Dirac operator and Ising model on a compact 2D random lattice

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

BI-TP2001/22

LPTOrsay01/89

DiracoperatorandIsingmodel

onacompact2Drandomlattice

arXiv:hep-lat/0110063v1 12 Oct 2001L.Bogacz1,2,Z.Burda1,2,J.Jurkiewicz2,A.Krzywicki3,C.Petersen1andB.Petersson1Fakult¨atf¨urPhysik,Universit¨atBielefeldP.O.Box100131,D-33501Bielefeld,Germany21InstituteofPhysics,JagellonianUniversityul.Reymonta4,30-059Krakow,Poland3LaboratoiredePhysiqueTh´eorique,B atiment210,Universit´eParis-Sud,91405Orsay,FranceAbstractLatticeformulationofafermionic eldtheoryde nedonaran-domlytriangulatedcompactmanifoldisdiscussed,withem-phasisonthetopologicalproblemofde ningspinstructuresonthemanifold.Anexplicitconstructionispresentedforthetwo-dimensionalcaseanditsrelationwiththeIsingmodelisdiscussed.Furthermore,anexactrealizationoftheKramers-Wannierdualityforthetwo-dimensionalIsingmodelonthemani-foldisconsidered.Theglobalpropertiesofthe eldarediscussed.

TheimportanceoftheGSOprojectionisstressed.Thisprojec-tionhastobeperformedforthedualitytohold.

Introduction

ThemasslessMajoranafreefermiontheorybelongstothesameuniversalityclassasthecriticalIsingmodelonaregularlattice[1,AnexplicitconstructionoftheMajorana-Dirac-Wilsonfermion eldtheoryonaran-domlytriangulatedplanewasintroducedinThistheorywasshowntobeequivalenttotheIsingmodelalsooutsidethecriticalregion.Inref.[5]Cartesiancoordinateswereassignedtothenodesofthelattice.Thedirec-tionsofthelinksandoftherelatedgammamatriceswereexpressedintheglobalframeoftheplane.Thisapproachworksforlatticesembeddedina

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

atbackgroundwhereonehasatone’sdisposalaglobalframeoftheunder-lyinggeometry[6,7,8].However,ifonewantstogeneralizeittoalatticeonacurvedbackgroundwherenoglobalframeexists,a eldoflocalframes

[9,10,11]hastobeintroduced.Thisbeingdone,onecanputfermionsonacurvedmanifoldwithanytopologyandonecaneventuallyattack,forexam-ple,problemsof eldtheoryonadynamicalgeometrylikethoseencounteredinstringtheoryorinquantumgravity[14,15,16,17,18].

Thisgeneralizationwaspartiallycarriedoutin[10,11]whereanex-plicitconstructionoftheMajorana-Dirac-Wilsonoperatorsoncurvedcom-pacttwo-dimensionallatticeswasintroduced.

Hereweextendthesestudies.Inparticular,wediscussthesigni canceoftheGSOprojection,whichasinstringtheoryalsohereplaysanimpor-tantphysicalrole[12,13].WeshowthatwithacarefultreatmentoftheglobalpropertiesoftheDiracoperatorandofthespinstructuresonthemanifoldonecan ndastrictmathematicalone-to-oneequivalencebetweenthepartitionfunctionoftheMajorana-WilsonfermionsandthatoftheIsingmodel.WeshowexplicitlythatinourdiscretizationoftheDiracoperatoronacompactmanifold,theGSOprojection-thesummationoverallspinstructures-doesremovethenon-contractiblefermionicloops,thatisthosenotcorrespondingtothedomain-wallsofthecorrespondingIsingmodel.Further,weshowthatforthedualitytoholdexactlyasaone-to-onemapbetweentheIsingmodelonatriangulationandonitsduallattice,asortofGSOprojectionhasalsotobedone.Di erentspinstructuresfortheIsing eldaresimulatedbyphysicalcutsproducedbytheintroductionofantifer-romagneticloops,whichmimicantiperiodicfermionicboundaryconditions.Thepaperisorganizedasfollows.Insection1wegiveanintroductiontotheproblemofde ningtheDiracoperatoronacompactmanifold.Itistext-bookmaterial[13,20].Werecallithereforcompleteness,tokeepthearticleself-contained.Insection2,weshowhowtoadaptthestandardWil-sondiscretizationschemeoffermionsontheregulartranslationallyinvarianthypercubiclattice[22]tothelocal-framedescription,whichcanbegeneral-izedtothecaseofirregularcurvedlattices.Insection3,usingasanexamplethestandardtoroidalregularlattice,wediscussthesignproblemandtheglobalpropertiesofthefermionic eldonacompactmanifold.Insection4wearguethatinthecaseofirregularlatticesthelocalframedescriptionisparticularlynatural,andtheninsection5weshowhowtoliftthiscon-structiontothespinorialrepresentation.Indoingthisweintroducerotationmatricesbetweenneighboringframeswhicharecrucialfortheconstruction.Inparticular,usingthespinorialrepresentationofthesematricesweareabletode neinsection6theDirac-Wilsonoperator.Thestandardde nitionofthepartitionfunctionrepresentingquantumamplitudesisrecalledinsection

