categories Jones-Wassermann subfactors for modular tensor Advances in Mathematics

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Advances in Mathematics

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Jones-Wassermann subfactors for modular tensor categories

Zhengwei Liu^a,∗, Feng Xu^b

aDepartmentofMathematicsandDepartmentofPhysics,HarvardUniversity, UnitedStatesofAmerica

bDepartmentofMathematics,UniversityofCalifornia,Riverside,UnitedStatesof America

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received10July2017 Receivedinrevisedform16 February2019

Accepted8August2019 Availableonline28August2019 CommunicatedbytheManaging Editors

Keywords:

Subfactors

Modulartensorcategories Dualities

Conformalnets Reconstructionprogram

Therepresentationcategoryof a conformalnetisa unitary modular tensor category. Weinvestigate thereconstruction program: whether all unitary modular tensor categories are representation categories of conformal nets. We give positiveevidence:thefruitfultheoryofmulti-interval Jones- Wassermann subfactors on conformal nets is also true for modulartensorcategories.Weconstructmulti-intervalJones- Wassermannsubfactorsforunitarymodulartensorcategories.

Weprove that these subfactors aresymmetrically self-dual.

Itgeneralizesandcategorifiestheself-dualityoffiniteabelian groups.Wecallthisdualitythemodularself-duality,because themodularity ofthemodulartensorcategory appearsina crucial way. For each unitary modular tensor category, we obtain a sequence of unitary fusion categories. The cyclic groupcasegivesexamplesofTambara-Yamagamicategories.

©2019ElsevierInc.Allrightsreserved.

* Correspondingauthor.

E-mailaddresses:[email protected](Z. Liu),[email protected](F. Xu).

https://doi.org/10.1016/j.aim.2019.106775 0001-8708/©2019ElsevierInc. Allrightsreserved.

1. Introduction

Subfactor theory provides an entry point into a world of mathematics and physics containing largeparts ofconformalfieldtheory, quantumalgebras andlowdimensional topology,see [9])and referencestherein.In[10] V.Joneshasdevisedarenormalization programbasedonplanaralgebrasasanattempttoshowthatallfinitedepthsubfactors arerelatedtoCFT,i.e.,thedoubleofafinitedepthsubfactorshouldberelatedtoCFT.

Moregenerally,theprogramisthefollowing:givenaunitarymodulartensorcategory (MTC) C [22,30], canwe construct a (complete rational) conformal net whose repre- sentation category is isomorphic to C? We shall call such a program “reconstruction program”,analoguetoasimilarprograminhigherdimensionsbyDoplicher-Roberts[1].

Given arationalconformalnetA,andletI be aunionof m≥1 disconnectedinter- vals.TheJones-WassermannsubfactoristhesubfactorA(I)⊂A(I) [13,21,31,32].This subfactoris relatedto permutationorbifoldand asimpleapplicationof orbifoldtheory shows thattheJones-Wassermannsubfactorisself-dual,see Remark6.19.

If the reconstruction program works, then for any MTC _C we can find a rational conformal net A such thatthe category of representations of Ais isomorphic to _C, it willfollowthatthereareself-dualJones-Wassermannsubfactorsforeachintegerm>1.

Henceapositive solutionto reconstructionprogramwouldimplythatwecanconstruct self-dualJones-Wassermannsubfactorsforeachintegerm>1 associated withany uni- tary MTC C.Thisisthemotivationforourpaper.

Our main result gives a construction of self-dual Jones-Wassermann subfactors for each integer m > 1 associated with any unitary MTC _C. The main difficulty is the proof of the self-duality for Jones-Wassermann subfactors for MTC. Furthermore, we prove thattheJones-Wassermann subfactorsare symmetricallyself-dual. The proofof theself-dualityandsymmetricalself-dualityessentiallyrequires themodularityof C,so we callitthemodular self-duality.Webelievethatourconstructionwill shednewlight onthereconstructionprogram.

Weconstructthe“m-interval”Jones-Wassermannsubfactorassociatedwithaunitary MTC C by aFrobenius algebraγm inC^m, the m^th tensorpower of C. The notionof intervals is natural in conformal nets, and it is important to understand the locality.

However, there is no notion of intervals in modulartensor categories. This is a major problem inthereconstructionprogram.

WegiveanexplicitformulafortheobjectandthemorphismoftheFrobeniusalgebra γ_m. When m = 2, the Jones-Wassermann subfactor defines the quantum double of _C [2,21,25–27].Forpeoplewhoareinterestedinthealgebraicaspects,theunitarycondition isnotnecessaryforourconstruction.Theunitaryconditionisimportantforanalysisand thereconstructionprogram.

Thebimodulecategory ofasubfactorisdescribedbyasubfactorplanaralgebra[11].

