A lattice approach to the conformal $\OSp(2S+2|2S)$ supercoset sigma model. Part I: Algebraic structures in the spin chain. The Brauer algebra

0.42 MB·English

by  Constantin Candu

A lattice approach to the conformal OSp(2S + 2|2S) supercoset sigma model
Part I: Algebraic structures in the spin chain. The Brauer algebra.
Constantin Candu(1) and Hubert Saleur(1,2)
ServicedePhysiqueThe´orique,CEASaclay,
GifSurYvette,91191,France(1)
DepartmentofPhysicsandAstronomy,UniversityofSouthernCalifornia
LosAngeles,CA90089,USA(2)
8 Abstract
0
Wedefine and study a latticemodel which weargue is inthe universality class of the OSp(2S +2|2S)
0
supercosetsigmamodelforalargerangeofvaluesofthecouplingconstantg2. Inthisfirstpaper,weanalyze
2 σ
indetailsthesymmetriesofthislatticemodel,inparticularthedecompositionofthespaceofthequantumspin
n chainV⊗LasabimoduleoverOSp(2S +2|2S)anditscommutant,theBraueralgebraB (2). Itturnsoutthat
L
a V⊗L isanonsemisimplemoduleforbothOSp(2S +2|2S)and B (2). Theresultsareusedinthecompanion
J L
papertoelucidatethestructureofthe(boundary)conformalfieldtheory.
2

] 1 Introduction
h
t
-
p The solution of the AdS × S5 worldsheet string theory is one of the cornerstones of the AdS/CFT duality
5
e program.Despitecontinuouseffortandprogressonclassicalaspectsinparticular[1],andthegenerallyaccepted
h
presenceofbothintegrabilityandconformalinvariancesymmetries,mostaspectsofthequantumtheoryremain
[
elusive.

1 Itisnaturaltotrytounderstandsomeaspectsofthisquantumtheorybyfirsttacklingsimplermodelswith
v
similarproperties. ThesocalledOSp(2S +2|2S)cosetmodel-specifically,asigmamodelonthesupersphere
0
OSp(2S + 2|2S)/OSp(2S + 1|2S) - is a very attractive candidate for such an exercise: like the AdS × S5
3 5
4 worldsheettheoryitisconformalinvariantanditstargetspaceisasupergroupcoset. Ofcourse,itlacksother
0 aspectssuchastheBRSTstructureofthestringtheory.
. Apart from the string theory motivation, models such as the OSp(2S + 2|2S) coset model are extremely
1
0 interestingfromthepureconformalfieldtheorypointofview.Indeed,theyaresigmamodelswhicharemassless
8 withoutanykindoftopologicalterm,andforalargerangeofvaluesofthecouplingconstantg2. Tomakethings
σ
0 morepreciseletusbrieflyremindthereaderofsomegeneralities. Superspheresigmamodelshavetargetsuper
v: spacethesupersphereSR−1,2S :=OSp(R|2S)/OSp(R−1|2S)andcanbeviewedasa“supersymmetric”extension
i ofthenonlinearO(N)sigmamodels(whichdiffersofcoursefromtheusualO(N)“supersymmetric”models).
X
Useascoordinatesarealscalarfield
ar φ:=(φ1,...,φR+2S)
wherethefirstRcomponentsarebosons,thelast2S onesfermions,andtheinvariantbilinearform
φ.φ′ = J φiφ′j
ij
X
whereJ istheorthosymplecticmetric
I 0 0
R
J = 0 0 −IS 
Idenotingtheidentity.Theunitsupersphereisdefin0edbIyS the0constraint
φ.φ=1
Theactionofthesigmamodel(conventionsarethattheBoltzmannweightise−S)reads
1
S = d2x∂ φ.∂ φ
2g2 Z µ µ
σ
1
TheperturbativeβfunctiondependsonlyonR−2S toallorders(see,e.g.,Ref[2]),andisthesameastheone
oftheO(N)modelwithN :=R−2S.Physicscanbereliablyunderstoodfromthefirstorderbetafunction
β(g2)=(R−2S −2)g4 +O(g6)
σ σ σ
The modelfor g2 positiveflows to strongcouplingfor R−2S > 2. Like in the ordinarysigma modelscase,
σ
thesymmetryisrestoredatlargelengthscales,andthefieldtheoryismassive. ForR−2S < 2meanwhile,the
modelflowstoweakcoupling,andthesymmetryisspontaneouslybroken. Oneexpectsthisscenariotowork
for g2 small enough, and the corresponding Goldstone phase to be separated from a non perturbative strong
σ
couplingphasebyacriticalpoint.
The case we areinterested in hereis R−2S = 2, where the βfunctionvanishesto all ordersin perturba-
tion theory, and the model is expectedto be conformalinvariant, at least for g2 small enough, the Goldstone
σ
phase beingreplacedby a phase with continuouslyvaryingexponentsnotunlikethe low temperatureKoster-
litz Thouless phase. How the group symmetry combines with the (logarithmic)conformalsymmetry in such
modelsislargelyunknown.Itisanessentialquestiontobesolvedbeforeanyseriousattemptstounderstanding
universalityclassesinnoninteractingdisordered2Delectronicsystemscanbecontemplated[3].
