On the structure of categorical duality operators

On the structure of categorical dualit y op erators Corey Jones ∗ , Xinping Y ang † Abstract W e systematically study categorical duality operators on spin (and any on) chains with resp ect to an in ternal fusion category symmetry C . W e parameterize duality op erators on the quasi-lo cal algebra in terms of data dep enden t on the asso ciated quan tum cellular automata (QCA) on the symmetric subalgebra B . In particular, a QCA α on B defines an inv ertible C - C bimo dule category M α , and the dualit y operators extending α form a simplex, with extreme p oin ts in bijective correspondence with the simple ob ject of M α . Then we consider the structure of external symmetries generated by a family of dualit y op erators, and sho w that if the UV mo dels are all defined on tensor product Hilb ert spaces, these categories necessarily flow to w eakly in tegral fusion categories in the IR. Con ten ts 1 In tro duction 1 2 Non-lo cal op erators in infinite v olume 4 2.1 Sectorization of non-lo cal op erators . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Microscopic SymTFT picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Charge categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Dualit y op erators 9 3.1 Example: Generalized Kramers-W annier . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Q-system mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2 Dualit y op erator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.3 UV-category of duality op erators . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The general picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.1 Bimo dule channels and top ological b oundary defects . . . . . . . . . . . . 16 3.2.2 P arameterizing duality op erators . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.3 Univ ersal tensor categories and emanan t symmetries . . . . . . . . . . . . 22 A An algebraic construction of generalized KW dualit y op erators 23 A.1 Kramers-W annier duality op erator . . . . . . . . . . . . . . . . . . . . . . . . . . 24 A.2 Z -graded tensor category from duality op erators . . . . . . . . . . . . . . . . . . 25 1 In tro duction Of recent interest are generalized symmetries of spin c hains [ SS24b , MYLG25 , Ina22 , BBPSN25 , BBSNT25a , BBSNT25b ]. F usion category symmetries, describing finite internal symmetries of ∗ Departmen t of Mathematics, North Carolina State Universit y , Raleigh, NC 27695, USA † P erimeter Institute for Theoretical Physics, W aterlo o, ON N2L 2Y5, Canada 1 quan tum field theories in (1+1)d, are no w fairly well-understoo d from a structural viewp oin t using the framework of SymTFT [ KWZ17 , FMT24 ], which has b een recently dev elop ed on the lattice [ BBSNT25a , BBSNT25b , EJ25 ]. In the op erator algebraic framework for the infinite v olume limit, SymTFT decomp ositions can b e completely c haracterized by a the symmetric subalgebra B (or in the SymTFT context, the ph ysical b oundary subalgebra) of the algebra A of all quasi-lo cal op erators, and their DHR bimo dules [ Jon24 , HJJY25 ]. This has led to formal in terpretations of the categorical Landau paradigm [ BBPSN23 ], as well as versions of several anomaly-enforced gaplessness results [ BPSNW25 , EJ25 ]. In the context of fusion category symmetry , there is a very in teresting phenomenon gen- eralizing the famous Kramers-W annier duality called c ate goric al duality [ CGR V + 25 , LCT26 , Ina26 , LDOV23 , LD V24 ]. A categorical duality operator is a (non-lo cal), lo calit y preserving op erator 1 whic h is not necessarily unitary but restricts to a unitary on the symmetric sec- tor. A duality op erator can thus exc hange inequiv alent symmetric gapp ed phases. In the case of Kramers-W annier, the underlying symmetry is the on-site Z 2 spin flip symmetry , and the Kramers-W annier op erator exchanges the trivial SPT with the symmetry breaking phase. This allo ws us in principle to systematically access non-trivial symmetric gapped phases from simpler ones. Th us an understanding of categorical dualities, and in particular classifying them up to symmetric finite-depth circuits, is a very interesting problem. An y duality op erator defines a quantum cellular automata (QCA) on the algebra B . Using the theory of DHR bimo dules (outlined b elo w), we pro ve the following theorem, which reduces the study of duality channels to QCA on the symmetric subalgebra B , whic h in turn hav e b een extensiv ely studied [ Jon24 , JSW26 , MLC24 ]. Theorem 1.1. Suppose we hav e a fusion category C acting on the lattice, with symmetric subalgebra B ⊆ A . 1. Asso ciated to α ∈ QCA( B ) is an in vertible C - C bimo dule category M α . 