Then-boxspaceoftheplanaralgebraofthem-intervalJones-Wassermannsubfactorfor C isgivenbythevectorspacehom_C^m(1,γ_mⁿ).Itturnsouttobenaturaltorepresentthese vectorsbya3Dpicture.Thisrepresentationidentifieshom_C^m(1,γⁿ_m) asaconfiguration

spaceConfn,mona2Dn×mlattice.Thereforetheconfigurationspace{Confn,m}m,n∈N

unifiestheJones-Wassermannsubfactorsforallm≥1.Itisanaturalcandidateforthe configurationspaceofa2Dlatticemodelthatcanbeusedinthereconstructionprogram.

Moreover,we showthatplanartangles canacton{Confn,m}m,n∈N intwodifferent directionsindependently.Inonedirectionmisfixed.Theseactionsaretheusualonesin theplanaralgebraofthem-intervalJones-Wassermannsubfactor.Intheotherdirection, nisfixed.These actionsrelatetheJones-Wassermannsubfactorwith differentintervals whichhavenotbeenstudiedbefore.

Thebi-directionalactionsofplanartanglesarecompatiblewiththegeometricactions onthe2D lattices.Wecall suchfamilyof vectorspacesabi-planar algebra.It isanew subjectinsubfactortheoryanditaddsoneadditionaldimensiontothetheoryofplanar algebras.

This3Drepresentationalsoleadstothediscoveryofanewm-n dualitybyidentifying theconfiguration spaces{Confn,m}m,n∈N and{Confm,n}m,n∈N,while themeaning of the actions of planar tangles in the two directions are completely different. It will be interestingtounderstandtheseadditionalsymmetriesinconformalfieldtheory.

When_C istherepresentationcategory ofafiniteabelian groupG,theconfiguration space Conf(_C)_2,2 becomes L²(G). Moreover, the modular self-duality coincides with the self-duality of G. The proof of the self-duality of G requires the discrete Fourier transformonG.ForgeneralMTCs,weconstructthestringFouriertransform(SFT)on theconfigurationspacehom_C^m(1,γ_mⁿ) toprovethemodularself-duality.Fromthispoint ofview,themodularself-duality andtheSFTgeneralizeand categorifytheself-duality andtheFouriertransformoffiniteabeliangroups.

Furthermore,weprovethatthem-intervalJones-Wassermannsubfactorofaunitary MTC C issymmetricallyself-dual,namelytheirplanaralgebrasareunshadedinTheo- rems6.18and6.20.Theprojectioncategoryoftheevenpartofasubfactorplanaralgebra isaunitaryfusioncategory [4,24].Theprojectioncategoryof anunshadedplanaralge- braisaZ₂-gradedunitaryfusioncategory,whosegeneratingoddobjectissymmetrically self-dual[17,23].Fromtheunshadedplanaralgebraofthem-intervalJones-Wassermann subfactorsofC,weobtainaZ2-gradedextensionofthefullsubcategoryofC^mgenerated bythe Frobeniusalgebra γm,such thatγm=τ⊗τ and τ is asymmetricallyself-dual oddobject. The object τ appeared as a twisted sector in orbifold theory of conformal nets,seeRemark6.19and[12].WhentheGrothendieckringof C isafinitecyclicgroup, we obtain the Tambara-Yamagami category [29] from the unshaded planar algebra of thetwo-intervalJones-Wassermansubfactorof_C, andτ istheonly oddsimpleobject.

Ingeneral,forany unitaryMTC C andany intervalm, m≥1,weobtainaZ2-graded unitaryfusioncategory[4],whichgeneralizesTambara-Yamagamicategories.

Themodulartransformation S wasstudiedintheearly workof ’tHooft inrational conformalfieldtheory as aHopf link,aunion of aWilsonloopand aDirac string[6].

Thishas been formalizedintheframeworkof conformalnetsby Rehrenin[28]. Inthe generaltheoryofMTCs,weprovethatthemodulartransformationSfortheMTC_C is identicaltotheSFTonthevectorspacehom_C²(1,γ₂²) inourconstruction.Thisprovides

a different point ofview to understandtheS matrix ofMTCs as a specialcase ofthe SFT. Theunshadedconditionof theplanaralgebraisnecessary todefine theSFT asa matrix onhom_C²(1,γ₂²).The SFTis always aunitary,so the modularity of theMTC, namely theS matrixisaunitary[30],iscrucial inthisidentification.

The recentprogressabout theFourieranalysis oftheSFT onsubfactorsin[8,15,19]

leadstomanyinterestinginequalitiesfortheS matrix.Theseunshadedplanaralgebras havebeenusedinthequon3Dlanguageforquantuminformation[18],wherethevector space hom_C²(1,γ₂²) is consideredas the1-quonspace.This newinterpretationleadsto many interestingalgebraic identitiesforthe S matrixinaMTC as theFourier duality ofquons[16].Acombinationoftheseworksleadstofurtherapplicationsinthestudyof MTC.

Acknowledgment.Zhengwei LiuwassupportedbyTempletonReligionTrustunderthe grant TRT159and byanAMS-Simons TravelGrant.Feng XuwassupportedbyNSF under the grantDMS-1764157 and anacademic senate grant from UCR. The authors wouldliketothankVaughanF.R.Jonesforstimulatingdiscussionsabouthisrenormal- ization program.