TheOSp(2S +2|2S)cosetmodelwasconsideredinparticularintwopapersbyMannandPolchinskiusing
the massless scattering and Bethe ansatz approaches. This is indeed a natural idea, since supersphere sigma
modelsare in generalintegrable, and, when massive (ie R−2S > 2) can be describedby a scattering theory
involvingparticlesinthefundamentalrepresentationofthegroup.TheS matrixiswellknown
Sˇ(θ)=σ (θ)E+σ (θ)P+σ (θ)I
1 2 3
Here,Iistheidentity,Pisthegradedpermutationoperator,andEisproportionaltotheprojectorontheidentity
representation.ForR,S arbitrary,factorizabilityrequiresthat
2iπ
σ (θ) = − σ (θ)
1 (N−2)(iπ−θ) 2
2iπ
σ (θ) = − σ (θ)
3 (N−2)θ 2
whereN = R−2S,whileσ itselfisdetermined,uptoCDDfactors,bycrossingsymmetryandunitarity. One
2
immediatelyobservesthatwhenN =2,theamplitudeσ cancelsout,leavingascatteringmatrixwithasimpler
2
tensorialstructure,sincethePoperatordisappears.Thiscorrespondstoaparticularpoint[4]onthesigmamodel
criticalline(where,amongotherthings,thesymmetryisenhancedtoSU(2S +2|2S)),therestofwhichisnot
directlyaccessiblebythisconstruction.1 Theideausedin[5] istoconsiderananalyticalcontinuationtoR,S
real,andanapproachtoR−2S =2withproperscalingofthemass. Thoughinterestingresultswereobtained,
theemphasisinthesepaperswasnotonconformalproperties.
Anotherlineofattack,moresuitedtotheconformalaspects,waslaunchedbyReadandSaleurin2001[4],
who proposedto use a lattice regularizationto controlthe integrablefeaturesof the model. They obtainedin
thiswaythe spectrumofcriticalexponentsforseveralrelatedsigmamodelsonsupertargetspaces, including
the OSp(2S +2|2S)cosetoneat a particular(critical)valueof the couplingg2. The resultsexhibitedseveral
σ
mysterious features, including a pattern of large degeneracies, and a set of values of the exponents covering
(modulointegers)alltherationals. Intwo subsequentpapers[6, 7], itwasarguedfurtherthatmanyalgebraic
propertiesoftheconformalfieldtheorycouldbeobtainedatthelatticelevelalready. Theseincludefusion,and
thestructureofconformal“towers”(seebelowforfurtherdetails).
The work we present in this paper and its companion is an attempt at understanding the conformal field
theoreticdescriptionoftheOSp(2S+2|2S)modelforallvaluesofthecouplingbyusingalatticeregularization.
Foremostinthelatticeapproachistheunderstandingofthealgebraicstructureofthelatticemodel-thealgebra
defined by the local transfer matrices and its commutant. While in the cases discussed in [7] most necessary
resultswerealreadyavailableinthemathematicalliterature,thesituationhereismuchmorecomplicated: ina
fewwords,wehavetodeal,insteadoftheTemperleyLiebalgebra,withtheBraueralgebrawhoserepresentation
theory,inthenonsemi-simplecase,isfarfromfullyunderstood. Animportantpartofourworkhasconsisted
in filling up the necessary gaps of the literature, sometimes rigorously, but sometimes at the price of some
1ThecaseR=2,S =0isspecialandallowsforanextensionoftheSmatrixtothewholeO(2)criticalline.
2
conjectures.Thisalgebraicworkisthesubjectofthefirstpaper,whichwerealizemightbeabithardtoreadfor
aphysicsreader. Wecapitalizeonthealgebraiceffortinthecompanionpaper,wheretheboundaryconformal
fieldtheoryforthecosetsigmamodelisanalyzedthoroughly.
InthesecondsectionofthispaperwediscussgeneralitiesaboutlatticeregularizationsofO(N)sigmamodels
in 2 dimensions and define the model we shall be interested in. In section 3, the transfer matrix, the loop
reformulationand the associated Brauer algebra are introducedand discussed. Section 4 is the main section,
wherethefulldecompositionoftheHilbertspaceofthelatticemodelundertheactionofOSp(2S +2|2S)and
B (2) is obtained. Our main resultcan be foundin eqs. (4.36) and (4.37). Section 5 discusses aspects of the
L
hamiltonianlimitandsection6containsconclusions. Technicalaspectsofrepresentationtheoryarediscussed
furtherintheappendices.
Forthereader’sconvenience,weprovideherealistofnotationsusedthroughoutthepaper:
• osp(R|2S)istheLiesuperalgebraofthesupergroupOSp(R|2S)
• B (N)istheBraueralgebraonLstringswithfugacityforloopsN =R−2S
L
• V = CR|2S isthemod2gradedvectorspaceCR⊕C2S withevenpartV = CR andoddpartV = C2S.
R|2S 0 1
WeshalloftendroptheindicesR,2S inV .
