TTwo-dimensional topological order and operatoralgebras
Yasuyuki Kawahigashi
Graduate School of Mathematical SciencesThe University of Tokyo, Komaba, Tokyo, 153-8914, Japane-mail: [email protected]
Kavli IPMU (WPI), the University of Tokyo5–1–5 Kashiwanoha, Kashiwa, 277-8583, JapanTrans-scale Quantum Science InstituteThe University of Tokyo, Bunkyo-ku, Tokyo 113-0033, JapanandiTHEMS Research Group, RIKEN2-1 Hirosawa, Wako, Saitama 351-0198,Japan
February 23, 2021
Abstract
We review recent interactions between mathematical theory of two-dimensionaltopological order and operator algebras, particularly the Jones theory of sub-factors. The role of representation theory in terms of tensor categories isemphasized. Connections to 2-dimensional conformal field theory are also pre-sented. In particular, we discuss anyon condensation, gapped domain wallsand matrix product operators in terms of operator algebras.
In quantum mechanics, an observable is represented by a self-adjoint operator on acomplex Hilbert space of states. It would be convenient if we are allowed to makealgebraic operations such as addition and multiplication on them, so we considera set of operators on a complex Hilbert space which is closed under addition andmultiplication. In order to add and multiply them freely, we restrict our attention to bounded linear operators, though many observables appearing quantum mechanicsare unbounded. One way to get a bounded linear operator from an unboundedone is exponentiation. We exponentiate an unbounded self-adjoint operator A to1 a r X i v : . [ m a t h - ph ] F e b et a (bounded) unitary operator exp( iA ). Since self-adjointness of an observableis important, we also require the set of operators is closed in the ∗ -operation. (Inmathematics, we write A ∗ for the adjoint of A rather than A † which is more commonin physics literature.) We further require that it is closed in a certain topology, andthen we call it an operator algebra .We have two natural choices for this topology. One is the norm topology whicharises from uniform convergence on the unit ball of the Hilbert space and the other isthe strong operator topology which arises from pointwise convergence on the Hilbertspace. (The strong operator topology is weaker than the norm topology.) Whenan operator algebra is closed in the norm topology, it is called a C ∗ -algebra . Whenan operator algebra is closed in the strong operator topology, it is called a vonNeumann algebra . The latter was introduced by von Neumann in connection toquantum mechanics and representation theory as the name shows. Both naturallyappear in mathematical physics. Being closed in a weaker topology is a strongercondition, so a von Neumann algebra is automatically a C ∗ -algebra by definition,but it is usually not very convenient to regard a von Neumann algebra as a C ∗ -algebra since a typical von Neumann algebra is often quite different from a typical C ∗ -algebra. For example, a typical C ∗ -algebra appearing in applications is oftenseparable (as a Banach space), but a von Neumann algebra is never separable unlessit is finite dimensional.Finite dimensional C ∗ -algebras and finite dimensional von Neumann algebrasare the same, and they are finite direct sums of M n ( C ), n × n -matrix algebras withcomplex entries. From this viewpoint, one can say that an infinite dimensionaloperator algebra is an infinite dimensional analogue of a matrix algebra.A commutative C ∗ -algebra is isomorphic to C ( X ) for some compact Hausdorffspace X . A commutative von Neumann algebra is isomorphic to L ∞ ( Y, µ ) for somemeasure space (
Y, µ ). From this viewpoint, one can say that a general operatoralgebra is a noncommutative version of a function algebra. We sometimes say forthis that a general operator algebra is a function algebra on a noncommutative space .Noncommutative geometry of Connes[15] is a far advancement of this idea.A simple von Neumann algebra, in the sense that it has only trivial closed two-sided ideals in the strong operator topology, is called a factor . A general von Neu-mann algebra is decomposed into a direct integral of factors, a generalized version ofa direct sum, just like a representation decomposes into a direct integral irreduciblerepresentations. A von Neumann algebra is a factor if and only