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

7.Inthissectionwealsolistthepropertiesofthemathematicalexpressionsencounteredincalculatingthepartitionfunction.Insection8wecalculatethepartitionfunctionusingthehoppingparameterexpansion.Thetopolog-icalloopsignproblememergesnaturallythere.Theissueofloopsignsisdiscussedinmoredetailinsection9wherethesignisde nedasafunctionofclassesofloophomotopies.Therelationbetweensignsofnon-contractiblefermionicloopsandofdomain-wallsinIsingmodelandthetopologicalas-pectofthedualityisdiscussedinsection10.Insection11wegivetwoanalyticexamples,calculatingthecriticaltemperatureoftheIsingmodelonthehoneycomblatticeandthecriticalvalueofthehoppingparameteronthedynamicaltriangulation,makinguseoftheexistenceoftheexactmapbetweentheIsingmodelandthefermionicmodel.Weclosewithashortdiscussion.

1Preliminaries

Theaimofthispaperistodiscretizeatheoryoffermionsonarandom,possibly uctuatinggeometry.Letus rstrecallsomebasicfactsaboutthecontinuumformulationofthisproblem.

ConsideraD-dimensionalcompactRiemannianmanifold,onwhichacoordinatesystemξµisde ned.Ifanonsingularchangeofcoordinatesξµ→′ξµisperformedatsomepointxonthemanifold,thenalineartransformationofthecomponentsofanyvectorortensor eldinthetangentspaceatxhasalsotobecarriedout,inordertoensuretheinvarianceofthetheoryundercoordinatetransformations.Forvectors,thematrixofthislineartransformationreads:′ ξµµAν(x)=

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

(orthonormality)ande1(x)∧e2(x)...∧eD(x)>0(orientability),wherethesymbols·and∧denotetheinternalandexternalproducts.

Expressedinagivencoordinatesystemξµ,theorthonormalityandori-entabilityconditionsread:

νgµν(x)eµa(x)eb(x)=δab,e(x)≡deteµa(x)=

2

whereσab=1ωµabσabψ,(4)

2

¯γµ µψ=dDξeψ

DD12¯(x)D(x,y)ψ(y).dxdyψ(5)

TheDiracoperatoronthemanifoldis

D(x,y)=δ(x y)γa(x)· a(x),(6)

or,lessformally,justγ·D.Weshalldiscretizethisoperatorinthenextsection.Beforedoingso,however,letusdiscussitstopologicalproperties.Locally,onecanalwaysde neacontinuouslyvarying eldofframes.However,doingthisgloballyforacompactmanifoldisusuallyimpossible.

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Whatcanbedoneinsteadinthiscaseistocoverthemanifoldwithopenpatches,ineachofwhichonecanseparatelyde neacontinuous eldofframes,andforanyregionofoverlappingpatchesUandVprovidetransitionmatricesforrecalculatingtheframeswhengoingfromonepatchtotheother:

[eU]a(x)=[RUV]ba[eV]b(x).(7)

Here,thetransitionfunctionRUVisaSO(D)rotationmatrix.Itfollowsthatthespinorsintheoverlappingregioncanberecalculatedas:

[ψU]α(x)=[RUV]βα[ψV]β(x).(8)

whereRUVisanimageofRUVinthespinorialrepresentation.InaregionwherethreepatchesU,V,Wintersect,thetransitionmatricesmustobviouslyful llthefollowingself-consistencyequations:

RUVRVWRWU=,RUVRVWRWU=.(9)

Thesecondequationcanbealmostautomaticallydeducedfromthe rstonebyrewritingitinthespinorialrepresentation.However,becausethespinorialrepresentationR→±Ristwo-valued,thesignsoftheR’sarenotautomatically xedbyR’s.Inotherwords,onehastoadjustinadditionthesignsofthetransitionfunctionsforthespinorsinsuchawaythattheconsistencyequationisful lledinanytripleintersectingpatch.