R|2S
• V⊗L isconsideredasaleftosp(R|2S)andrightB (N)bimodule
L
• λ⊢ Lstandsfor“λisapartitionofL”andλ′isthepartitionλtransposed
• Sym(L)andCSym(L)arethesymmetricgrouponLobjectsanditsgroupalgebra
• T (q)istheTemperleyLiebalgebrawithfugacityforloopsq+q−1
L
• dandDaregenericelementsofB (N)andOSp(R|2S)
L
• X = {µ ⊢ L−2k | k = 0,...,[L/2]}isthesetofpartitionslabelingtheweightsof B (N). X (S) ⊂ X
L L L L
selectsthoseofthemwhichdorealizeonV⊗L
• Associateweightsλ,λ∗arelabelsofOSp(R|2S)irrepswhicharenonequivalent(identical)andbecome(split
intotwo)isomorphic(nonequivalent)irrepsundertherestrictiontothepropersubgroupOSp+(R|2S)ofsu-
permatriceswithsdetD=+1
• H (S) = {λ ∈ X (S) | λ ≤ S } is the set of hookshape partitionslabelingthe weightsof osp(R|2S)
L L r+1
irrepsappearinginV⊗L andY (S)= H (S)∪H (S)∗
L L L
• ∆ (µ)arestandardorgenericallyirreduciblerepresentationsofB (N)
L L
• S(λ),g(λ),G(µ),B (µ),D (j)areirreduciblerepresentationsofCSym(L),osp(R|2S),OSp(R|2S),B (N)
L L L
andT (q)respectively.
L
• Ig(λ),IG(µ),IB (µ)aredirectsummandsofV⊗L asaosp(R|2S),OSp(R|2S)andB (N)module
L L
• sc arethesupersymmetricgeneralizationofO(N)symmetricfunctions
λ
• χ′(d),χ (d),sch (D)arecharactersof∆ (µ),B (µ)andG(λ).
µ µ λ L L
2 The OSp(R|2S) lattice models: generalities
2.1 The models andtheir loopreformulations
LatticediscretizationsofO(N) sigmamodelshavehada longhistory. Thesimplestwayto gois obviouslyto
introduce spins taking values in the target manifold - the sphere O(N)/O(N − 1) - on the sites of a discrete
lattice, with an interaction energyof the Heisenbergtype E = −J S~ .S~ (where . stands for the bilinear
<ij> i j
O(N)-invariant quadratic form). This is however difficult to studyPtechnically, as the number of degrees of
freedomoneachsiteisinfinite. Apossiblewaytogoistodiscretizethetargetspace,leadingtovarioustypesof
3
“cubicmodels”[8].Anotherwaywhichhasprovedespeciallyfruitfulintwodimensionshasbeentoreformulate
theproblemofcalculatingthepartitionorcorrelationfunctionsgeometricallybyusingthetechniquesofhighor
lowtemperatureexpansions,thusobtaininggraphswithcomplicatedinteractionrulesandweightsdeterminedby
propertiesoftheunderlyinggroups.Thesimplestoftheseformulationsappearedin[9]wheretheauthorsstudied
theO(N)modelonthehoneycomblatticeintwodimensions,andreplacedmoreovertheterm eβJS~i.S~jbyits
<ij>
considerablysimplerhightemperatureapproximation (1+KS~ .S~ ),K =βJ[10].ExpaQndingthebrackets,
<ij> i j
insaythecalculationofthepartitionfunction,onecanQdrawgraphsbyputtingabondbetweenneighboringsites
iand jwheneverthetermS~ .S~ ispickedup.Theintegraloverspinvariablesleavesonlyloops,withafugacity
i j
equaltoNasthereareN colorsonecancontract.Notethatbecauseoftheverylowcoordinationnumberofthe
honeycomblattice,onlyself-avoidingloopsareobtained. Thisleadstothewellknownself-avoidingloopgas
partitionfunction:
Z = KENL (2.1)
SAL
XG
where the sum is taken over all configurationsG of self avoiding, mutually avoiding closed loops in number
L, covering a total of E edges. Note that once an expression such as (2.1) is written down, it is possible to
analyticallycontinuethedefinitionofthemodelforNanarbitraryrealnumber.Barringtheuseofsuperalgebras,
onlyNintegergreaterorequaltoonehasawelldefinedmeaningasaspinmodel(thecaseN =1coincideswith
theIsingmodel2). Intwodimensions,theMerminWagnertheorempreventsspontaneoussymmetrybreaking,
so for N integer, critical behavior can only occur for N = 1,2. Analysis of the same beta function suggests
howeverthatlatticemodelsdefinedbysuitableanalyticcontinuationshouldhaveaGoldstonelowtemperature
phaseforallN < 2,thoughitsaysnothingaboutwhetherthisphasemightendbyasecondorfirstorderphase
transition.
