A relative tensor product of subfactors over a modular tensor category
aa r X i v : . [ m a t h . OA ] M a y A relative tensor product of subfactorsover a modular tensor category ∗ Yasuyuki Kawahigashi † Graduate School of Mathematical SciencesThe University of Tokyo, Komaba, Tokyo, 153-8914, JapanandKavli IPMU (WPI), the University of Tokyo5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japane-mail: [email protected]
September 17, 2018
Abstract
We define and study a certain relative tensor product of subfactors over a modulartensor category. This gives a relative tensor product of two completely rational het-erotic full local conformal nets with trivial superselection structures over a commonchiral representation category. In particular, we have a new realization of fusion rulesof modular invariants. This also gives a mathematical definition of a composition oftwo gapped domain walls between topological phases.
The theory of subfactors due to Jones [21] has been a very powerful tool in conformal fieldtheory. We study some aspects of full conformal field theory from a viewpoint of subfactorsand modular tensor categories. (We consider only unitary modular tensor categories inthis paper.)We are interested in a subfactor N ⊂ M with finite Jones index [ M : N ]. In conformalfield theory, it is often useful to formulate a subfactor N ⊂ M in terms of a Q -systemΘ = ( θ, w, x ) where θ is an endomorphism of a type III factor N with separable predualand w ∈ Hom(id , θ ), x ∈ Hom( θ, θ ) as in [31]. When θ is an object of an abstract modulartensor category C , we say Θ is a Q -system on C . (Note that any modular tensor categoryis realized as a subcategory of End( N ) for a type III factor N .) It is also often called a C ∗ -Frobenius algebra on C . When we have x = ε ( θ, θ ) x , where ε denotes the braiding,we say that the Q -system Θ is local. It is also often said that it is commutative. We sayΘ is Lagrangian if we have (dim θ ) = dim C . (See [11, page 153] for the origin of this ∗ Keywords: conformal field theory, modular tensor category, modular invariant, subfactor; MSC: 81T40,46L37, 18D10 † Supported in part by Research Grants and the Grants-in-Aid for Scientific Research, JSPS. { A ( I ) } be a completely rational local conformal net in the sense of [26], [24], andlet C be the Doplicher-Haag-Roberts representation category of { A ( I ) } . (It is a modulartensor category by [26].) A maximal full conformal field theory in the sense of [25] is givenby a local Lagrangian Q -system on C ⊠ C opp as in [25], where “opp” means the oppositemodular tensor category for which the braiding is reversed. (Also see [4, Proposition 6.7].)Let θ = L λ ∈ Irr( C ) ,µ ∈ Irr( C opp ) Z λµ λ ⊠ ¯ µ be the object of such a Q -system on C ⊠ C opp , where“Irr” means the set of equivalence classes of simple objects in the modular tensor category.The matrix Z = ( Z λµ ) is then a modular invariant in the sense that it commutes withthe S - and T -matrices arising from C as in [4, Proposition 6.6]. Suppose we have twosuch modular invariants ( Z λµ ) and ( Z µν ). Then the matrix product Z Z clearly satisfiesthe properties of the modular invariant except for the normalization condition Z = 1where 0 denotes the identity object of the modular tensor category C . It is sometimespossible to have a decomposition Z Z = P i Z ,i into modular invariants Z ,i . Suchdecomposition rules of matrix products have been studied under the name of fusion rulesof modular invariants in [13], [15, Section 3.1], [16, Remark 5.4 (iii)]. We have a machineryof α -induction for subfactors as in [32], [6], [7], [8],[9], and it produces a modular invariantas in [7]. It gives a Q -system on C ⊠ C opp as in [35], and this is a general form of amaximal full conformal field theory on C ⊠ C opp as in [4, Proposition 6.7]. The results in[15, Section 3.1], [16, Remark 5.4 (iii)] say that a braided product of Q -systems on C givesa fusion rule of the