Thisisaglobaltopologicalproblem.Ifitissolvableontheentiremani-fold,themanifoldissaidtoadmitaspinstructure.Intwoandtreedimen-sions,thequestionoftheexistenceofaspinstructurereducessimplytothemanifoldorientability;inhigherdimensionstheproblemismorecomplex.Anotherimportantquestionis:howmanynon-equivalentspinstructuresareadmittedonagivenmanifold?Intwodimensions,theansweris22g,wheregisthegenusofthemanifold[13].Thisnumberisrelatedtothenumberofpossiblesignchoicesforindependentnon-contractibleloopsonthemanifold.

Agooddiscretizationschemeshouldre ectallthesetopologicalproper-ties.Aswillbeseen,theexplicitconstructionfortwo-dimensionalcompactmanifoldstobeproposedinthepresentpaperdoesful llthisrequirement.TheDiracoperator(6)canbeexpressedinlocalcoordinatesasγµ µ,oralternativelyinframecomponentsasγa a,i.e.withoutreferencetolocalcoordinates.Theconstructionproposedinthispaperis,infact,coordinate-free:weshallexpresseverythinginframeindicesa,withoutreferringtocoordinateindicesµ.

Inthelatticeconstruction,thenearestneighborrelationthatmimicsthestructureofthecontinuumformulationwillbegivenbyalocalvector:at

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

eachpointiontheduallatticeweshallde nelocalvectorsnjipointingtothethreeneighboringverticesj.Tocalculatederivatives(di erences)inthedirectionofnjiweshalldecomposeitinthelocalframeeia.Similarly,allvector,tensorandspinorindicesofobjectsfromthetangentspaceswillbeexpressedintheselocalorthonormalframes.Liftingtheconstructionfromthevectortothespinorrepresentationoftherotationgroup,weshallstoretheinformationaboutnearestneighborsintheformofrotationmatrices.Werefertothemastothe‘basicrotations’,anddenotethembytheletterB.Theadvantageofusingrotationsisthatwecanexpresstheminthespinorialrepresentation,B→B.

2Thediscretizationscheme

Letusstartwithadiscussionoffermionsonaregular atlattice,usingtheWilsonformulation[22].Then,weshallseehowtogoover,aftersomemodi cations,tothecaseofirregularlattices.

TheDirac-Wilsonactionforfreefermionsreads:

S= K

2 Ψ¯ Ψ .(10)

wherethemulti-index describesthenodepositiononthelattice,and µisoneoftheDdirectionsofthelattice.Thegammamatricesγµarerigidlyassociatedwiththesedirections:

{γµ,γν}=2δµν.(11)

IntheEuclidean

mannvariablesΨ¯sector,theDirac eldisrepresentedbyindependentGrass-

αandΨα,α=1,...,N.Inparticular,forD=2,the

dimensionofthespinorrepresentationisN=2.Inthefollowing,spinorindiceswillusuallybeimplicit;weshallwritethemexplicitlyonlywhennecessary.

Weshallnowrewritetheaction(10)inacoordinate-freeformwhichcanbeextendedtothecaseofirregularlattices.

Insteadofusingthemulti-index todescribethevertexposition,weassociatewitheachvertexasinglelabel,sayi,whichisacoordinate-freeconcept.Obviously,theparticularchoiceofalabeldoesnothaveanyphysicalmeaningandthetheoryhastobeinvariantunderrelabelings.Thephysicalinformationwillbeencodedinthenearestneighborrelations.

Usingtheselabels,theactioncanbecastintothefollowingform:

S= K Ψ¯iH1

ijΨj+ ij

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Figure1:Ahypercubiclatticewithtranslationalsymmetryandaglobalframethat xesthecoordinatedirectionsfortheentirelattice.Alterna-tively,onecanuselocalframesthatvaryfrompointtopoint.Thishastheadvantageofbeinggeneralizabletoacurvedbackground.

wherethe rstsumrunsoverorientedlinksconnectingnearestneighborsonthelattice.ThehoppingoperatorHijisde nedas

Hij=1

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Onecanintroduceindependentorthonormalframesasin g.1.Ateachlatticepointionehasapairoforthonormalvectors(ei1,ei2).Inparticular,onatorusthelocalframeseiacanbeobtainedfromtheglobalframeEabylocalrotations:

(15)eia=[Ri]b

aEb.