Model (2.1) lacks interaction terms which would appear with less drastic choices of the lattice and the
interactions: these are the terms where the loops intersect, either by going over the same edge, or over the
same vertex, maybe many times. It has often been argued that such terms are irrelevant for the study of the
criticalpointsoftheO(N)modelsintwodimensions.Mostoftheinteresthasfocusedonsuchcriticalpointsfor
N ∈[−2,2],whichhavegeometricalapplications-inparticularthecaseN =0isrelatedwiththephysicsofself-
avoidingwalks. Itturnsouthoweverthatintersectiontermsarecrucialfortheunderstandingoflowtemperature
phases. Indeed,themodel(2.1)doeshaveasortofGoldstonephaseforN ∈[−2,2]calledthedensephase,but
its propertiesare notgeneric, and destroyedbythe introductionof a smallamountof intersections. A simple
waytoseethatthedensephaseisnotgenericisthattheexponentsatN = 2arealwaysthoseoftheKosterlitz
Thoulesstransitionpoint: model(2.1)doesnotallowonetoenterthelowtemperaturephaseoftheXYmodel.
Also, model(2.1) has a first order transition for N < −2, which is notthe behavior expectedfrom the sigma
modelanalysis.
Itwassuggestedin[11]thatmodel(2.1)canberepairedbyallowingforsomeintersections. Theminimal
scenarioone can imagine is to define a similar modelon the square lattice, and allow for self intersectionsat
verticesonly,soeithernone,oneortwoloopsgothroughthesamevertex.Theresultingobjectsareoftencalled
trails. Thisgivesthenewpartitionfunction
Z = KENLwI
T
XG′
wherethesumistakenoverallconfigurationsG′ ofclosedloops,whichvisitedgesofthelatticeatmostonce,
innumberL,coveringatotalofE edges,withI intersections.
Thephasediagramofthismodelhasnotbeenentirelyinvestigated. Itisexpectedthatatleastforwsmall
enough, the critical behavior obtained with K = K , w = 0 is not changed (though K is), while the low
c c
temperaturebehaviorK >K willbe.
c
In [11] a yet slightly different version was considered corresponding, roughly, to the limit of very large
K, whereallthe edgesof thelattice arecovered. Thepartitionfunctionofthisfullypackedtrailsmodelthen
dependsononlytwoparameters:
Z = NLwI (2.2)
FPT
XG′′
2ThegroupO(2)isdifferentfromSO(2)≃U(1)becauseoftheadditionalZ2freedominchoosingthesignofthedeterminant.
4
Figure1: Dense intersectingloopcoveringofa lattice with annulusboundaryconditions. Illustrationof bulk
(B),contractible(C),even(E)andodd(O)loops.Periodicimaginarytimerunsvertically
a b c
Figure2:VerticesoftheZ symmetricsixvertexmodel
2
and an example of allowed configuration is given in figure 2.1. Numerical and analytical argumentssuggest
strongly that this model has all the generic properties of the O(N) sigma model in the spontaneously broken
symmetry phase (such that one might derive from analytic continuation in N of the RG equations), for all
N ≤2.
Notethatexpression(2.2)canbeobtainedverynaturallyifinsteadofputtingthedegreesoffreedomonthe
vertices, one putsthem on the edgesof the lattice. In thiscase, the minimalformof interactioninvolvestwo
edgescrossingatonevertex.InvarianceundertheO(N)groupallowsforthreeinvarianttensorsasillustratedon
thefigure8,whileisotropyandinvarianceunderanoverallscalechangeoftheBoltzmannweightsleavesone
withasinglefreeparameter,thecrossingweightw. Graphicalrepresentationofthecontractionsontheinvariant
tensorsreproduceseq.(2.2),aswillbediscussedbelow.
ForN < 2,model(2.2)flowstoweakcouplingintheIR,andthereforeitisexpectedthatthecriticalprop-
ertiesofthecorrespondinglowtemperature(Goldstone)phasedonotdependonw,afactcheckednumerically
in[11]. Thecase N = 2isexpectedtobedifferent: asmentionedalreadyintheintroduction,thebetafunction
of the correspondingsigma modelis exactly zero so the couplingconstantdoes notrenormalize. It is indeed
easytoseethattheloopmodel(2.2)withN = 2isequivalenttothe6vertexmodelwitha= b= 1+w, c = 1.
Considertheverticesofthe6vertexmodelasrepresentedonfigure2. Wechoseisotropicweightsa=b, c. We
candecideto splittheverticesofaconfigurationintopiecesoforientedloopsasrepresentedonfigure3. For
eachvertex,therearetwopossiblesplittings, andwe assumethattheyarechosenwith equalprobability. The
loopsobtainedbyconnectingallthepiecestogetherprovideadensecoveringofthelattice,andcomewithtwo
possibleorientations,henceafugacityoftwooncetheorientationsaresummedover. Theloopscanintersect,
withaweightwgivenfromtheobviouscorrespondence:
a b 1+w
= =
c c 2
andthus
a2+b2−c2 2
∆= =1−
2ab (1+w)2
WenotethatthereareindeedthreeinvarianttensorsforthecaseofO(2). Thecorrespondingprojectorsare
E, 1(I −P), 1(I −E +P). Theyprojectrespectivelyontwoonedimensionalrepresentations,andonasingle
2 2
twodimensionalone.
5
= +
= +
= +
Figure3: Themappingofthesixvertexmodelontotheorientedloops
Theparameter∆coverstheinterval[−1,1]asw∈[0,∞].Changing∆iswellknowntochangetheexponents
of6vertexmodel,andthereforeeq.(2.2)for N = 2exhibitsacriticalline,whichisinfactintheuniversality
classofthecontinuousXY(O(2))modelinthelowtemperature(KosterlitzThouless)phase.