corresponding Q -systems on C ⊠ C opp . In this way, we indirectly havean irreducible decomposition of a certain relative tensor product of two local irreducibleLagrangian Q -systems on C ⊠ C opp .One typical example of such fusion rules is given as follows. Let C be the modulartensor category corresponding to the WZW-model SU (2) . Then by [34, Page 202] (andalso by [27, Theorem 2.1] and [4, Proposition 6.7]), we have exactly three irreducible localLagrangian Q -systems on C ⊠ C opp and they are labeled with A , D , E as in [10]. (Theselabels are for the modular invariant matrices. The label A corresponds to the identitymatrix.) Their nontrivial fusion rules are as follows by [13, Section 5.1], [16, Remark 5.4(iii)]. D ⊗ D = 2 D ,D ⊗ E = E ⊗ D = 2 E ,E ⊗ E = D ⊕ E . We would like to extend this relative product to the irreducible local Lagrangian Q -systems on C ⊠ C opp2 and C ⊠ C opp3 in this paper where C , C , C can be different. Thissetting corresponds to a heterotic full conformal field theory.The author thanks M. Bischoff, L. Kong, R. Longo, K.-H. Rehren and Z. Wang foruseful discussions and comments. Parts of this work were done at Instituto SuperiorT´ecnico, Universidade de Lisboa and Microsoft Research Station Q at Santa Barbara.The author thanks the both institutions for their hospitality.2 A relative tensor product of Q -systems We consider a Q -system Θ = ( θ, w, x ) where θ is an endomorphism of a type III factor N with separable predual and w ∈ Hom(id , θ ), x ∈ Hom( θ, θ ). We adapt [5, Definition 3.8],which means that such a Q -system corresponds to an inclusion N ⊂ M where M may notbe a factor. We have N ′ ∩ M = C if and only if the Q -system Θ is irreducible.We recall the following proposition in [33]. (Also see [12, Proposition 3.7, Corollary3.8].) Proposition 2.1
Let
Θ = ( θ, w, x ) be an irreducible local Q -system where θ is of theform L λ ∈ Irr( C ) ,µ ∈ Irr( C opp2 ) Z λµ λ ⊠ ¯ µ for some modular tensor categories C , C . Then itis Lagrangian if and only if we have the modular invariance property S C Z = ZS C and T C Z = ZT C for the matrix Z = ( Z λµ ) , where S C , S C , T C , T C are the S -matrix for C , S -matrix for C , T -matrix for C and T -matrix for C , respectively. This was first raised as a problem in [36, Section 3] in the context of full conformalfield theory, and proved by M¨uger [33] and an unpublished manuscript of Longo and theauthor. This is valid in a general context of a modular tensor category. (Also see [4,Proposition 5.2].)Let ( θ, w, x ) be a Q -system where θ is of the form L λ ∈ Irr( C ) ,µ ∈ Irr( C ) ,ν ∈ Irr( C ) Z λµν λ ⊠ µ ⊠ ν for some modular tensor categories C , C . By applying the functor T to the C componentas in [28, Section 4.1], [4, Section 4.2], we obtain a new Q -system T (Θ) = ( T ( θ ) , w T , x T )where T ( θ ) = L λ ∈ Irr( C ) ,µ ∈ Irr( C ) ,ν ∈ Irr( C ) Z λµν λµ ⊠ ν . (We need the braiding structure of C in order to define x T .) Note that even if Θ is irreducible, T (Θ) is not irreducible ingeneral.Let ( θ, w, x ) be another Q -system where θ is of the form L λ ∈ Irr( C ) ,µ ∈ Irr( C ) Z λµ λ ⊠ µ for some modular tensor categories C , C . By applying [20, Corollary 3.10], we have a new Q -system ( θ , w , s ) with θ = L λ ∈ Irr( C ) Z λ λ where 0 denotes the identity object of C .We call it the restriction of Θ to C .Now let Θ = ( θ , w , x ) and Θ = ( θ , w , x ) be Q -systems where θ = M λ ∈ Irr( C ) ,µ ∈ Irr( C opp2 ) Z λµ λ ⊠ ¯ µ on C ⊠ C opp2 and θ = M µ ∈ Irr( C ) ,ν ∈ Irr( C opp3 ) Z µν µ ⊠ ¯ ν on C ⊠ C opp3 for some modular tensor categories C , C , C .Let Θ ⊠ Θ be the tensor product of the two Q -systems for which the object is givenby M λ ∈ Irr( C ) ,µ ∈ Irr( C opp2 ) ,µ ′ ∈ Irr( C ) ,ν ∈ Irr( C opp3 ) Z λµ Z µ ′ ν λ ⊠ ¯ µ ⊠ µ ′ ⊠ ¯ ν. By applying the T functor to the C components, we obtain a new Q -system whose objectis M λ ∈ Irr( C ) ,µ ∈ Irr( C opp2 ) ,µ ′ ∈ Irr( C ) ,ν ∈ Irr( C opp3 ) Z λµ Z µ ′ ν λ ⊠ ¯ µµ ′ ⊠ ¯ ν.