2. Configurationspaces

2.1. Modulartensor categories

We refer the readers to [30] for basic definitions about modular tensor categories (MTC). Allcategories inthis paper are strict. Suppose C is a unitary MTC.Let Irr be the set of (representatives of) simple objects of _C and the unit is denoted by 1.

Take X˜ =

X∈Irr(C)

X. For anobject X, itsdual object is denotedby X. Its quantum dimensionisd(X).Letμ=

X∈Irr

d(X)² betheglobaldimensionof C.

ThemodularconjugationJ_C on C isahorizontalreflection.WehavethatJ_C(X)=X.

Moreover, for objects X, Y, Z in_C, J_C : hom_C(X ⊗Y,Z)→ hom_C(Y ⊗X,Z) is an anti-linear algebroid isomorphism.The adjointoperator∗on C isavertical reflection.

We havethatX^∗ =X. Moreover,∗: hom_C(X⊗Y,Z)→hom_C(Z,X⊗Y) is ananti- linearalgebroid anti-isomorphism.Thecontragredient map ρπ on C is arotationbyπ.

We havethat ρ_π(X) = X. Moreover, ρ_π : hom_C(X ⊗Y,Z) → hom_C(Z,Y ⊗X) is a linearalgebroidanti-isomorphism. Furthermore

J_C =ρπ◦ ∗. (1)

ForobjectsX,Y in C,theinner productofα,β ∈hom_C(X,Y) isdefinedas

β, α= α

β^∗ ,

whichisanelementinhom_C(1,1)∼=C.

We can identify the morphism spaces hom_C(Z,X ⊗Y) and hom_C(1,X ⊗Y ⊗Z) as follows: For a morphism α ∈ hom_C(1,X ⊗Y ⊗Z), we obtain a morphism α˜ = (1_X⊗1_Y ⊗φ_Z⊗Z)(α⊗1_Z) inhom_C(Z,X⊗Y), where φ_Z⊗Z ∈hom_C(1,Z⊗Z) is the dualitymap.

Notation2.1(Frobenius reciprocity).Diagrammaticallywerepresentα˜as

˜ α

:=

α

Notation2.2.ForanobjectX in C,wedenoteanortho-normal-basisofhom_C(1,X) by ON B_C(X),or ON B(X) forshort.Wedenoteanortho-normal-basisofhom_C(X,1) by ON B^∗_C(X),orON B^∗(X) forshort.

FortwoobjectsX andY,wehavetheresolution oftheidentity:

1X⊗1Y = =

Z∈Irr,α∈ON B(X⊗Y⊗Z)

d(Z) α^∗

α (2)

Lemma 2.3. Suppose Y and Z are objects in C and X ∈ Irr. Let ON B(Y,X), ON B(X,Z)be ONBof hom_C(Y,X)andhom_C(X,Z).Then

{

d(X)βα : X∈Irr, α∈ON B(Y, X), β∈ON B(X, Z)} isanONB of hom_C(Y,Z).

Proof. Forαi∈ON B(Y,X),βi∈ON B(X,Z),i= 1,2,

d(X) β1α1, β2α2=d(X) α₂ β2

β₁^∗ α^∗₁

= β1, β2 α₂

α₁^∗

= β1, β2 α1, α2.

Ontheotherhand,

X∈Irr

dim hom_C(Y, X) dim hom_C(X, Z) = dim hom_C(Y, Z).

Therefore,thestatementholds.

Fig. 1.Grid(n,m) forn= 4,m= 3.

2.2. Configurationspaces

Now letus definetheconfigurationspaceonafinite2D-latticewiththetargetspace C.Eachconfigurationhasthree parts:Z-, X-,Y- configurations.

WeuseGrid(n,m) torepresentthegridZ_n×Z_m× {±1}.Weallocatetheverticesof thegrid at(i,j,±1),1≤i≤nand0≤j ≤m−1 in the3Dspacewhichareindicated by thebullets inFig.1. To simplifythenotations, wedraw picturesfor n= 4, m= 3.

Thereadercanfigureoutthegeneralcase.

ForthelatticeLat=Z_n×Z_m,aZ-configurationisamapfromthesitesofthelattice to simpleobjectsin_C.Wedenotethesimpleobjectatthesite(i,j) asX_i,j.Wedenote this Z-configuration by X_i,j and represent it in the 3D space by assigning the object Xi,j totheline from(i,j,1) to(i,j,−1) asinFig.2:

Fig. 2.Z-Configuration.

Wedenote X_i,j =X_1,j⊗ · · · ⊗X_n,j andX_i,j=X_i,0⊗ · · · ⊗X_i,m−1.Moreover, d(X_i,j) :=

1≤i≤n,0≤j≤m−1

d(Xi,j), d(X_i,j) :=

1≤i≤n

d(X_i,j),

Fig. 3.X- andY-Configurations.

d(X_i,j) :=

0≤j≤m−1

d(X_i,j).