if it does not de-compose into a direct sum of two nonzero von Neumann algebras. A von Neumannalgebra is a factor if and only if its center is the complex number field C .The von Neumann algebra B ( H ), the set of all bounded linear operators on aHilbert space H , is a factor, but this is not very exciting. A much more interestingfactor is constructed as follows. A matrix x ∈ M n ( C ) is mapped to (cid:18) x x (cid:19) ∈ M n ( C ). With this embedding, we have an increasing sequence of matrix algebras, M ( C ) ⊂ M ( C ) ⊂ M ( C ) ⊂ · · · . We take an increasing union (cid:83) ∞ n =1 M n ( C ). This has a natural representation on a2ilbert space and we take the closure of the image in the strong operator topology.This gives a factor and is called the hyperfinite II factor .When one factor is N contained in another factor M , we say it is a subfactor .Its study is somehow analogous to an algebraic study of a subgroup H ⊂ G and asubfield K ⊂ L . We have a notion of the index [ G : H ] for a subgroup H ⊂ G andthe degree of extension [ L : K ] for a field extension L ⊃ K . We have a similar notionof the Jones index [ M : N ] for a subfactor N ⊂ M . One important new feature isthat it is a positive real number larger than or equal to 1, or infinity, and can easilytake a non-integer value. We are interested in algebraic studies of subfactors withfinite Jones index. Subfactors of the hyperfinite II factor have very rich structures.Theory of subfactors produced a new topological invariant, the Jones polynomial [33]for knots , which is expected to be useful for topological quantum computations[61].
We consider a certain 2-dimensional status of matters and a typical example is thinliquid on a large plane. A point on the plane can have a special status by excitation .An excited point behaves like a particle ( quasi-particle ) and is called an anyon .Suppose we have finitely many anyons and study exchanges among them. Thenatural group for exchanges is the braid group and we have braid group statistics.(If the original dimension where quasi-particles live is 3, then the natural grouprepresenting such an exchange is the permutation group, and we have a boson ora fermion .) An anyon is a more general version of a boson and a fermion and wehave this name because a phase arising from an exchange of quasi-particle can take any value. A modular tensor category gives a mathematical description of such asystem of finitely many anyons. Each irreducible object of a modular tensor categorycorresponds to an anyon. Two anyons are fused and produce new anyons. This is afusion of anyons. (See Bakalov-Kirillov[4] for details of modular tensor categories.) Aso-called non-abelian anyon is expected to produce a topological quantum computer,but such an anyon has not been observed experimentally yet.Noncommutative geometry[15] is a certain generalization of Riemannian geom-etry using operator algebras. An operator algebra with a certain extra structure isregarded as a noncommutative manifold . It has been used to study the fractionalquantum Hall effect . This is also related to anyons.A topological phase is also understood in terms of homotopic classification of gapped Hamiltonians and we have a version called symmetry protected topologicalphases . It has been recently studied extensively in the context of C ∗ -algebras andindex theorems by Ogata[54].We next explain the notion of a fusion category, which is a more general versionthan a modular tensor category. Recall that a finite group G consists of the followingingredients.1. An associative multiplication 3. The identity element3. The inverse elementsFor a finite group G , the set of its finite dimensional unitary representations havethe following structures.1. Irreducible decomposition into finitely many ones2. An associative tensor product3. The identity representation4. The contragredient representationAbstract axiomatization of such a set gives a notion of a fusion category . That is,each object behaves like a representation, and we have only finitely many irreducibleobjects. We have direct sums and irreducible decompositions, tensor products of ob-jects, and the trivial object and the dual object of each object, which behaves likea contragredient representation. We have morphisms between objects, like