ThespinorcomponentsΨiaretransformedbytheserotationsintotheircom-ponentsinthelocalbasesψi:

ψiα=[Ri]β

αΨiβ, 1¯α=Ψ¯βψiRii αβ,(16)

wherethematricesRibelongtothehalf-integerrepresentationoftherota-tionsRi:ab1RiγaR (17)i=[Ri]bγ.

Incomponent-freenotationtheequations(15),(16)and(17)read:

ei=RiE,ψi=RiΨi,¯i=Ψ¯iR 1,ψi1RiγR i=Riγ.(18)Usingthisnotation,ingthelocalframes,wecanwritetheaction(12)as:

S= K ij ¯iHijψj+1ψ

21 1Ri[1+nij·γ]R iRiRj.

Here,Uijisamatrixallowingtorecalculatethecomponentsofaspinorgoingfromaframejtotheframei.Inotherwords,itisasortofaconnectionmatrixthatperformsaparalleltransportofspinorsbetweenneighboringvertices.

Sofar,equation(20)iswritteninahybridnotation,becausethespinorsarealreadyexpressedinthelocalframeseiwhereasnijandγarestillwrittenintheglobalframeE.However,applying(17)to(20)one nds:

1a 1abRinij·γR i=nij,aRiγRi=nij,aRbγ=nij·γ(i) Uij (20)(21)

whereinthelocalbasisthevectornijhasthecomponents

anij,b=nij,aRb,(i)(i)(22)

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

di erentfromtheglobalframecomponentsnij,a.Thenewbracketedindex(i)nowdi erentiatesbetweendi erentlocalframeswherethecomponents

(i)(j)ofthevectorarecalculated;thus,nijreferstothesamevectorasnij,

butwithcomponentsexpressedinadi erentframe.Intuitively,whattheequationmeansissimplythatthecomponentsofavectorinarotatedbasiscanbealternativelycalculatedbyperformingtheinverserotationonthevectoritselfwhilekeepingthebasis xed.

Animportantpointisthatthecrossoverfromtheglobaldescriptiontothelocaloneasin(21)preservesthenumericalvaluesoftheγamatrices.Inotherwords,γ1associatedwiththelocaldirectionei1atapointihasthesamenumericalvalueasγ1associatedwiththeej1atanyotherpointj,andlikewiseforγ2.(i)Usingthecomponentsnijofthenearestneighborvectorinthelocalframei,wecannowwrite(20)as

Hij=1

2 1+(i)nij·γUij= 1

2 1+(j)nij·γ. (24)

Thesedi erentexpressionsforHijcorrespondtodi erentwaysofcalculating¯iHijψjin(19).Onemethodisto rstparalleltransportthehoppingtermψ

thespinorψjfromjtoi,gettingUijψj,andthentocalculatethecorrespond-ingscalarintheframei,asisdoneonthelefthandsideof(24).Sometimesitisconvenienttoreplacenij= njiinordertochangethedirectionofthevectorbetweenindicesiandj,asisdoneinthesecondexpression.Alterna-¯ifromitoj,whichgivesψ¯iUij,tively,onecan rsttransportthespinorψ

andthencalculatethecorrespondingscalarintheframej,asisdoneontherighthandside,etc.Alltheseexpressionsareequivalentandcanbededucedfromeachother,sothatthemostconvenientoneisalwayschosen.

Theadditionalupperindexinthebracketsmakesformulaevisuallylesstransparentbutremovesthelogicalambiguitywhichotherwisemightleadtoconfusion.Wewillthereforeextendthisnotationtoallobjectsoccurring(j)(i)inourconstruction.Forexample,ψj=Uijψjmeansthatthespinorψjis

(j)(i)¯iistransported¯i¯i=ψUijmeansthatψtransportedfromjtoi.Similarly,ψ

fromitoj.Thereisnosummationovertherepeatedindices.Theonlyexceptionwillbemadeforobjectscalculatedintheframebelongingtothepointwheretheyarethemselvesde ned,sinceinthiscaseleavingoutthe

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

upperindexdoesnotcauseanyambiguity.Forexample,wewillwriteψi(i)insteadofψi.