2. The set of (unital) dualit y op erators which restrict to α on B form a simplex, and the extreme p oin ts are in bijective corresp ondence with the simple ob jects of M α . In particular, any classification of QCA on B up to symmetric circuits provides a classification of dualit y channels up to symmetric circuits. Another ma jor motiv ation for studying Kramers-W annier type dualities is their role in extending the paradigm for non-in v ertible symmetry from the SymTFT picture described ab o ve to something more general. This larger class of symmetries, motiv ated explicitly by the Kramers- W annier example, can b e realized by duality op erators but the fusion rules of these operators migh t “mix with translation”. This giv es rise to infinite fusion rules on the lattice, which nev ertheless flow to finite fusion rules in the IR [ SSS24 ]. The class of symmetries realized this w ay has b een termed “emanant” [ CS23 , SS24a ]. An in teresting observ ation is the class of fusion categories that can arise emanan tly in the IR is strictly larger than the class that can b e realized directly in the UV. Indeed the Kramers- W annier op erator flows under R G to fusion category symmetry with Ising fusion rules, whic h has an ob ject with quan tum dimension √ 2. Non-in tegral fusion categories cannot b e realized as internal symmetries on the tensor pro duct Hilb ert space lattice in the UV [ EJ25 ]. In fact, it has b een conjectured that an y of these generalized fusion categorical symmetries emerging in the IR must b e weakly integral, namely that the squares of the dimensions of all simple ob jects are integers [ LCT26 , Ina26 ]. Part of the motiv ation of this pap er is to demonstrate this conjecture is true, under the assumption that the generalized symmetry op erators on the 1 in the infinite v olume limit, non-lo cal operators can be characterized as arbitrary completely positive maps on the quasi-lo cal algebra. 2 lattice are duality op erators with resp ect to a fixed in ternal symmetry , extending the setup of the Kramers-W annier op erator. When viewing duality op erators as symmetries in the UV, we will use the term external symmetries to contrast with the initial internal symmetries constraining RG. Op erationally , a dualit y op erator is external if it acts by a non-trivial QCA on the b oundary algebra. External symmetries are designed to include not only spatial symmetries but also our duality op erators. Ho wev er, it is not entirely clear to what extent external symmetries are describ ed by an explicit tensor category , rather than just a fusion ring (or more precisely a hypergroup). One p oin t that distinguished external symmetries from internal ones is that the UV fusion rules on the lattice do not, in general, match the IR fusion fusion rules, even when the category in question is anomaly-free (equipp ed with a fib er functor). Moreo ver, external symmetries describ ed b y dualit y op erators cannot necessarily b e consisten tly defined on any fixed finite region in a quan tum spin chain. F or example, the non-inv ertible symmetry in Rep † ( D 8 ) SPTs can be realized as the Z 2 × Z 2 dualit y op erator on the cluster spin chain [ SS24b ]. While the lattice symmetry operators satisfy the Rep † ( D 8 ) fusion rules, the dualit y op erators do not satisfy the zipping condition [ LXY25 ], and th us do not naturally form a tensor category in the UV. When the IR fusion category X is anomalous, the UV external symmetry structure necessarily digresses from X even more, meaning that the fusion rules must mix with translation. T o clarify this situation, w e argue that any (internal or external) symmetries realized either in the UV or IR b y duality op erators m ust necessarily b e a quotient of a univ ersal tensor category , whic h w e construct. As a corollary , we argue that any emanant fusion category symmetry emergin in the IR from a symmetry constrained RG flo w of external symmetries on a tensor pro duct Hilb ert space lattice must b e weakly integral. Theorem 1.2. Let B ⊆ A b e a physical b oundary subalgebra with fusion category C , and supp ose Φ 1 , . . . Φ n are duality op erators generating an external symmetry . Then there is a canonical F n -graded extension of the category C , denoted as C # F n . 1. The normalized fusion rules asso ciated to { Φ i } are a quotient of the fusion rules of the tensor category C # F n . 2. If the duality op erators { Φ i } flow to an internal symmetry in the IR describ ed by a fusion category X , then X is a quotien t of C # F n . Corollary 1.3. If A ∼ = ⊗ Z M d ( C ) is a tensor product quasi-local algebra with in ternal symmetry C , then an y X emerging in the IR from an RG flow of duality op erators is weakly integral. The rest of the paper is organized as follows: in Section 2 , we lay out the formal definitions of our framew ork, explaining the connection b et ween non-lo cal op erators and C ⋆ -corresp ondences in infinite v olume limit. In Section 2.2 , we define the ph ysical b oundary subalgebra in the con- text of a microscopic SymTFT decomp osition with symmetry category C and a choice of charge category D ( 2.3 ). In Section 3 , we give an op erator algebraic form ulation of duality op erators. W e start with a specific realization ( 3.1 ) of generalized Kramers-W annier duality op erators in the Q-system mo del as a motiv ating example to study the UV-category of duality op erators. In Section 3.2 , w e parametrize the dualit y op erators as top ological in terfaces b et w een a top ological b oundary extension and its t wisted one, which leads to the universal tensor category ( 3.2.3 ), an F n -graded extension of C , in the UV. Ac kno wledgements The authors would like to thank Dac huan Lu, Sahand Seifnashri, Nathanan T an tiv asadak arn for useful conv erstaions. X.Y. is in particular grateful to Chenqi Meng for an early stage discussion on duality op erators. C.J. was supp orted by NSF DMS-2247202. Researc h at Perimeter Insti- tute is supp orted in part by the Go vernmen t of Canada through the Department of Innov ation, 3 Science and Economic Developmen t and by the Pro vince of Ontario through the Ministry of Colleges and Universities. 