inter-twiners between representations. For two representations π and σ of a group G ,the two tensor products π ⊗ σ and σ ⊗ π are trivially unitarily equivalent, but suchequivalence is not assumed for a general fusion category. In this sense, the tensorproduct operation in a fusion category is noncommutative in general. Fusion cate-gories give a special subclass of tensor categories. (See Etingof-Nikshych-Ostrik[20]for a general theory of fusion categories. See Bischoff-Kawahigashi-Longo-Rehren[7]for tensor categories and operator algebras.) Each object of a fusion category has adimension, which is a positive real number larger than or equal to 1. A dimensionis additive with respect to a direct sum and multiplicative with respect to a tensorproduct.We have an important class of fusion categories for which the above commutativ-ity of tensor products holds in some mathematically nice way. Such commutativityis called braiding because it is similar to reversing a crossing of two wires. (A wirelabels an object in a fusion category.) A braiding naturally comes in a pair — over-crossing and undercrossing. It is more interesting if these two are really different.If this is the case in some appropriate sense, this gives a notion of a modular tensorcategory , as mentioned above for anyons. So objects in a modular tensor categorybehave more like representations of a group, and they also behave like particles.The Kitaev toric code [34] gives one example of a modular tensor category. In thisexample, we have four irreducible objects representing four anyons. Each irreducibleobject has dimension 1 in this example, and the tensor product rules (also called fusion rules ) are given by the group multiplication of Z / Z × Z / Z . (This groupstructure can give a trivial braiding, but here we have a different, nontrivial braidingstructure.)Another example is the Fibonacci category and it has two anyons, the trivial onelabeled as 1 and another one labeled as τ . The fusion rules are given by τ = 1 ⊕ τ .This is expected to be related to fractional quantum Hall liquids. This is also related4o the Jones polynomial at the deformation parameter q = exp(2 πi/ n irreducible objects. Thebraiding produces an n -dimensional unitary representation π of SL (2 , Z ), the modu-lar group. (See Bakalov-Kirillov[4].) We set S = π (cid:18) − (cid:19) and T = π (cid:18) (cid:19) .A matrix Z is called a modular invariant if it satisfies the following, where theindex 0 means the trivial object, often called the vacuum , which behaves like a trivialrepresentation.1. Z λµ ∈ { , , , . . . } .2. Z = 1.3. ZS = SZ , ZT = T Z .When a modular tensor category is given, the number of modular invariants isalways finite, and they are sometimes explicitly classified. See Cappelli-Itzykson-Zuber[12] for such a classification. A typical appearance of a modular invariantis 2-dimensional conformal field theory. Use of modular invariants in classifica-tion in operator algebraic conformal field theory has been given in Kawahigashi-Longo[42][43].
We now recall a mathematical (axiomatic) framework of quantum field theory. Ourbasic ingredients are as follows (as in the
Wightman axioms ).1. The spacetime2. The spacetime symmetry group3. Quantum fields (operator-valued distributions) on the spacetimeThis setting is well-studied and has a long history, but is difficult to handle froma mathematical/technical viewpoint, and we consider a different approach based onoperator algebras instead here. We now consider a family of von Neumann algebrasparameterized by spacetime regions. Each von Neumann algebra is generated byobservables on a spacetime region. Though observables can be easily unboundedoperators, we consider von Neumann algebras of bounded linear operators as before.Our idea is that relative positions of this family of von Neumann algebras encodephysical information of a quantum field theory. This approach is called algebraicquantum field theory as in Haag[28].A 2-dimensional conformal field theory is a quantum field theory with conformalsymmetry on the 2-dimensional Minkowski space. It decomposes into an extension5f a tensor product of two quantum field theories living on the compactification ofthe light rays { x = ± t } and each is called a chiral conformal field theory (a chiralhalf ). It is a quantum field theory on the spacetime S with Diff( S )-symmetry.