Usingthisnotation,theWilsonactionbecomes:

S= K ij 1¯ψi2 i¯iψi.ψ(25)

Contraryto(10),thisformoftheWilsonactioncannowbegeneralizedtoanyrandomirregularlattice.Italsomakesdirectcontactwiththecontinuumformalism(5).Finally,notethatitisinvariantunderachangeofthelocalframes:

ei→Riei,Ψi→RiΨ,1¯i→Ψ¯iR Ψi,1Uij→RiUijR j.(26)whereRiarearbitrarylocalrotations,andRiarethecorrespondingmatricesinthespinorialrepresentation.

3Atopologicalproblem

Letusreturntotheconsequencesofthefactthatthe(spinorial)half-integerrepresentationoftherotationgroupisactuallyonlyarepresentationuptoasignfactor.

Intwodimensions,theSO(2)groupcanbeparametrizedbyasingleparameterφ∈[0,2π).Foragivenvalueofthisparametertherotationmatrixisgivenby:

R(φ)=eφ =cos(φ)+ sin(φ)= cos(φ)sin(φ)

sin(φ)cos(φ) (27)

where abisthestandardantisymmetricmatrixwith 12=1.

ThecorrespondingmatrixR(φ)inthespinorialrepresentationisR(φ)=ie

σφ22=cos(φ/2)+ sin(φ/2)=cos(φ/2)sin(φ/2)

sin(φ/2)cos(φ/2) (28)

where =iσ2isanantisymmetrictensorthatisnumericallyidenticalwiththeonein(27).Thedi erence,ofcourse,isthatthetensorinequation(27)hasframeindices abwhereastheonein(28)hasspinorialindices αβ.

Inorderto xtheglobalsignofR(φ),onshouldcontroltheangleφintherange[0,4π)ratherthantheusual[0,2π).Thiswouldrequirechanging

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Figure2:Rotationofalocalframeby2π.Eventhoughtheresultingframecon gurationisobviouslythesameasbefore,spinorcomponentscanchangetheirsignduetothesignambiguity.

continuouslytheangleandcalculatingtheoverallchangedφkeepingtrack

ofthenumberof‘fullcircles’.However,thiscannotbedoneheresincetherelativeanglesbetweentheframeseiaaredeterminedinthefundamentalrange[0,2π)only.

Thesignambiguityalsohastopologicalconsequences.Consideroncemoretheregular,toroidal, atlatticeandchooseonitaconstant eldofidenticalframes(see g.2).We rstsetUij=foralllinks.Trivially,ifatavertexitheframeisrotatedby2π,theframecon gurationdoesnotchange.However,becauseRi(2π)= inthespinorialrepresentation,alllinksemergingfromiacquireanegativesignUji= accordingtothetrans-formationlaw(26).Theresulting‘sign eld’isdi erentfromtheoriginalonebutatthesametimeequivalenttoit.Byrepeatingthisprocedureinotherverticesonecanproducemanydi erent,butequivalent,signcon gurationsforthesame eldofframes.

Itiseasytoseethatalocalrotationofaframeby2πpreservestheoverallsignofallelementaryplaquettes,i.e.theproductofsignsofalllinksontheplaquette’sperimeter.Thus,foranycon gurationobtainedfromtheoriginalone,allelementaryplaquetteshaveapositiveoverallsign.Weshallrequirethistobetrueingeneral,i.e.foranycon gurationoflocalframesonthelatticethesignofallelementaryplaquettesissetto+1;thisensuresthatspinorsremainunchangedbyparalleltransportaroundanyelementaryplaquette.Thisrequirementisdictatedbytheunderlyingcontinuumtheory,inwhichparalleltransportofaspinoraroundaclosedloopinalocally at

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

PL∩P

L∪P L

Figure 3: A

small deformation of a loop L (bold line) by an elementary loop P (dashed line), resulting in the loop L′ .

patch leaves the spinor intact. Later on, for curved lattices, we shall modify this constraint so as to adjust it to the case where there is a de cit angle inside an elementary plaquette. Assuming that all elementary plaquettes have a positive sign we can prove now some simple topological theorems concerning the signs of loops on the lattice. It is convenient to de ne an auxiliary operation for loops on a lattice, to be called a small deformation of a loop. To deform a loop L, we pick an elementary plaquette P which shares at least one common link with L, and substitute the intersection L∩ P by the complementary part of P, resulting in a new loop L′= L∪ P L∩ P (see g. 3).1 As with elementary plaquettes, we can de ne the overall sign of a loop as the product of signs of all links on the loop. One easily checks that the sign of the deformed loop L′ is the same as that of L– namely, the addition of P to L cannot change the sign because P has a positive sign by default, and the removal of the intersection L∩ P cannot change the sign because each link is‘removed twice’ (once from P and once from L), so that the total number of removed links is always even. Any contractible loop can be obtained from the elementary loop by a sequence of small deformations. Thus all contractible loops have positive signs.Somewhat more precisely, we also have to require that the intersection L∩ P be connected, so as to avoid situations in which a small deformation splits a loop into two or more parts.1