AnX-configurationwithboundaryX_i,jisamorphismajinhom_C(1,X_i,j).Wedenote theboundary byX(a_j):=X_i,j. A Y-configuration with boundary X_i,j isa morphism bi inhom_C(X_i,j,1).We denote theboundary by X(bi) :=X_i,j. Werepresent them in the3D space inFig.3. Moreover, wecall aj =a0⊗ · · · ⊗am anX-configuration with boundaryX_i,j andb_i=b₁⊗ · · · ⊗b_n aY-configurationwithboundaryX_i,j.Wedefine theX-configuration spacewithboundaryX_i,j as

ConfX(X_i,j) :=

m−1 j=0

hom_C(1, X_i,j).

WedefinetheY-configurationspacewithboundaryX_i,j as

Conf_Y(X_i,j) :=

n i=1

hom_C(X_i,j,1).

Wecall a_j⊗b_i aconfiguration with boundaryX_i,j, denotedby X(a_j⊗b_i):=X_i,j. Werepresent itinthe 3Dspace as inFig.4: We define theconfiguration spaceon the n×m2D-latticeLattobetheHilbertspace

Conf(Lat) =Conf(_C)_m,n: =

Xi,j∈Irr^nm

Conf_X(X_i,j)⊗Conf_Y(X_i,j)

=

Xi,j∈Irr^nm

⎛

⎝^m−1

j=0

hom_C(1, X_i,j)⊗ n

i=1

hom_C(X_i,j,1)

⎞

⎠,

where each hom space is considered as a Hilbert space. We simply use the notation aj⊗b_i to representanelement inConf(Lat).

Fig. 4. Configurations for n = 4, m = 3: We use the small square (orthogonal to the Z-axis) at the coordinate(i,j,0) toindicatethevertex(i,j) inthelatticeZm×Zn.Moreover,theboundariesofa_j and biareseparatedonoppositesidesofthesmallsquares.(Forinterpretationofthecolorsinthefigure(s),the readerisreferredtothewebversionofthisarticle.)

2.3. Duality

WhenweconsiderLat=Zn×Zmasalatticeonatorus,itsduallatticeLat isalso Zn ×Zm and the configuration space on thedual lattice is Conf(C)m,n. We allocate the vertices of the corresponding Grid(n,m) at (i+ 1/2,j−1/2,±1), 1≤ i ≤ n and 0≤j≤m−1 inthe3Dspace.

We defineabilinearform LL ontheconfigurationspacesof thelatticeandthedual latticeConf(Lat)⊗Conf(Lat)=Conf(C)_m,n⊗Conf(C)_m,n.Fora_j⊗b_iwithboundary X_i,j inConf(Lat),anda

j⊗b_i withboundaryX_i,j inConf(Lat),thebilinearformLL is definedas

LL(aj⊗b_i, a_j⊗b_i) =μ^{(1−n)(m−1)2}

d(X_i,j)d(X_i,j )

(3) When m = 0 or n = 0, we define the configuration space as the ground field. We define LLas themultiplicationofthetwoscalars.

Theorem 2.4. Forany unitary MTC C, theconfigurations spaces of thelattice and the dual lattice aredual toeach other.Preciselythemapfrom Conf(Lat)tothedual space of Conf(Lat)inducedby LL(−,−)isan isometry.

Wefirst provethecasefor m=n= 2. Weprovethe generalcaseby abi-induction in§6.

Proof for the casem=n= 2: Whenm=n= 2,thediagraminEquation(3) becomes theHopflinkandLLdefines theS matrixof C.

Bythemodularityof C,namelytheS matrixisaunitary[30],themapinducedbyLL isanisometry.

Proposition 2.5. SupposeV is aHilbert space and {αi}is an ONB.Let V be the dual spaceof V.Forf ∈V,alinear functionalon V,

r(f) =

i

f(αi)αi (4)

isindependentof thechoiceof thebasis.

Proof. Itfollows directlyfrom thedefinition.

Themap r:V^∗ →V is ananti-isometrywhichiswell-known asthe Rieszrepresen- tation.Thereforeweobtainananti-isometryD:Conf(Lat)→Conf(Lat) thatwecall thedualitymap:

Definition2.6(duality maps).Wedefine D+(x) =

x∈B

LL(x, x)x, D₋(x) =

x∈B

LL(x, x)x,

whereB isanONBofConf(Lat) andB isanONBofConf(Lat).

ThereforeTheorem2.4isequivalent tothefollowingProposition.

Proposition2.7. ThemapD+isananti-linearisometryfromConf(Lat)toConf(Lat), andD₋ isitsinverse.

Definition 2.8.We use 1n,m to denote the trivial configuration whose Z-, X-, Y- configurationsareall1.Wedefine

μn,m: =D₋(1n,m). (5)

Definition 2.9.WedefineLas alinearfunctionalonConf(Lat) as L(x)=LL(x,1_n,m).