(Now the space and the time are mixed into one dimension).An operator algebraic formulation of a chiral conformal field theory is called a(local) conformal net in Kawahigashi-Longo[42]. This is based on operator algebrasof observables on S . We have another mathematical formulation of conformal fieldtheory, based on Fourier expansions of operator-valued distributions on the circle S , which is called a vertex operator algebra . (This name means an algebra of vertexoperators and is not directly related to operator algebras.) See Kawahigashi[36][39]as general references on conformal nets and vertex operator algebras. These two aresupposed to be different mathematical formulations of the same physical theory, sothey should be equivalent, at least under some nice assumptions. We have shownthat we can pass from a vertex operator algebra to a local conformal net and comeback under some mild assumptions as in Carpi-Kawahigashi-Longo-Weiner[13]. Thisrelation of the two has been much studied recently.Under the standard set of axioms of a conformal net, each von Neumann algebraappearing in one is always isomorphic to the unique Araki-Woods factor of typeIII . So each von Neumann algebra contains no information about a conformal fieldtheory, but as a family of von Neumann algebras, it does contain information aboutconformal field theory.Representation theory of a conformal net is called a theory of superselection sec-tors . Each representation (of the family of von Neumann algebras on another Hilbertspace) is called a superselection sector as in Doplicher-Haag-Roberts[18][19]. Theygave a tensor product structure there as a composition of DHR endomorphisms andit has a braiding in the setting of conformal field theory as in Fredenhagen-Rehren-Schroer[24][25]. Under some finiteness assumption, called complete rationality , weget a modular tensor category of representations as in Kawahigashi-Longo-M¨uger[44].Representation theory of a vertex operator algebra is theory of modules . Wehave a tensor product structure and a braiding for modules. Under some finitenessassumption, we also get a modular tensor category of modules as in Huang[29][30].Under nice identification of a local conformal net and a vertex operator algebraas in Carpi-Kawahigashi-Longo-Weiner[13], Gui[27] has identification of the corre-sponding representation categories for many examples. Recall a 2-dimensional topological order is described with a modular tensor category.Suppose we have two topological orders described with two modular tensor categories C and C . The exterior tensor product C (cid:2) C opp2 , where “opp” means reversing thebraiding, gives a new modular tensor category. The number of new anyons is theproduct of the two numbers of anyons.We have a physical notion of a gapped domain wall between the two topological6rders and it is mathematically defined to be an irreducible local Lagrangian Frobe-nius algebra with the base space (cid:76) Z λµ λ (cid:2) ¯ µ in C (cid:2) C opp2 , where Z λµ = 0 , , , . . . gives a generalized modular invariant . (See Kawahigashi[37] for a mathematicallyprecise formulation.)Lan-Wang-Wen[46] conjectured that if we have a generalized modular invariantmatrix Z with Z = 1 for modular tensor categories C and C , and the entriesof matrix Z satisfy some inequalities about multiplicities , then there would exist acorresponding gapped domain wall.However, subfactor theory easily disproves this conjecture. Actually, the chargeconjugation matrix ( δ λ ¯ µ ) gives a counterexample for some modular tensor category C = C arising from a finite group by a recent work of Davydov[16]. We have alsogiven a correct form of the conjecture using Witt equivalence in Kawahigashi[37]In physics literature, we have a notion of composition of two gapped domainwalls and its irreducible decomposition. We would like to formulate this notionmathematically.Suppose we have three topological orders described with three modular tensorcategories C , C and C , respectively. We further assume to have two irreduciblelocal Lagrangian Frobenius algebras with the base space (cid:76) Z λµ λ (cid:2) ¯ µ in C (cid:2) C opp2 and (cid:76) Z µν µ (cid:2) ¯ ν in C (cid:2) C opp3 .We would like to have a new Frobenius algebra with