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Figure4:Anon-contractiblelooponatoroidallatticewithaconstantframe.ThesinglelinksdrawnasboldlinesallhavetransitionmatricesUji= ,whereasallotherlinkshaveUji=;asaconsequence,theloophasanegativeoverallsign.

Thisisnot,however,thecasewithnon-contractibleloops,whichcantakeeithersign.Anexampleofaloopwithnegativesignisshownin g.4:ifwechooseUji= foronecompleterowoflinksonthelattice(asinthe gure)andUji=everywhereelse,thenanyloopthatencirclesthelatticeintheydirectionpassesthroughexactlyonelinkwithnegativesign,andthushasanegativeoverallsign2.

Obviously,twosigncon gurationsareequivalentifonecantransformoneintotheotherbyasequenceoflocalrotationsRi(2π)= .Becauselocalrotationsdonotchangethesignofanyloop,acon gurationwithatleastoneloopofnegativesigncannotbeequivalenttoacon gurationthathasonlyloopsofpositivesign.Inotherwords,thetwosigncon gurationsaretopologicallydistinct.

Now,usingsmalldeformationswecaneasilyprovethatallnon-contractibleloopsencirclingthetorusinthesamedirectionmusthavethesamesign.Thismeans,forexample,thatitissu cienttocalculatethesignofjustone‘vertical’loop(whichencirclesthelatticeintheydirection)toknowthesignofallotherverticalloops.Moregenerally,thesignofaloopisnotapropertyofasingleloopbutratherofallloopsinthesamehomotopyclass,i.e.thosethatcanbeobtainedfromeachotherbyasequenceofsmall

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Figure5:(Left)Alatticewithtoroidalboundaryconditions.(Right)Alat-ticewiththeboundaryconditionsofaKleinbottle.Thearrowsindicatethedirectionsinwhichtheoppositeedgesaretobetakenwhenjoinedtogether.deformations.Onthetorustherearetwoindependentnon-trivialhomotopyclassesofloops(‘vertical’and‘horizontal’)and,therefore,fourdistinctpos-siblesigncon gurations.These,inturn,correspondtofourdistinctspinstructures.

Thestatementcanbegeneralizedbyobservingthatthereare2gindepen-dentclassesofnon-contractibleloopsonasurfacewithgenusg,whichmeansthatthereare22gdi erentsigncon gurationsandthusthesamenumberofspinstructures.Inparticular,alatticewithsphericaltopologyadmitsonlyonespinstructure.

Ontheotherhand,onanon-orientablelatticeonecannotgloballyde nea eldoforientableframes.Anexampleofsuchalatticeistheso-calledone-sidedtorusorKleinbottle,whichisconstructedinthesamewayasthestandardtorusbuthasdi erentboundaryconditions,asshownin g.5.Itispossibletoshowthataframetransportedalongaclosedpathwouldhavechangeditshandednessafteracompletetouraroundthelattice.Becausetheredoesnotexistsa eldoforientableframes,onecannotinthiscasede neaspinstructureoraDiracoperator.

4Localframesonarandomlattice

Theform(10)oftheWilsonactionisparticularlysimplenotonlybecauseofthesimpletopologyofthetorus,whichallowsforthede nitionofaglobalframe,butalsobecauseoftheregulargeometryofthelatticewhichevery-

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

whererepeatsthesamesimplemotif.Onanirregularlattice,localanglesandlinklengthschangefrompointtopoint.Thismustbere ectedinthecon-structionofthehoppingterm,whichdependsontheselocaldetailsthroughthecovariantderivative.

Tomakethegeometricalpartofthediscussionassimpleaspossible,andtominimizethenumberoflocaldegreesoffreedomofthelattice,werestrictthediscussiontoequilateralrandomtriangulations.Thisgreatlyreducesthenumberoflocaldegreesoffreedom,makingthediscussionmoretransparentandallowingustofocusontheinterestingtopologicalpartoftheproblem.Letusmention,however,thatthepresentedconstructioncanbeeasilygeneralizedtothecaseofvariablelinklengthsandangles.