All what was said so far can be easily generalized to the case of spins taking values on a supersphere
OSp(R|2S)/OSp(R−1|2S). The fugacity of loops is now equal to R− 2S: this combination is the number
ofbosonicminusthe numberof fermioniccoordinates,andfollowsfromthe usualfactthatwhen contracting
fermionsalongaloop,aminussignisgenerated3,seesec.3.3. Theloopmodelformulationthereforeprovides
a convenient graphical representation of the discrete supersphere sigma models for all R − 2S, in particular
R−2S ≤2whereinterestingphysicsisexpectedtooccur. Thisphysicswasexploredin[11],andtheexpected
resultswereobtainedforR−2S <2. ThepurposeofthispaperistoexplorethemorechallengingR−2S = 2
case.
Ofcourse,atthenaivelevelofpartitionfunctionsandwithoutworryingaboutboundaryconditions,itlooks
asifthereisnodifferencebetweentheO(N)spinmodelanditssuperspherecousinsprovidedR−2S = N. The
pointisthattheobservablesofthemodelsaredifferentor,attheveryleast,comewithdifferentmultiplicities.
Indeed, consider for instance correlation functionsof spin variables. In the O(2) case, the spin has only two
componentsS1,S2, so one cannot build a totally antisymmetric tensor on three indices. This means that the
correspondingoperator(whichhasa nicegeometricalinterpretationto be givenin the nextpaper)willnotbe
presentinthiscase,thoughitwillbeintheOSp(2S +2|2S)modelwhenS >0. Notethatingeneral,correlators
involvingspinswithin the first R bosonicandthe first 2S fermioniclabelswill be the same foranychoiceof
groupOSp(R′|2S′)withR′−2S′ =R−2S andR′ ≥R.4 Thisisimmediatelyprovedbyperformingagraphical
expansionofthecorrelator:variablesoutsideofthesetofthefirstRbosonicandthefirst2S fermioniclabelsare
notgettingcontractedwiththespinsinthecorrelators,andcancelagainsteachotherintheloopcontractions.
Astandardtricktoextractthefulloperatorcontentofamodelistostudythepartitionfunctionwithdifferent
boundary conditions. Consider for instance the spin model on an annulus with some symmetry preserving
boundaryconditionsinthespacedirection.Withwhatwewillcallperiodicboundaryconditions(corresponding
totakingthesupertraceoftheevolutionoperator)inthetimedirection,representationsofOSp(2S +2|2S)will
always be counted with their superdimension, and the partition function will be identical with the one of the
O(2) case. But if we take antiperiodicboundaryconditions, we will get a modified partition function (in the
sense of [4]) which is a trace over the Hilbert space instead of a supertrace, counts all observables with the
multiplicities(notsupermultiplicities),andwillturnouttobeaverycomplexobject.
A goodalgebraicunderstandingof the lattice modelwillbe essential to make furtherprogress, and, since
theareaislargelyunexplored,thiswilloccupyusformostoftherestofthisfirstpaper.
3ThegeneralizationofresultsforO(N)modelstothecaseoforthosymplecticgroupsdatesbacktotheworkofParisiandSourlas[12].
4Provided,ofcourse,thattheboundaryconditionsimposedontheR′bosonicand2S′fermionicdegreesoffreedomarethesamein
bothcases.
6
3 Transfer matrices and algebra
3.1 Transfer matrices
As discussed briefly in the introduction, the OSp(R|2S) spin model we consider is most easily defined on a
squarelatticewithdegreesoffreedom(states)ontheedgesandinteractionstakingplaceatvertices. Thesetof
statesoneveryedgeisacopyofthebasespaceVofthefundamentalOSp(R|2S)representation.Interactionsata
vertexcanbeencodedinalocaltransfermatrixtactingonV⊗2andcommutingwiththeOSp(R|2S)supergroup
action.Wecalltanintertwinerandwritet∈End V⊗2.
OSp(R|2S)
TheBoltzmannweightsofthemodelarecomponentsofthetransfermatrixalongabasisofintertwiners. A
naturalchoiceof basis arethe projectorsontoOSp(R|2S)irreduciblerepresentationsappearingin the decom-
positionofthetensorproductoftwofundamentalOSp(R|2S)representations. Tofindthemonecanapplythe
same(anti)symmetrizationandtracesubstractiontechniquesusedforreducingO(N)tensorrepresentations. If
e ,...,e isamod2gradedsetofbasisvectorsinV withgradingg,thedecompositionofV⊗2willread
1 R+2S
1 2J
e ⊗e = e ⊗e +(−1)g(i)g(j)e ⊗e − ij Jkle ⊗e (3.1)
i j 2 i j j i R−2S k l
+1e ⊗e −(−1)g(i)g(j)e ⊗e + Jij Xk,l Jkle ⊗e.
2 i j j i R−2S k l
(cid:16) (cid:17) Xk,l
Here J is the OSp(R|2S)invarianttensor, Jij = (J−1) , and g(i) = 1 (resp. g(i) = 0) if i is fermionic(resp.
ij ij
bosonic). Each of the threetermson the l.h.s. of (3.1) transformsaccordingto an irrepsofOSp(R|2S),or, in
otherwords,belongstoasimpleOSp(R|2S)module.