the base space (cid:76) ( (cid:80) µ Z λµ Z µν ) λ (cid:2) ¯ ν . That is, the matrix part is just given by a matrix multiplication .We can construct a Frobenius algebra with the base space (cid:76) ( (cid:80) µ Z λµ Z µν ) λ (cid:2) ¯ ν byconsidering a tensor product functor and taking an intermediate Frobenius algebrabut this is reducible in general. We have a notion of irreducible decomposition of aFrobenius algebra corresponding to irreducible decomposition of an operator algebra.We show that after irreducible decomposition, each Frobenius algebra is localand Lagrangian in Kawahigashi[38]. Being Lagrangian is shown to be equivalent tomodular invariance property in M¨uger[51].On the matrix level, we thus have a decomposition (cid:80) µ Z λµ Z µν = (cid:80) i Z ,iλν . α -induction and anyon condensation For a modular tensor category C and a Frobenius algebra, we have a machinery of α -induction , similar to the induction procedure in the classical representation the-ory. If C corresponds to a rational conformal field theory and it has an extension(like a conformal embedding), then the vacuum representation of the larger con-formal field theory gives a commutative Frobenius algebra in C . Our setting is ageneralization of this, and then from a representation of the smaller conformal fieldtheory, we get something like a representation of the larger conformal field theory.This process of α -induction depends on a choice of a positive or negative braiding,and we use symbols α ± λ for the induced “representations” arising from an object λ in the modular tensor category C . See B¨ockenhauer-Evans-Kawahigashi[8][9] foran operator algebraic formulation. The Frobenius algebra corresponds to the dualcanonical endomorphism in B¨ockenhauer-Evans-Kawahigashi[8][9]. The intersection7f the images of positive and negative α -inductions give a new modular tensor cat-egory. In the case of an extension of a conformal field theory like a conformalembedding, this new modular tensor category is the representation category of thelarger conformal field theory. That is, positive and negative α -inductions produceonly something like a representation, but the intersection of their images really givea genuine representation. This procedure corresponds to condensation of anyons ,as in the branching rule of Bais-Slingerland[3], and objects in the Frobenius algebracorresponds to condensed anyons in Bais-Slingerland[3]. (If we have a commutativeFrobenius algebra, then the condensed anyons have to be bosons.) They make anew vacuum in the new modular tensor category.Many researchers are interested in commutative Frobenius algebras, but the re-sults and methods in B¨ockenhauer-Evans-Kawahigashi[8][9] apply also to noncom-mutative Frobenius algebras. Commutativity of a Frobenius algebra is called lo-cality in B¨ockenhauer-Evans-Kawahigashi[8][9], since this corresponds to localityof an extended chiral conformal field theory. The results in B¨ockenhauer-Evans-Kawahigashi[8][9] are stated in terms of operator algebras, but they can be trans-lated into a language of modular tensor categories alone.The α -induction machinery produces a modular invariant Z λµ = (cid:104) α + λ , α − µ (cid:105) , where λ, µ are irreducible objects in C as in B¨ockenhauer-Evans-Kawahigashi[8] and anirreducible local Lagrangian Frobenius algebra in C (cid:2) C opp as in Rehren[58]. Thelatter coincides with the one constructed in Fr¨ohlich-Fuchs-Runkel-Schweigert[26] ina more categorical language, as shown in Bischoff-Kawahigashi-Longo[6].A general fusion category has no braiding, but we have a general procedurecalled the Drinfel (cid:48) d center as in Drinfel (cid:48) d[17] which produces a new modular tensorcategory from a given fusion category.The α -induction produces a new fusion category for one choice of + or − braidings. Its Drinfel (cid:48) d center Drinfel (cid:48) d[17] is given by the tensor product of theoriginal modular tensor category and the extended one as in B¨ockenhauer-Evans-Kawahigashi[10]. This mathematical result coincides with a result called boundary-bulk duality in physical literature like Kong[45]. See Table 1 of Kong[45] for moreinterpretation of α -induction in terms of anyon condensation.See Bischoff-Jones-Lu-Penneys[5] for another aspect of anyon condensation