3y restricting this Q -system to C ⊠ C opp3 , we obtain a new Q -system whose object is M λ ∈ Irr( C ) ,µ ∈ Irr( C ) ,ν ∈ Irr( C opp3 ) Z λµ Z µν λ ⊠ ¯ ν. Definition 2.2
We call the above Q -system the relative tensor product of Θ and Θ over C and write Θ ⊗ C Θ . From the definition, it is easy to see the following.
Proposition 2.3
The relative tensor product operation is associative.
To apply this notion to a full conformal field theory, we need the following.
Proposition 2.4
If two Q -systems are both local, then the relative tensor product Θ ⊗ C Θ is also local. Proof.
For notational simplicity, we may treat C ⊠ C opp3 as a single modular tensorcategory, so we simply write C for C ⊠ C opp3 as if C were the trivial modular tensorcategory Vec of finite dimensional Hilbert spaces.Locality of the tensor product Q -system Θ ⊠ Θ is represented as in Fig. 1. (Wefollow the graphical convention of [7, Section 3], but compose morphisms from the bottomto the top, which is a converse direction to the one in [7, Section 3].) In this picture, thetriple points on the left hand side denote x , x , x , respectively. The second braiding onthe right hand side is reversed because we have C opp2 for this component. ✻ ✻ ✻ ✻ ✻ ✻❖ ❖ ❖✗ ✗ ✗ ■ ■✒✒ ✒■ L = L ⊠ ⊠ ⊠ ⊠ λ ′′ λ λ ′ ¯ µ ′′ ¯ µ ¯ µ ′ µ ′′ µ µ ′ λ ′′ λ λ ′ ¯ µ ′′ ¯ µ ¯ µ ′ µ ′′ µ µ ′ Figure 1: Locality (1)From Fig. 1, we connect the wires ¯ µ ′′ and µ ′′ , the wires ¯ µ and µ , and the wires ¯ µ ′ and µ ′ on the both hand sides so that the wires connecting ¯ µ and µ go over the onesconnecting ¯ µ ′ and µ ′ . Then we obtain Fig. 2. Then the Reidemeister move II on the mostright picture of Fig. 2 produces Fig. 3.Fig. 3 represents the locality of Θ ⊗ C Θ . (cid:3) ✻❖ ✗ ■ ✒ L = L ⊠ ⊠ ❘■ ■ ❘■ ■ λ ′′ λ λ ′ µ ′′ µ µ ′ λ ′′ λ λ ′ µ ′′ µµ ′ Figure 2: Locality (2) ✻ ✻❖ ✗ ■ ✒ L = L ⊠ ⊠ ❘■ ■ ❘■ ■ λ ′′ λ λ ′ µ ′′ µ µ ′ λ ′′ λ λ ′ µ ′′ µ µ ′ Figure 3: Locality (3)We consider the irreducible decomposition Θ ⊗ C Θ = L i Θ i , which is a finite sum.By [6, Corollary 3.6], this coincides with the corresponding factorial decomposition M = L i M i where the Q -system Θ ⊗ C Θ corresponds to an inclusion N ⊂ M and the oneΘ i corresponds to N ⊂ M i .We first list the following lemma. See [11, Definition 5.1] for the definition of Wittequivalence. Lemma 2.5
Let C , C be Witt equivalent modular tensor categories and Θ = ( θ, w, x ) be an irreducible local Q -system where θ is of the form L λ ∈ Irr( C ) ,µ ∈ Irr( C opp2 ) Z λµ λ ⊠ ¯ µ .Then there exists an irreducible local Lagrangian Q -system ˜Θ = (˜ θ, ˜ w, ˜ x ) where ˜ θ is of theform L λ ∈ Irr( C ) ,µ ∈ Irr( C opp2 ) ˜ Z λµ λ ⊠ ¯ µ with ˜ Z λµ ≥ Z λµ for all λ ∈ Irr( C ) , µ ∈ Irr( C opp2 ) and ˜ Z = Z = 1 where denotes the identity objects of C and C . Proof.