Then

L(aj⊗b_i) =μ^{(1−n)(m−1)2}

d(X_i,j) ,

(6) and

μ_n,m=

α∈B

L(α)α, (7)

where B beisanONBofConf(Lat).

In§3,westudytheactionsofrotationsandreflectionsinX- andY-directionsonthe latticesandtheinducedactionstheconfigurationspaces.

In §4, we fix m and study the structure of the configuration space for different n.

We prove that μ_3,m defines a Frobenius algebra and these configuration spaces ad- mit the action of planar tangles in the X-direction. Thus For each m ≥ 1, {Sn,+ = Conf(C)_m,n}n∈N definesasubfactorplanaralgebra.Thisdefines them-intervalJones- WassermannsubfactorfortheunitaryMTC_C. WeprovethisresultinTheorem 4.13.

In§5,wedescribetheactionofplanartanglesonthedualspace{Sn,−}.

In §6, we construct a planar algebra *-isomorphism from S_·,+ to S_·,−. That means {Sn,±} defines a self-dual subfactor planar algebra, i.e., the m-interval Jones- Wassermannsubfactorfor C isself-dual.WeprovethisresultinTheorem6.18.Further- more,weprovethatthe*-isomorphismcommuteswithstringFouriertransform(SFT).

That means {Sn = Conf(_C)_m,n}n∈N defines an unshaded subfactor planar algebra, i.e., them-intervalJones-Wassermann subfactorfor C issymmetrically self-dual.(Itis called symmetricallyself-dual,sincetheSFTisasymmetricmatrix[17].)Weprovethis resultinTheorem6.20.Moreover,thedualitymapisrelatedtotheSFToftheunshaded planaralgebra.

Remark2.10.Ifwe fix n, insteadof m, then allthe resultsalso work.So we alsohave the action of planar tangles on the configuration spaces in the Y-direction. Therefore theconfigurationspaces{Conf(C)m,n}m,n∈N}admittheactionofplanartanglesintwo differentdirections.

3. Actionsonconfiguration spaces

3.1. Automorphismson thelattice

NotethatthelatticeZm×Zn isinvariantunderthefollowing actions:

• Theclockwise 2π/nrotationaroundtheY-directionρ₁:(i,j)→(i−1,j).

• ThereflectionintheX-directionθ1: (i,j)→(n+ 1−i,j).

• Theclockwise 2π/mrotationaroundtheX-directionρ2:(i,j)→(i,j+ 1).

• ThereflectionintheY-directionθ₂ (i,j)→(i,m−1−j).

NowletusdefinetheinducedactionontheconfigurationspaceConf(C)n,m. Fork= 1,2,theinducedactionsontheZ-configurationsare

ρk(X)i,j=X_ρ−1 k (i,j), θk(X)i,j=J_C(X_θ−1

k (i,j)).

ForanX-configurationa_k,we define

ρ₁(a_j) = ,

θ1(aj) =J_C(aj), ρ₂(a_j) =a_j,

θ2(aj) =

= .

ForaY-configurationbi,wedefine ρ1(bi) =bi,

θ1(bi) = = ,

ρ2(bi) = ,

θ₂(b_i) =J_C(b_i).

Definition 3.1.Foraconfigurationa_j⊗b_i,we define

ρ₁(a_j⊗b_i) = (ρ₁(a₀)⊗ · · · ⊗ρ₁(a_m−1))⊗(b₂⊗ · · · ⊗b_n⊗b₁), θ1(aj⊗b_i) = (θ1(a0)⊗ · · · ⊗θ1(am−1))⊗(θ1(bn)⊗ · · · ⊗θ1(b1)), ρ₂(a_j⊗b_i) = (a_m−1⊗a₁⊗ · · · ⊗a_m−2)⊗(ρ₂(b₁)⊗ · · · ⊗ρ₂(b_n)), θ2(a_j⊗b_i) = (θ2(am−1)⊗ · · · ⊗θ2(a0))⊗(θ2(b1)⊗ · · · ⊗θ2(bn)).

The actions onthe configurationspace are definedbytheir linearor anti-linear exten- sions.

Note thatthisdefinitioncoincidewiththegeometricactionsontheconfigurationsin Fig.4.Therefore,theirrelationsalsoholdontheconfigurationspace.

Proposition 3.2.Ontheconfigurationspace, ρ₁ andθ₁ commutewithρ₂ andθ₂,andfor k= 1,2,ρkθk=θkρ⁻_k¹,ρ^m_k = 1,θ²_k= 1.

3.2. Automorphismson thedualpairof lattices

Similarly,wealsodefinethefouractionsontheduallatticeLatandtheconfiguration spaceConf(Lat).