IntroducetheidentityI,thegradedpermutationoperatorP(alsoknownasbraidoperator),andE theTem-
perleyLieboperator(proportionaltotheprojectoronthetrivialrepresentation),
I kl =δkδl, P kl =(−1)g(i)g(j)δkδl, E kl = JklJ . (3.2)
ij i j ij i j ij ij
IntermsofprojectorsontoirreducibleOSp(R|2S)modules,eq.(3.1)maybewritteninamoreelegantwayas
1 2 1 1
I = I+P− E + (I−P)+ E.
2 R−2S ! 2 R−2S
LetPdenoteasusualtheinversionofspace,TtheinversionoftimeandCthechargeconjugationwiththe
matrix J. One can check directly from definition (3.2) that P is C , P and T invariant, while E is P and
12 12 12
C T invariant. Moreover,E andI transformintoeachotherundertheπ/2rotationofthelatticeR,while E
12 12
andParerelatedbythecrossingsymmetryC T
1 1
R: Eijkl −→ Jjj′Ek′lij′Jk′k =Iijkl
C1T1 : Eijkl −→ Jii′Ek′i′jlJk′k = Pijkl.
TakeI,E andPasbasisofintertwinersinEnd V⊗2. Thelocaltransfermatrixgenerallydependson
OSp(R|2S)
threeindependentweightsw ,w andw . However,onahomogeneousandisotropiclatticeonecannormalize
I E P
w =w =1andleaveonlytheweightw=w . Finally,thelocaltransfermatrixtakestheform
I E P
t(w)=I+wP+E. (3.3)
Ona diagonallattice withopenboundariesrepresentedinfig. 4choosethe timein verticaldirection. The
notation of sites at a fixed time is such that the left edge i and right edge i+1 meet at vertex i. Let t(w) ∈
i
End V⊗L denote a transfer matrix acting nontrivially only at vertex i according to eq. (3.3). From the
OSp(R|2S)
figureitisclearthatoddandeventimesareinequivalent. ThetransfermatrixT,propagatingonestepforward
atequivalenttimes,maybewrittenasaproductT =YX ofonelayertransfermatrices
[(L−1)/2] [L/2]
X = t , Y = t , (3.4)
2i 2i−1
Yi=1 Yi=1
7
Y
X
Figure4:TheonelayertransfermatricesX andY representedonadiagonallatticeofwidth6.
schematicallyshowninfig.4.
The simplestway to define a partitionfunctionthatdependson the wholespectrumof the transfermatrix
T is by taking the trace of T at a certain power β. Selecting other boundaryconditionswith some nontrivial
symmetrygenerallyamountstorestrictingthewholespaceofstatesofthemodeltoasubspacecompatiblewith
thesymmetryofchosenboundaryconditions. Whatexactlywe meanby “symmetryofboundaryconditions”
willbeexplainedlaterinsec.3.3. Forthemomentletusjustsaythatitisconvenienttoconsideramoregeneral
classofboundaryconditions,calledquasiperiodic,inwhichTβis“twisted”bytheactionofanelementDofthe
supergroup.Definethequasiperiodicpartitionfunctiontobe
Z =str D⊗LTβ. (3.5)
D V⊗L
Wemusttakethesupertraceineq.(3.5)ifwewantthequasiperiodicpartitionfunctiontobewelldefined. For
instance,whenD= J2wegettheusualtracepartitionfunctionandwhenDequalstotheidentitymatrixweget
thesupertracepartitionfunction.
NotethatbecauseDisasupermatrix,thetensorproductinD⊗L hastobegraded,thatis
D⊗2·η⊗ξ = D·η⊗D·ξ ⇒ D⊗2 ij =(−1)g(k)(g(j)+g(l))Di Dj
kl k l
(cid:16) (cid:17)
Afterinsertingthelocaltransfermatrixfromeq.(3.3)ineq.(3.4)andexpandingthetransfermatrixT,the
quasiperiodicpartitionfunctionreadsasasumofweightedproductsofE’sandP’s. Suchlinearlyindependent
i i
productsmustbeconsideredaswordsofatransfermatrixalgebra,whileintertwinersE andP aregenerators
i i
ofthisalgebra.InthenextsectionweidentifythisalgebraasarepresentationoftheBraueralgebra.
3.2 The Brauer algebra
ForanabstractintroductiontotheBraueralgebraseeref.[13,14]whileinthecontextofosp(R|2S)centralizer
algebraseeref.[15]. WecollectinthissectionsomewellknownfactsabouttheBraueralgebraweshallusein
thenextsections.
LetE andP,i=1,...,LactnontriviallyasE andPineq.(3.2)onlyatthesitesV ⊗V ofV⊗L. Onecan
i i i i+1
checkthatforP andE sodefinedthefollowingrelationshold:
i i
P2 =1, E2 = NE, EP = PE = E, (3.6)
i i i i i i i i
PP = P P, EE = E E, EP = E P,
i j j i i j j i i j j i
PP P = P PP , EE E = E,
i i±1 i i±1 i i±1 i i±1 i i
PE E = P E, EE P = EP .
i i±1 i i±1 i i i±1 i i i±1
Inthesecondlineoftheserelationsiand jaresupposedtobenonadjacentsites.
Relations(3.6)(isoneofthemanywaysto)definetheB (N)Braueralgebra(alsodenotedsometimesbythe
L
namesof braid-monoidalgebraordegenerateBirman-Wenzel-Murakamialgebra[16, 17, 18]). Note thatthis
algebradependsonasingle,generallycomplex,parameterN,andcontainsthemaybemorefamiliarTemperley
Liebalgebra,generatedbyE’salone,andthesymmetricgroupalgebra,generatedbyP alone.
i i
For N fixedand Lbigenough,theOSp(R|2S)spinmodelsprovidehighlyunfaithfulrepresentationsofthe
BraueralgebraB (N). Thisisbecause,inV⊗L,thegeneratorsP andE satisfyadditionalhigherorderrelations
L i i
Rontopof (3.6).5 Forasimpleexample,considertheO(2)spinmodelonalatticeofwidth3. Theprojector
5This situation is similar to what happens for models with SL(N) symmetry in the fundamental representation, and corresponding
quotientsoftheHeckealgebra.
8
Figure5: ThegraphicalrepresentationofthewordP P E P inB (N).
5 3 1 2 6
1 L 1 i i+1 L 1 i i+1 L
I E P
i i
Figure6:GraphicalrepresentationofgeneratorsI,E andP.
i i
onto the antisymmetric tensor of rank 3 is zero, thus, R contains the additional relation 1+ P P + P P =
1 2 2 1
P +P +P P P . OurspinmodelsingeneralproviderepresentationsofthequotientalgebrasB (N)/R. The
1 2 1 2 1 L
setofrelationsRcanbeexplicitlydescribedforS =0,see[19,20]andreferencestherein,andwehavelittleto
sayaboutthecaseS >0.
The first step in understanding the spectrum of the transfer matrix T brings up the question of B (N) ir-
L
reduciblerepresentations, and of their multiplicities in T for a particular choice of R and S. This leads us to
discussingsomeresultsabouttherepresentationtheoryoftheBraueralgebra.
Themostnaturalrepresentationtobeginwithistheadjointrepresentation. Itadmitsadiagrammaticrepre-
sentationintermsofgraphson2Lpointsinwhicheveryvertexhasdegree1. Usuallyoneordersthe2Lpoints
ontwohorizontalparallellinesasshowninfig.5. LetB denotethevectorspacespannedonthe(2L−1)!!such
L
diagrams.
Theproductd ∗d oftwodiagramsd andd isperformedbyputtingd ontopofd andreplacingeach
1 2 1 2 1 2
oftheloopsintheresultingdiagramwith N. Definediagrammaticallytheidentity I andthegeneratorsE, P
i i
asrepresentedinfig.6. Onecancheckthatthegraphicalrepresentationofgeneratorswiththemultiplication∗
ofdiagramssatisfyalloftheeqs.(3.6). TheleftactionofgeneratorsonB viathemultiplication∗ofdiagrams
L
providetheadjointrepresentationofB (N).
L
From the graphicalrepresentation we see that B has a series of invariantsubspaces B = BL ⊃ BL−2 ⊃
L L L L
··· ⊃ Bτ, where Bm is spanned on diagrams with fewer than m vertical lines and τ = L mod 2. The vector
L L
spacespannedondiagramswithexactlymverticallinesmaybedefinedasa B (N)modulebythecosetB′m =
L L
Bm/Bm−2. TheleftactionofB (N)onthismodulesmaybeseenasamodifiedmultiplication∗ ofdiagrams
L L L m
d ∗d , ifithasmverticallines
d ∗ d = 1 2
1 m 2 0, otherwise

Under the left action of the algebra theposition of horizontal lines in the bottom of a diagram does not
change.Foragivenconfigurationofthehorizontallinesinthetopofadiagramandagivenpatternofintersec-
tionsofverticallinesthereare(L−m−1)!!Cmpossibilitiesofchoosingtheconfigurationofhorizontallinesin
L
thebottomofthediagram.ThissimplymeansthatB′mdecomposesintoadirectsumof(L−m−1)!!Cmequiv-
L L
alentmodules. ThecosetrepresentativeB′′m oftheseequivalentleftmodulesisspannedonm!(L−m−1)!!Cm
L L
graphsonLpointswitheveryvertexhavingdegree0or1andalabelingwithnumbers1,...,moffreevertices.
Anexampleofsuchalabeledgraphisshowninfig.7. Ifthelabellingsareomittedtheresultinggraphiscalled
apartialdiagram.
ThelabelingofthemfreepointsofalabeledgraphisapermutationπinthesymmetricgroupSym(m). The
labeledgraphswillprovidearepresentationoftheBraueralgebra,whichisirreducibleforgenericvaluesofN,
ifwetakethelabellingsπinanirreduciblerepresentationofSym(m). Wecallsuchrepresentationsgenerically
irreducible. Letµbeapartitionofm,whichwewriteasµ ⊢ m. Inamorealgebraiclanguagethedefinitionof
genericallyirreducibleleftmodulestranslatesto
∆ (µ):= B′′m⊗ S(µ), µ⊢m, (3.7)
L L Sym(m)
whereS(µ)isanirreducibleSym(m)module.Inviewoflaternumericalanalysiswegivebelowabasisin∆ (µ)
L
anddescribetheactionofB (N)onthisbasis.
L
9
2 1 4 3
Figure7:Thetopofthediagramrepresentedinfig.5. Thelabelingpermutationis(2,1,4,3).
Let p⊗ π denote the labeling of a partial diagram p with the permutation π, v ,...,v be a set of basis
1 fµ
vectorsinS(µ)andρ (σ)bethematrixofthepermutationσintherepresentationρ . Anaturalbasisin∆ (µ)
µ µ L
isgivenbyallpairs p⊗v. Theactionofadiagramd ∈ B (N)onabasisvectoris
i L
d∗ p⊗ρ (σ−1)v, ifd∗phasmfreepoints
d·p⊗v = m µ i (3.8)
i 0, otherwise,

where σ is the labeling of d ∗ g andg is the partial diagram p labeled with the identity permutation. The
dimensionsd of∆ (µ)is f (L−m−1)!!Cm.
µ L µ L
Insimplewords,agenericallyirreduciblemoduleisaspanongraphsonLpoints,obtainedbychoosingm
pointsamongL,pairingalltheothers(thisgivesthemultiplicity(L−m−1)!!sinceintersectionsareallowed),
choosingforthemunpairedonesarepresentationofthepermutationgroupandsettingtozerotheactionofany
Brauerdiagramthatreducesthenumbermofunpairedpoints.
Thegenericallyirreduciblerepresentationslabeledbyµ ⊢ L−2k,k = 0,...,[L/2]appearinthedecompo-
sitionoftheadjointrepresentationwithmultiplicitygivenbytheirdimensiond whenB (N)issemisimple.
µ L
Letusconcludewithafewwordsaboutthereducibilityofgenericallyirreduciblemodules∆ (µ).Forinteger
L
N and a numberof strings L > N the Brauer algebrais notsemisimple and, as a consequence,certain of the
modules∆ (µ)becomereducible,thoughtheyremainindecomposable.6Theirreduciblecomponentsappearing
L
insuchreduciblemodules∆ (µ)arefarfrombeingunderstood(thesituationismuchworsethaninthecaseof
L
thenonsemisimpleTemperleyLiebalgebra[21,22]). Numericalcomputationsbasedonthediagonalizationof
thetransfermatrixinthediagrammaticrepresentationofB (N)restrictedto∆ (µ)decreasesinefficiencyvery
L L
fastwithincreasingL, comparedtotheidealcasewherethetransfermatrixisrestrictedtoanirrepsof B (N).
L
ThisisbecauseforbigLandµfixedthenumberofirreduciblecomponentsin∆ (µ)“goeswild”andthereare
L
alotof“accidentaldegeneracies”inthespectrumofthetransfermatrixrestrictedto∆ (µ).
L
However, a significant progress in this direction has been recently made in [23, 24]. Let us note that, as
described in [23], the content of (at least some) ∆ (µ)’s can be computed by repeated applications of Frobe-
L
nius reciprocity applied to the short exact sequence of [14] describing the structure of the induced modules
B (N)⊗ ∆ (µ).
L+1 BL(N) L
In the end we recall the basic results for the Temperley Lieb algebra, to allow a quick comparison with
Brauer. TemperleyLiebalgebradiagramsareasubsetofBraueralgebradiagramssubjecttotheconstraintthat
no intersections between edges are allowed. The dimension of the algebra is given by the Catalan numbers
(2L)!/L!(L+1)!. Themainlineofreasoningforfindinggenericallyirreduciblemodulesfollowsthesameway,
except there is no available action of the symmetric group on vertical lines. Therefore, the analogue of the
labeledgraphswillbethepartialdiagrams,inwhichnofreepointsmaybetrappedinsideanedge.Thenumber
ofsuchgraphsisCn −Cn−2,wheren=(L−m)/2isthenumberofedges.Thegenericallyirreduciblemodules
L−1 L−1
D (m)areparametrizedbythenumberm= L,L−2,... offreepointsinthegraphs.
L
ThepresentedfactsabouttheBraueralgebrashouldbeenoughtounderstandtheloopgasreformulationof
OSp(R|2S)spinmodel,whichwegiveinthenextsection.
3.3 LoopreformulationofOSp(R|2S)spinmodels: the algebraicpointofview
The emergence of dense intersecting loops becomes transparent if we take the local transfer matrices in the
adjointrepresentationoftheBraueralgebra. Thissimplyamountstoreplacingineq.(3.3)thegeneratorsI, E
i
andP definedbyeq.(3.2)withthediagramsinfig.6. Theadjointlocaltransfermatrixisrepresentedinfig.8.
i
We now define a loop model on a diagonallattice represented in fig. 4, with reflecting boundarieson the
left and right (ie, free boundary conditions in the space direction) and identified boundaries on the top and
6Weadoptherethephysicist’shabitofcallingindecomposableamodulewhichisreduciblethoughnotfullyreducible. Thereforethe
setofindecomposablesdoesnotcontaintheirreducibles,unlikeinmostofthemathliterature.
10