Let ˜ C be the modular tensor category arising as the ambichiral category fromthe Q -system Θ as in [8, Theorem 4.2]. (Note that the ambichiral objects correspond to5yslectic/local modules in the terminology of [11], [12].) By [9, Corollary 4.8], C ⊠ C opp2 isWitt equivalent to ˜ C , which means that ˜ C is Witt equivalent to the trivial modular tensorcategory Vec. By [23, Theorem 2.4], we have an irreducible local Lagrangian Q -system on˜ C . Composing this with the original Q -system Θ, we have an irreducible local Lagrangian Q -system ˜Θ on C ⊠ C opp2 . It has the modular invariance property by Proposition 2.1, and˜ Z λµ ≥ Z λµ and ˜ Z = Z = 1 are clear. (cid:3) Theorem 2.6
If the Q -systems Θ and Θ are both Lagrangian, so is each Θ i . Proof.
Set Z λν = P µ Z λµ Z µν and let L λ ∈ Irr( C ) ,ν ∈ Irr( C opp3 ) Z ,iλν λ ⊠ ¯ ν be the object for Θ i .By Proposition 2.1, being Lagrangian for Θ i is equivalent to modular invariance property S C Z ,i = Z ,i S C and T C Z ,i = Z ,i T C for Z ,i , where S C , S C , T C , T C are the S -matrixfor C , S -matrix for C , T -matrix for C and T -matrix for C , respectively.Note that C and C are Witt equivalent, and so are C and C . Hence C and C are also Witt equivalent and each Θ i has a Lagrangian extension ˜Θ i whose object is L λ ∈ Irr( C ) ,ν ∈ Irr( C opp3 ) ˜ Z ,iλν λ ⊠ ¯ ν by Lemma 2.5 and we have S C ˜ Z ,i = ˜ Z ,i S C and T C ˜ Z ,i =˜ Z ,i T C by Proposition 2.1. By Lemma 2.5, we may write ˜ Z ,iλν = Z ,iλν + ˆ Z ,iλν , where eachˆ Z ,iλν is a non-negative integer.Since the matrix P i Z ,i also has the modular invariance property, the matrix ˆ Z = P i ˆ Z ,i also has the modular invariance property. This implies P λν S C , λ ˆ Z λν S C ,ν = ˆ Z ,but Z = P i Z ,i = 0 and S C , λ > S C ,ν >
0. We thus have ˆ Z λν = 0 for all λ ∈ Irr( C )and ν ∈ Irr( C ). This proves the modular invariance property S C Z ,i = Z ,i S C and T C Z ,i = Z ,i T C for Z ,i , as desired. (cid:3) Note that the use of modular invariance in the last paragraph of the above proof is thesame as in [17, p. 726 (5.2)].This relative tensor product of Q -systems looks similar to that of bimodules, but theexample of the A - D - E modular invariants mentioned in the Introduction shows thattheir fusion rules do not give a fusion category since the rigidity axiom is not satisfied.We have interpreted an irreducible local Lagrangian Q -system on C ⊠ C as a gappeddomain wall between topological phases represented with C and C in [23, Definition 3.1].(See [18], [19], [30] for physical treatments of gapped domain walls.) From this viewpoint,the above relative tensor product gives a mathematical definition of the composition ofgapped domain walls mentioned in [30, Fig. 1 (d)]. (Note that irreducibility of a Q -systemis called stability of a gapped domain wall in [30].) A mathematical definition of such acomposition has been studied in [29], [1]. It would be interesting to compare the abovedefinition with theirs.Another construction of fusion product with some formal similarity has been definedin [3]. It would be interesting to find direct relations to their construction.6 eferences [1] Y. Ai, L. Kong and H. Zheng, Topological orders and factorization homology,arXiv:1607.08422.[2] B. Bakalov and A. Kirillov, Jr., “Lectures on tensor categories and modular func-tors”, American Mathematical Society, Providence (2001).[3] A. Bartels, C. L. Douglas and A. Henriques, Conformal nets III: Fusion of defects,to appear in Mem. Amer. Math. Soc. [4] M. Bischoff, Y. Kawahigashi and R. Longo, Characterization of 2D rational local con-formal nets and its boundary conditions: the maximal case,
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