Proposition3.3. Forx∈Conf(Lat)and x∈Conf(Lat),wehave that

LL(x, x) =LL(ρk(x), ρk(x)), k= 1,2 (8) LL(x, x) =LL(θ1(x), ρ1θ1(x)), (9) LL(x, x) =LL(ρ1θ2(x), θ2(x)). (10) Proof. Itisenoughtoprovethecasex=a_j⊗b_i,x=a

j⊗b_i.

RecallthatLLisdefinedbyacloseddiagram inthe3Dspaceas showninEquation (3).Applyingtherotationonthe3Ddiagram,weobtainEquation(8).

Ifweconsiderthe3Ddiagram asanelement inC,thenwehavethat

LL(a_j⊗b_i, a

j⊗b_i)

=μ^{(1−n)(m−1)2}

d(X_i,j)d(X_i,j )

=μ^{(1−n)(m−1)2}

d(X_i,j)d(X_i,j )

=LL(θ1(x), ρ1θ1(x))

Theproof ofEquation(10) issimilar.

Bydefinitions,ρ_k, θ_k,k= 1,2,preserve1_n,m.Take x = 1_n,minProposition3.3,we obtainthat

Proposition 3.4. Foranyxin Conf(Lat),k= 1,2, L(ρ_k(x)) =L(x), L(θk(x)) =L(x).

Proposition 3.5. Thefouractions ρ_k,θ_k,k= 1,2,preserve μ_n,m.

Proof. ThestatementfollowsfromProposition3.4andthedefinitionofμn,minEquation (7).

4. Jones-WassermannsubfactorsforMTC

4.1. Identification

Suppose_C isaunitaryMTC.TakeIrr^m={X =X₀⊗· · ·⊗X_m−1:X_j∈Irr, 1≤j ≤ m−1}.LetC^mbethem^thDelignetensorpowerofC,denotedbyC^m:=CC· · ·C. We canconsider X as anobjectX :=X0· · ·Xm−1 inC^m.Then thesetIrr^mis identicaltothesetof(representativesof)simpleobjectsof_C^m.TheobjectsX in_C and X inC^m havethesamequantumdimension,d(X) =d(X)=

m−1

j=0

d(Xj).

Definition4.1.ForaunitaryMTCC andm∈Z+,wedefine

N_X = dim hom_C(X₀⊗ · · · ⊗X_m−1,1); (11) γ=γm=

X∈Irr ^m

NXX. (12)

Proposition4.2. Recallthat μistheglobal dimensionof C.Form≥1, d(γ) =μ^m−1.

Proof. Itisobvious form= 1.Whenm≥2, d(γ) =

X∈Irr^m

N_Xd(X)

=

X∈Irr_m−1,Y∈Irr1

dim hom_C(X, Y )d(X)d(Y) by Frobenius reciprocity

=

X∈Irr m−1

d(X) ²

=μ^m−1.

Notation4.3.Forafixedm,we takeδ=μ^m−12 .

ForeachX,letON B_C^∗(X) be anONBofhom_C(X, 1).Thenwecanusethebasisto representthemultiplicity ofsimpleobjectsinγ.

γ=

X∈Irr^m,b∈ON B^∗_C(X)

X(b), (13)

whereX(b)=X=X0· · ·Xm−1.

Therepresentationiscovariantwithrespect tothechoiceoftheONB:Foranobject Y in C^m and a morphism y ∈ hom_C^m(Y,N_XX), we take two ONB B(1), B(2) of hom_C(X, 1).Thenweobtaintwo representations

y=

b1∈B(1)

y(b1), y(b1)∈hom_C^m(Y, X(b1)),

y=

b2∈B(2)

y(b₂), y(b₂)∈hom_C^m(Y, X(b₂)).

Thecovarianceoftherepresentationmeansthat y(b) =

b

b, by(b).

Note that

hom_C^m(1, γⁿ) =

Xi,j∈Irr^nm

bi∈ON B^∗_C(X_i,j),1≤i≤n

hom_C^m(1, X_i,j(b_i)), (14)

where

hom_C^m(1, X_i,j(b_i)) =^m−1j=0 hom_C(1, X_i,j(b_i)).

Wecallhom_C^m(1,γⁿ) the n-boxspaceofγ.Foranyaj ∈hom_C(1,X_i,j),0≤j≤m−1, we havea_j(b_i)∈hom_C^m(1,X_i,j(b_i)).

Definition 4.4.Wedefineamap Φ: hom_C^m(1,γⁿ)→Conf(C)n,m asalinearextension of

Φ(aj(b_i)) =aj⊗b_i.

ThedefinitionisindependentofthechoiceoftheONBb_i,sincetherepresentationis covariant.Moreover,

a_j(b_i), c_j(d_i)= a_j, c_j b_i, d_i= a_j⊗b_i, c_j⊗d_i.

So Φ is an isometry. Therefore we can identify the vectors in the two Hilbert spaces hom_C^m(1,γⁿ) and Conf(_C)_n,m. Weusethenotation

a_j(b_i) to representanelement inhom_C^m(1,γⁿ).