Onanequilateraltriangulation,thelocalgeometryiscompletelyencodedintheconnectivityofthelattice;allotherdetailsare xedbythesimplegeometryoftheequilateraltriangle.Inparticular,thede citangleatavertexiisdeterminedsolelybyitsorderqi: i=(6 qi)π/6.

Thelocalcurvatureofthelatticeisconcentratedintheverticesofthetriangulation.Thegeometrybecomessingularinthesepointsandthereforeitisdi culttoprovideauniquede nitionofatangentspaceatthevertices.Itismoreconvenienttode netangentspacesatthedualpointsofthelattice,i.e.atthecentersofthetriangles.Insideeachtrianglethegeometryislocally atandthusnaturallyspansatangentspace.Wethereforelocatealllocalframes,andalsoallfermionic elds,atthecentersofthetriangles.Eachpointiwherea eldisde nedhasthenthreeneighbors,eachofwhichatthesamedistancefromi.Thevectorspointingtotheneighborsarealsoequallyspacedintheangularvariable,i.e.theyareseparatedbyangles2π/3.

Beforede ningthefermionic elds,however,letusdiscussthepropertiesofthe eldoforientedorthonormallocalframesonsucharandomtriangu-lation.Anexampleofatriangulationdecoratedwithframesisshownin g.6.

Ateachtriangleilivetwoorthonormalvectorsei1andei2suchthateia·eib=δab.Apartfromtheinternalproductthereisalsoanexternalone∧,whichenablesonetochooseframeswiththesamehandednessei1∧ei2>0foralltriangles.Nowconsidertwoneighboringtrianglesiandj,eachendowedwithitsownframeeiandej.Theinteriorsofthetwotrianglestogetherforma atpatchofthetriangulation.Onecanthinkofthetwoframesasbeingtwoalternativeframesforthesamepatch.Onecancalculatecomponentsofourobjectsineitheroneofthem,andeasilyrecalculatethemwhengoingfromonetotheother.TothispurposeintroduceSO(2)transitionmatricesUijandUjisuchthat:

UijUji=,ei=Uijej,

ej=Ujiei.(29)

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Figure6:Asmallpieceofarandomtriangulationwithlocalframes.Ujkisthetransitionmatrixbetweentheframesatkandj,andqiistheorderofthevertexi.

Onecanrepeatthesamecalculationforanypairofneighboringtrianglesanduseittotransportaframebetweenanytwopointsi1andinalonganopenpathC=(i1,i2,...,in):

ein=Uinin 1...Ui3i2Ui2i1ei1=U(C)ei1.(30)

Sincewestudyatheorywhosecontentisindependentofthechoiceofframes,weareinterestedinthepertinenttransformationlawsandinquantitiesin-variantunderlocalSO(2)rotationsoftheframes:ei→e′i=Riei.TheobjectU(Cji)=Ujk...Uniforanyopenpathbetweeniandjtransformsas: 1U(Cji)→U′(Cji)=RjU(Cji)Ri,(31)asonecanseefrom(29).Inparticular,foraclosedpathLibeginningandendingatthesametrianglei,U(Li)transformsas

1U(Li)→U′(Li)=RiU(Li)Ri,(32)

andhenceTrU(Li)isaninvariant.Moreover,thisinvariantdoesnotdependonthechoiceoftheinitialpointioftheloop,andisthusapropertyoftheloopLitself.Itisageometricalquantityrelatedsimplytothetotalangle dαbywhichatangentvectorisrotatedwhentransportedalongtheloop.Ona atlattice,thisangleisamultipleof2π.Onacurvedlatticethe

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

situationissomewhatmorecomplicated.Inparticular,foranelementaryloopLqsurroundingavertexoforderq,theloopinvariantis

1

3=cos(6 q)π

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

3 =cosqπ

3.(40)

asclaimedin

(33).

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

5Thespinorialrepresentation

ThenextstepistolifttheconnectionsUijtothespinorialrepresentation,Uij→Uij.Wecontinuetousetheconventionofdenotingallrotationma-tricesinthespinorialrepresentationbycalligraphicletters:U→Uforconnections,B→Bforbasicframerotations,T→TforturnsandF→Ffor ips.