andfusion categories. A vector ( v j ) has one index j . This is pictorially represented with one circle withone leg labeled with j as in Fig. 1. A matrix ( a jk ) has two indices j, k . This ispictorially represented with one circle with two legs labeled with j, k as in Fig. 1.Similarly, we can consider circles with 3 legs, 4 legs, and so on. Such an object iscalled a tensor .The ( j, l ) entry of the matrix product of ( a jk ) and ( b kl ) is given by (cid:80) k a jk b kl .This is pictorially represented with two circles corresponding to ( a jk ) and ( b kl ) whereone leg of one and one leg of the other, both labeled with k , are concatenated as8 j aj k Figure 1: A vector v j and a matrix a jk in Fig. 2. Such concatenation is called a contraction . We are interested in tensorswith 3, 4 and 5 legs here. a bj l Figure 2: A matrix product ( ab ) jl Consider a tensor with 3 legs and denote it by a ljk . We consider a state repre-sented as (cid:88) l ,l , ··· ,l n Tr( a l a l · · · a l n ) | l l · · · l n (cid:105) where a l m is a matrix and a 3-tensor as in Fig. 3. Such a state was first considered byFannes-Nachtergaele-Werner[23] and is called a matrix product state (MPS) today. a a al l l n (cid:88) l ,l , ··· ,l n | l l · · · l n (cid:105) Figure 3: A matrix product stateNext consider a tensor with 4 legs and denote it by b lmjk . We consider a matrixrepresented as (cid:88) l ,l ,...,l n ,m ,m ,...,m n Tr( b l m · · · b l n m n ) | l l · · · l n (cid:105) (cid:104) m m · · · m n | as in Fig. 4 and we call it a matrix product operator (MPO) .Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11] considered analgebra of matrix product operators arising from a system of physically nice tensornetworks of 4-tensors. Such an algebra is called a matrix product operator algebra(MPOA) . They considered a category of matrix product operators and presented agraphical method to construct an interesting system of anyons . They have shownthat we get a modular tensor category and discussed its physical consequence.This paper has caught much attention of physicists. They are aware of its simi-larity to an old work of Ocneanu[21]. We now show that their construction is reallythe same as Ocneanu’s mathematically as in Kawahigashi[40].9 l ,l ,...,l n ,m ,m ,...,m n a a am m m n l l l n | m m · · · m n (cid:105) (cid:104) l l · · · l n | Figure 4: A matrix product operatorLet N ⊂ M be a subfactor with finite Jones index and finite depth. We set N = M − , M = M and apply the Jones tower/tunnel construction as in Definitions9.24, 9.43 of Evans-Kawahigashi[22] to get · · · ⊂ M − ⊂ M − ⊂ M ⊂ M ⊂ M · · · , where the prime denotes the commutants. We then consider a double sequence A jk = M (cid:48)− k ∩ M j of finite dimensional C ∗ -algebras. They form commuting squares as in Section 9.6 of Evans-Kawahigashi[22].Recall that a sequence A ⊂ B ∩ ∩ C ⊂ D is called a commuting square if the restriction of E B to C is equal to E A where E A and E B are conditional expectations as in Theorem 5.25 of Evans-Kawahigashi[22].Look at the Bratteli diagrams of a commuting square. (An inclusion of fi-nite dimensional C ∗ -algebras gives a Bratteli diagram as in page 67 of Evans-Kawahigashi[22].) Choose one edge from each diagram corresponding to each ofthe four inclusions. We then get a complex number from the 4 edges. This is anotion of a bi-unitary connection as in Definition 11.3 of Evans-Kawahigashi[22]considered by Ocneanu and Haagerup. (Also see Asaeda-Haagerup[2].) In the caseas above, we have an especially nice bi-unitary connection called a flat connection as in Ocneanu[52], Kawahigashi[35], Section 11.4 of Evans-Kawahigashi[22]. (Wealso get this by fixing two labels of quantum j -symbols . See Chapter 12 of Evans-Kawahigashi[22] for relations between quantum 6 j -symbols and flat connections.) Ifthe original subfactor is hyperfinite and has finite Jones index and finite depth, thenthis flat conenction recovers the original subfactor entirely by Popa’s theorem[56].We further have a finite system of flat connections corresponding to the directsummands of N (cid:48) ∩ M n +1 . They produce a tensor of 4 legs. (We have the number 4since a commuting square has 4 inclusions.) Though the normalization conventionsfor a bi-unitary connection and a 4-tensor are slightly different, the difference is onlyup to the Perron-Frobenius eigenvector entries as in Fig. 11 of Kawahigashi[40].Such a tensor exactly fits in the setting of Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11]. 