Definition 4.5. Induced by the isometry Φ, the four actions ρ_k, θ_k, k = 1,2, and the contractions∧k,k≥0,arealsodefinedonhom_C^m(1,γⁿ),stilldenotedbyρk,θk.

Recall that the multiplicity of X_−,j in γ is represented by b in ON B^∗(X_−,j). We needananti-isometricinvolutiononON B^∗(X_−,j) tospecifythedualofX_−,j(b).Tobe compatible with the geometric interpretation of theconfiguration in the3D space, we define thedualbyθ₁ asfollows:

Definition 4.6.ForanobjectX_−,j(b),wedefineitsdualobjectas X_−,j(θ₁(b)).

Note thatθ₁²(b)=b, thusX_−,j(b) = X_−,j(b). ByFrobeniusreciprocity, themodular conjugation on C^mis givenbyθ1.

The element inhom_C^m(1,γⁿ) isusually representedby adiagram onthe 2Dplane.

TobecompatiblewiththeisometryΦ,wesimplifythe3Dpicturesforconfigurationsby theirprojectionsontheplaneY = 0 asfollows:

(1) TheconfigurationinFig.4issimplifiedas itsprojectionontheplaneY = 0,

b1

• b2

• b3

• b4

• a_j

.

(2) Inducedbytheisometry Φ,Lbecomesalinearfunctionalonhom_C^m(1,γⁿ),

L(a_j(b_i)) :=L(a_j⊗b_i) =δ¹⁻ⁿ d(X_i,j)

b•₁

b•₂

b•₃

b•₄ aj

,

wherewesimplifythediagraminEquation(6) byitsprojectionontheplaneY = 0.

4.2. Contractions

The multiplication on _C^m defines a map from hom_C^m(γⁿ,γ^k) ⊗hom_C^m(γ^k,γ^l) to hom_C^m(γⁿ,γ^l). Applying Frobenius reciprocity, we obtain a contraction ∧k : hom_C^m(1,γ^n+k)⊗hom_C^m(1,γ^k+l)→ hom_C^m(1,γ^n+l).Then ∧k isalso definedon the configurationspacesinducedbyΦ.Wegivethedefinitionindetailhere.

Remark4.7. Thenotation ∧k comesfrom thegraded multiplicationin[5].

Suppose X_i,j, Y_i,j, Z_i,j are Z-configurations of size n×m, ×m, k×m. For X- configurationsaj∈hom_C(1,X_i,j⊗Z_i,j) andcj ∈hom_C(1,θ1(Z_i,j)⊗Y_i,j),0≤j≤m−1, wedefinethek-stringcontraction,k≥0,as

aj∧kcj:=

X_1,j · · · X_n,j Z₁ · · · Zk Z_k

· · · Z1 Y_1,j · · · Y_,j

aj cj

. Moreover,a_j∧kc_j:= (a₁∧kc₁)⊗ · · · ⊗(a_m∧kc_m).Suppose

bi ∈hom_C(1, X_i,j), for 1≤i≤n;

bi ∈hom_C(1, Z_i,j), forn+ 1≤i≤n+k;

di∈hom_C(1, θ1(Z_i,j)), for 1≤i≤k;

di∈hom_C(1, Y_i,j), fork+ 1≤i≤k+,

namelyb_i∈ConfY(X_i,j⊗Z_i,j) andd_i∈ConfY(θ1(Z_i,j)⊗Y_i,j).Wedefinethek-string contraction∧k ontheconfigurationsaj⊗b_i andcj⊗d_i as

(a_j⊗b_i)∧k(c_j⊗d_i) = k s=1

θ₁(d_k+1−s), b_n+s

a_j∧kc_j

⊗ _n

i=1

b_i⊗

k+

i=k+1

d_i

. (15)

Definition 4.8.We define the k-string contraction on the configuration spaces ∧k : Conf(C)n+k,m⊗Conf(C)k+,m → Conf(C)k+,m as a linear extension of Equation (15).

When = 0, we can identity Conf(C)n+k,m as operators from Conf(C)k,m to Conf(C)n,mcorrespondingtoFrobeniusreciprocity.Moreover,thecompositionofthese operators isassociative.SoweobtainaC^∗-algebroidforafixedm.

Proposition 4.9. Recall that ρ2 and θ2 are actions in the Y-directions. They commute with thecontraction∧k in theX-direction.

Proof. Recallthatρ2 isanisometryanditcommuteswithθ1 byProposition3.2,so ρ2(aj⊗b_i)∧kρ2(cj⊗d_i)

= k s=1

θ1ρ2(dk+1−s), ρ2(bn+s)

ρ2(aj)∧kρ2(cj)

⊗ _n

i=1

ρ2(bi)⊗ k+

i=k+1

ρ2(di)

= k s=1

θ1(dk+1−s),(bn+s)ρ2(a_j∧kc_j)⊗ _n

i=1

ρ2(bi)⊗ k+

i=k+1

ρ2(di)

=ρ2((aj⊗b_i)∧k(cj⊗d_i)).