Thestartingpointoftheconstructionisthedecomposition(36).Ifwewriteitinthespinorialrepresentation,eachmatrixthatoccursinthisequa-tionisdeterminedonlyuptoasign:e φ→±e φ/2(28).Theideaisnowtoa xthespinorialrepresentationofallmatricesontherighthandsideof

(36)withapositivesign:

B=e φ→B=e φ/2

→F=e π/2= ,F=e π=(41)(42)

andkeepthesignsji=±1asaseparatevariableforeachlink:

Uij→Uij=sij[Bj] 1 Bi,(i)(j)Uji→Uji=sji[Bi] 1 Bj.(j)(i)(43)Wedemandthatparalleltransportofaspinoralongagivenlinkandbackdoesnotchangethespinor.Weseethatthisisindeedthecase,i.e.wehaveUjiUij=if

sjisij= 1.(44)

Usingasimilarcalculationastheonewhichledto(40)one ndsthatinthespinorialrepresentationtheloopinvariantforanelementarylooparoundavertexis1.(45)2

where qisthede citangle,andSLqisasign±.Thefactorone-halfin

theargumentofthecosinefollowsfrom(42).Thetotalsignoftheloop,denotedbySLq,dependsonthechoiceofsignssijin(43)andhastobecalculated.WerequirethatthesignssijarechoseninsuchawaythatforeachelementaryloopthesignSLqispositive:

SLq=1.(46)

Notethatforq=6thisrequirementisnatural,becausetheplaquetteis at, 6=0,andasdiscussedbeforefora atpatchtheparalleltransportshouldbetrivial:U(L6)=.ThusindeedweshouldhaveSL6=1.Alsoforotherq’stherequirementcanbemotivated.Thegeometryofanelementaryplaquettecorrespondstothegeometryofa atcone,whichhasasingularity

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

Figure8:Theinternalgeometryofasetoftrianglesaroundavertexisthesameasthataroundthepeakofacone:itis ateverywhereexceptforasinglepointwherethecurvatureisconcentratedinasingularity.Wecandeterminethesignofanylooparoundtheconeifwe rstregularizethissingularityby‘ attening’thecone,and ndS=+1.

atthepeak.Theelementaryloopencirclesthissingularityatsomedistancerfromthepeak.Onecanregularizethesingularitybysmoothingthepeak,i.e.replacingitbyadi erentiablesurface(see g.8).

Indoingso,onedeformsonlyaverysmallregionwithinadistanceof aroundthepeak,where r.Nowimaginethatweshrinktheloop,continuouslydecreasingitsradius.ThenTrU(r)and (r)bothchangecon-tinuouslywithr.Inthelimitr→0,theloopendsuponthetopoftheregularizedpartofthegeometrywhichis at.Thus,againS=+1inthelimitofr→0.Thisalreadyissu cienttohavepositivesignforallvaluesofr,becauseinthecourseofcontinuouschanging,thede citangle waschangingcontinuouslyandhencethesignScouldnothavejumpedbetweennegativetopositivevalueswithoutmakingUdiscontinuous.Inotherwords,Smustkeepthevalue+1forallr.

Becausetheregularizedzonecanbemadearbitrarilysmall,weassumethatthetriangulatedlattice,whichcorrespondstothelimit →0,inheritsthepropertyoftheregularizedgeometry:thesignofanyelementaryloopisSLq=+1foranyq.

InordertoenforcetheconstraintSLq=+1foreachplaquette,onehas

toestablisharelationbetweenSLqandthesignsoflinkssji.Inanalogyto

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimens

(37),onecan

calculate

theloopinvariantinthespinorialrepresentationas:

TrU(L)=Trn k=1Uik+1ik=

ksik+1ik·Tr

kTik.(47)

Comparingthistotheresultpertinentforthefundamentalrepresentation

(37),one ndsthatanadditionalproductoflinksignsappears,asexpected.Butthereisalsoanothersourceofsignshiddenin(47).IthasitsorigininthespinorialrepresentationoftheturnmatricesT→T.Surprisingly,andincontrasttothefundamentalrepresentation,theproductofbasicrotationsdependsonthepositionoftheframe.Moreprecisely,calculatingtherotationcorrespondingtotheturntakenbythepathatikonegetsanadditionalsignzik:(i)(ik) 1(±)ikπTik=Bikk[B]F=zeiik 1k+1

2φik+1ik ,Bik 1ik=e1

2(φik+1ik φik 1ik+π) =e1

2( 4π/3+π)=e π

2(2π/3+π) =e5π6 .(52)

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