10et all the four inclusion diagrams be the Dynkin diagram A n . Number thevertices as 1 , , . . . , n and set ε = √− (cid:0) π √− / n + 1) (cid:1) . We define a bi-unitaryconnection as in Fig. 5, where µ denotes the Perron-Frobenius weight, that is, µ ( j ) =sin( jπ/ ( n +1)) / sin( π/ ( n +1)). (This also works for the Dynkin diagrams D n , E , E and E . See Fig. 11.32 of Evans-Kawahigashi[22].) This is actually a “square root”version of the one considered in Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11]. Wlj mk = δ kl ε + (cid:115) µ ( k ) µ ( l ) µ ( j ) µ ( m ) δ jm ¯ ε Figure 5: A flat connection on the Dynkin diagram A n The above situations are very similar to interaction round the face (IRF) solvablelattice models , like the one due to Andrews-Baxter-Forrester[1]. For an IRF model,we fix a diagram, choose four edges from it and make a square with them. Then acomplex value called the
Boltzmann weight , depending on a spectral parameter, isassigned to each such square. These values satisfy certain compatibility conditionssuch as the
Yang-Baxter equation . The crossing symmetry of an IRF model alsocorresponds to a flat connection. When the diagram is the Dynkin diagram of type A n , our formula for the flat connection is essentially the same as the restrictedsolid-on-solid (RSOS) model in Pasquier[55].We have a construction of the Drinfel (cid:48) d center as in Drinfel (cid:48) d[17] which gives amodular tensor category from a fusion category. In subfactor theory, the first suchconstruction was Ocneanu’s asymptotic inclusion as in Evans-Kawahigashi[21], Sec-tion 12.6 of Evans-Kawahigashi[22] which means N ∨ ( N (cid:48) ∩ M ∞ ) ⊂ M ∞ constructedfrom a hyperfinite II subfactor N ⊂ M with finite Jones index and finite depth.We later have the Longo-Rehren subfactor construction[49] and Popa’s symmetricenveloping algebra construction[57].In connection to 3-dimensional topological quantum field theory, Ocneanu fur-ther introduced a tube algebra as in Evans-Kawahigashi[21], Section 12.6 of Evans-Kawahigashi[22], Izumi[31] which is a finite dimensional C ∗ -algebra arising froma fusion category. Its each direct summand describes an “anyon” of the modulartensor category given by the Drinfel (cid:48) d center construction as in Drinfel (cid:48) d[17].Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11] have a similarconstruction of the anyon algebra and it describes a system of anyons.Suppose we start with a subfactor N ⊂ M with finite Jones index and finitedepth. We then have a family of flat connections as above. (The number of flat con-nections is equal to the dimension of the center of N (cid:48) ∩ M n +1 for sufficiently large n .)We can next show that this family produces a 4-tensor satisfying all the settings ofBultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11] simply by chang-ing the normalization constants arising from the fourth root of the Perron-Frobenius11igenvector entries as in Fig. 11 of Kawahigashi[40]. Theorem 6.1
Ocneanu’s tube algebra for the fusion category arising from a sub-factor and the anyon algebra of Bultinck et al. arising from its flat connections areisomorphic. In particular, the two fusion rules are identical and the Verlinde formulaalso holds for the setting of anyon algebra.
We have seen that we can construct a tensor network satisfying the require-ments of Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11] from asubfactor. It is known any fusion category is realized from a subfactor. Usuallya “representation” of such a category is a bimodule over II factors or an endo-morphism of a type III factor as in Longo[47][48], but we can also realize such a“representation” as a flat connection due to the open string bimodule constructionof Asaeda-Haagerup[2]. Theorem 6.2
Suppose a tensor satisfies a setting of Bultinck et al. If it realizes afusion category, then this fusion category is realized by a tensor of flat connectionsarising from a subfactor as above. Then the resulting tube algebras and the modulartensor categories are the same for the original and new systems.
In studies of topological phases, a gapped Hamiltonian has been important. Itmeans a family of self-adjoint matrices where the lowest eigenvalues are always sepa-rated ( “gapped” ) from the next eigenvalues. Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11] studied such a gapped Hamiltonian where the ground stateis realized with a projected entangled-pair state (PEPS) , which is a two-dimensionalversion of a matrix product state. They have a certain expression of a matrix prod-uct operator, which is called a projector matrix product operator (PMPO) since itis also a projection, and the PEPS naturally lives in the range of several PMPOs.We are thus interested in identification of these PMPOs from an operator algebraicviewpoint. (See Cirac-Perez-Garcia-Schuch-Verstraete[14] for a general theory ofPEPS.)Our setting is as follows. We have a 4-tensor corresponding to a bi-unitaryconnection. With this bi-unitary connection, we construct a double sequence offinite-dimensional string algebras A kl with A = C , A kl ⊂ A k +1 ,l and A kl ⊂ A k,l +1 as in Section 11.3 of Evans-Kawahigashi[22] (with any choice of the starting ver-tex ∗ ) and this produces hyperfinite II factors A ∞ ,l and A k, ∞ as the closures of (cid:83) k A kl and (cid:83) l A kl in the strong operator topology on appropriate Hilbert spaces.In particular, we have two subfactors A , ∞ ⊂ A , ∞ and A ∞ , ⊂ A , ∞ , both withfinite Jones index as in Theorem 11.9 of Evans-Kawahigashi[22]. Suppose one of thetwo subfactors has finite depth. Then so does the other by Sato[59]. In this case,we get a finite family of bi-unitary connections { W a } as in Section 3 of Asaeda-Haagerup[2], Section 2 of Kawahigashi[41], corresponding to irreducible A , ∞ - A , ∞ bimodules arising from the subfactor A , ∞ ⊂ A , ∞ . Each W a gives a corresponding4-tensor and they give a PMPO of length k as in Section 3.1 of Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegeman-Verstraete[11], Section 3 of Kawahigashi[41]. Weare interested in identifying the range of this projection. (Note that the range of12his projection is much smaller than the entire Hilbert space.) Then with subfactortechnique based on flat fields of strings of Ocneanu[53], Theorem 11.15 of Evans-Kawahigashi[22], and Definition 3 of Asaeda-Haagerup[2] gives the following as inTheorem 3.3 of Kawahigashi[41]. Theorem 6.3
The range of the k th PMPO in this setting is naturally identifiedwith the k th higher relative commutant A (cid:48)∞ , ∩ A ∞ ,k for the subfactor A ∞ , ⊂ A ∞ , arising from the original bi-unitary connection. The ranges of the k th PMPOs give an increasing sequence of finite dimensionalHilbert spaces indexed by k and they are Hilbert spaces we would like to study inconnection to 2-dimensional topological order. The k th higher relative commutantsgive an increasing sequence of finite dimensional C ∗ -algebras with a positive definiteinner product arising from a trace and they have been extensively studied for almost40 years in subfactor theory. The above identification is expected to be useful tostudy further relations between the two theories.Take the Dynkin diagram A as an example and consider the bi-unitary connec-tion as in Fig. 5. Since the graph A has four edges, the dimension of the Hilbertspace on which the k th PMPO acts is 4 k . Since this bi-unitary connection is flat,the Bratteli diagram of the higher relative commutants is given as in Fig. 6. Forexample, the row for the 6th higher relative commutant says 5 , ,
4, which meansthe algebra is M ( C ) ⊕ M ( C ) ⊕ M ( C ). (The top row is counted as the 0th.) Thedimension of this algebra is 5 + 9 + 4 = 122. An elementary computation showsthat the dimension of the k th higher relative commutants is (3 k − + 1) / k . If we take the Dynkin diagram A n for a large n ,the difference is even more drastic.1125 125 139 14 14Figure 6: The Bratteli diagram of the higher relative commutantsNote that the subfactor for the higher relative commutants here is A ∞ , ⊂ A ∞ , while we had the other subfactor A , ∞ ⊂ A , ∞ for the initial finite depth assumption13bove. The relation between the two subfactor was studied in Sato[60], and they arecharacterized as mutually anti-Morita equivalent subfactors. (In the above exampleof the Dynkin diagram A , the two subfactors are the same.)A recent general treatment of such tensors and tensor categories is given inLootens-Fuchs-Haegeman-Schweigert-Verstraete[50]. Though the language is a littlebit different, this is also closely related to subfactor theory. Acknowledgements
This work was partially supported by JST CREST program JPMJCR18T6 andGrants-in-Aid for Scientific Research 19H00640 and 19K21832.
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