Theproof forθ₂issimilar.

Lemma 4.10.Whenk= 1,wehave

• • • •

b1 bn d1+1 d1+

· · · · · ·

a_j c_j

=

b∈ON B(Zj) • • • • • •

b₁ b_n b θ1(b) d1+1 d1+

· · · · · ·

a_j c_j

.

Proof. WeapplyEquation(2),theresolutionofidentityinC,to1Zj inthefirstdiagram.

Only the component equivalent to 1remains non-zero, since the first diagram has no boundaryontheleft.This componentgivestheseconddiagram.

4.3. Frobenius algebras

Definition4.11.Wedefineμ_n= Φ⁻¹(μ_n,m),forn≥1,andμ₀= 1.Then μn=

α∈B

L(α)α, whereB be isanONBofhom_C^m(1,γⁿ).

Moreover,μ1isthecanonicalinclusionfrom1 toγ,andμ2 isthecanonicalinclusion from1 toγ⊗γ whichdefinesthedualofobjects.ByProposition3.5,fork= 1,2,

ρ_k(μ_n) =μ_n, (16)

θk(μn) =μn. (17)

Letusprovethatμ₃ definesaFrobenius algebra.

Lemma4.12.Forn≥2,≥1,

μ_n∧1μ=δ⁻¹μ_n+−2. (18) Proof. SupposeX_i,j,Y_i,j,Zj areZ-configurationsofsize(n−1)×m,(−1)×m,1×m respectively.ByLemma2.3,

d(Zj)aj∧1cj : Zj ∈Irr^m, aj ∈ON B(X_i,j⊗Zj), cj∈ON B(Zj⊗Y_i,j)

forms an ON B(X_i,j ⊗Y_i,j). Take b_i = b₁⊗ · · · ⊗b_n and d_i = d₁⊗ · · · ⊗d, where b_i∈ON B(X_i,j) andd_i ∈ON B(Y_i,j).ByLemma4.10,wehave

L(

d(Z_j)a_j∧1c_j(b_i⊗d_i))

d(Z_j)a_j∧1c_j(b_i⊗d_i)

=δ

b∈ON B^∗(Zj)

L(aj(b_i⊗b))L(cj(θ1(b)⊗d_i))aj∧1cj(b_i⊗d_i)

=δ

⎛

⎝

b∈ON B^∗(Zj)

L(aj(b_i⊗b))aj(b_i⊗b)

⎞

⎠

∧1

⎛

⎝

b∈ON B^∗(Zj)

L(cj(θ1(b)⊗d_i))cj(θ1(b)⊗d_i)

⎞

⎠.

Summingoveraj(b_i),cj(d_i),wehave

μ_n+−2=δμ_n∧1μ.

Formorphismsv∈hom_C^m(1,γ),w∈hom_C^m(γ,γ²),wecall(γ,v,w) aQ-system[20], or (γ,v,w,v^∗,w^∗) aFrobeniusalgebra[24] inthe*-category C^m,if

(w⊗1γ)w= (1γ⊗w)w;

(v^∗⊗1_γ)w= 1_γ = (1_γ⊗v^∗)w;

(w^∗⊗1_γ)1_γ⊗w=ww^∗.

Theorem 4.13.By Frobenius reciprocity, we can identify μn+k as a morphism μ^k_n ∈ hom_C^m(γ^k,γⁿ).Then(γ,μ₁,δ²μ¹₂)isaQ-systemin _C^m.

Proof. ByEquations(16),(17) and(18),

(μ¹₂⊗1γ)μ¹₂=μ¹₃= (1γ⊗μ¹₂)μ¹₂; (μ¹₀⊗1γ)μ¹₂=δ⁻²1γ = (1γ⊗μ¹₀)μ¹₂; (μ²₁⊗1_γ)1_γ⊗μ¹₂=μ_2,2=μ¹₂μ²₁.

So(γ,μ1,δ²μ¹₂) isaQ-systemin C^m.

Notation 4.14.Since μ₁ is the canonical inclusion, we simply denote this Q-system or Frobenius algebraby(γ,μ3).

Corollary 4.15.Fork≥1,n≥k+ 1,≥k,

μn∧kμ=δ⁻¹μn+−2k; μ_k∧kμ_k = 1.

Definition 4.16.Wecallthesubfactorassociated withtheFrobeniusalgebra(γ,μ3) the m-intervalJones-Wassermannsubfactorof C.

Themodularityis notused intheconstructionoftheJones-Wassermannsubfactors, butitiscrucialintheproofoftheself-dualityoftheJones-Wassermannsubfactor.The formulaof(γ,μ₃) intermsofthe3Dconfigurationisintuitiveintheproofofself-duality.

Remark 4.17.For an X-configuration aj with boundaryX_i,j, we project it to the 2D space, suchthattheboundarypointsareorderedonaline as

X1,0, . . . , X1,m−1, X2,0, . . . , X2,m−1, . . . Xn,0, . . . , Xn,m−1: