theyaredualtothen−1-faces),andtheedgeijbelongstoallTp,wehaveNYp=1ρTpTp+1ij=12Proof.ForthesimplicialmanifoldMeverypairofn-simplicesT,T′suchthati,j∈TTT′canbejoinedby”path”T0=T,T1,...,Tm=T′wherei,j∈TkTTk+1forallk=0,...,m−1,andTkTTk+1are(n−1)-simplicesforallvaluesofk.WehavebydefinitionρTT′ij=k=m−1Yk=0ρTkTk+1ijOurtrivialsetAoftherelationsforthesequantitiesfollowsfromthesamerelationsforthequantitiesμTij,μT′ijasabove.Inordertoprovethesetofrela-tionsBwepointoutthatanysuchclosedpathinthedualcelldecompositioncanbeobtainedasaproductofelementarypathsT0...Tmcorrespondingtothesimplicialstarsofeveryn−2-simplicesij⊂σ⊂St(ij).WehavehereSt(σ)=T0[...[TmandtherelationρT0T1ij...ρTmT0ij=μT0ijμT1jiμT1ij...μT0ji=0Lemmaisproved.Problem:IsitpossibletorecoverthewholeDGconnectionfromtheMinimalSubsetoftheabeliangauge-invariantcoefficientsρTT′ij?Whichin-variantsofDGconnectionshouldbeaddedifitisimpossible?Wearegoingtosolvethisproblemforthe2Dand3Dmanifoldsn=2,3wherethewholesetoftheadditionalinvariantsiseasytofindout:letuschooseanysetoftheclosedcombinatorial”framed”pathsγ1,...,γb1,a1,...,ator1representingthebasisofthehomologygroupH1(M,Z).Wedefinethefol-lowingabeliangauge-invariantTopologicalquantities:μ(γ)=YγμTll,l+1foreveryclosed”framedpath”γkconsistingofedgesthroughthevertices0,1,...,l,l+1,...,mk=0,equippedbysuchtriangles(n-simplices)Tlthat[l,l+1]⊂Tl.WearegoingtoprovebelowthefollowingTheorem1Foranyn≥2thesetofinvariants{ρTT′ij,μ(γk),μ(as)}3iscompletewherei6=j,ρTT′ij,[ij]⊂TTT′.Forthecompactoriented2-manifoldstheonlynontrivialrelationonthesequantitiesisY[ij]∈MρTT′ij=1,∂T=[ij]+...,∂T′=[ji]+...,T\T′=[ij]Here[ij]meansalledgesinthemanifoldM,the2-simplicesT,T′areorientedasprescribedbytheglobalorientation.Forthecompactorientedn-manifoldsthecompletesetofrelationsonthesequantitiescanbedescribedinthefollowingway:therearetrivialrela-tionsAforevery2-simplex[ijl]=Δ⊂TTT′whereρTT′ijρTT′jlρTT′li=1andtherelationsBforeveryclosedpathinthePoincaredualcelldecomposi-tioncorrespondingtotheboundariesofthedual2-cells.Forthedescriptionofthenontrivialrelationswechoosethesetofintegral2-chainsz1,...,zb2,u1,...,utor1}wherez1,...,zb2isthebasisofthegroupH2(M,Z)andusrepresentthebasisofcyc