2 Non-lo cal op erators in infinite v olume In the op erator algebraic picture of quantum field theories, a standard approach to describing theories in the infinite v olume limit is in terms of the algebras of lo cal op erators. The primary mathematical ob ject here is the quasi-lo cal C ⋆ -algebra A , consisting of (the completion of ) all lo cal op erators taken together. On the lattice, A is typically the infinite tensor of finite dimensional matrix algebras lo calized at lattice sites. More formally , an abstract quasi-lo c al algebr a ov er a metric space ( X, d ) is a (unital) C ⋆ - algebra A , together with a distinguished family of unital subalgebras A F ⊆ A , where F ⊆ X ranges ov er all metric balls in the space (other choices of “nice” subsets also are used dep ending on the context). W e assume that if F ∩ G = ∅ , then [ A F , A G ] = 0 and S F A F is dense in A . The subalgebra A F consists of the op erators lo c alize d in the region F . Though tec hnically sp eaking the elements of A are norm-limits of lo cal op erators (and are properly called quasi-lo cal op erators), we will slightly abuse terminology and refer to the elemen ts of A as lo cal. W e often assume some version of Haag duality : for a “nice” region F (e.g. a ball), the lo cal operators whic h commute with all the lo cal op erators lo calized outside F are lo calized in F . This allows us to detect where something is lo calized by what it commutes with. A state can b e defined in this setting as a (normalized) p ositive linear functional φ : A → C , whic h in turn furnishes a Hilb ert space representati on of A which caries φ as a v ector state. The other v ector states can b e viewed as lo cal p erturbations of φ , and in the case where φ is pure the whole Hilbert space representation is called a sup ersele ction se ctor or simply sector. The existence of inequiv alent sectors is the key feature of the infinite v olume limit underlying the explanatory p ow er of asymptotic reasoning with regard to macroscopic phases and phase transitions. One important conceptual issue when working in the infinite v olume limit is the problem of c haracterizing non-lo cal op erators. These op erators map b et ween differ ent Hilb ert space sectors, hence cannot b e naturally realized as limits of lo cal op erators in general. F rom this p erspective, a natural definition of a gener al , a not necessarily lo cal op erator is a completely p ositiv e (cp) map Ψ : A → A . This con tains the lo cal op erators a ∈ A via Φ a ( b ) := aba ∗ , and w e see that Φ a ◦ Φ a ′ = Φ aa ′ , giving us op erator comp osition. W e again sligh tly abuse terminology and call a conv ex combinations of this type of cp map a lo cal op erator, and refer to general cp-maps as simply op erators. Describing a state as a p ositiv e functional φ : A → C , a cp map Ψ acts on φ by φ ◦ Ψ. In fact cp maps are the most general op erators on the the state space of a C ⋆ -algebra compatible with the natural top ology , conv ex structure, and coupling to larger systems, justifying our use of cp maps as op erators. F rom this p ersp ectiv e, it is natural to say what it means for an op erator to b e lo calized in a region I : Φ is lo calized in I if for all a ∈ B I c , a Φ(1) = Φ( a ) = Φ(1) a . Clearly for an y op erator b ∈ B I , Φ b is localized I . Ho wev er, in general there could b e non-local op erators Φ that are nevertheless lo calized in the finite region I , whic h is precisely the situation with string op erators asso ciated to any ons in top ologically ordered quantum man y-b ody systems. This will b e the jumping off p oin t for SymTFT and physical b oundary subalgebras, which we elab orate in the net subsection. 2.1 Sectorization of non-lo cal op erators An important feature of states in infinite volume whic h we hav e alluded to previously is se c- torization . Informally , tw o abstract pure states φ and ψ in infinite volume lie in the same 4 sup ersele ction se ctors 2 if they are related to each other by a quasi-lo cal p erturbation. More formally , we can describ e this relationship in terms of the represen tation theory of A . Asso ciated to any state φ is its GNS Hilb ert space L 2 ( A, φ ), which carries a b ounded rep- resen tation of the quasi-lo cal algebra A . L 2 ( A, φ ) is defined by starting with a formal state v ector Ω ϕ , and forming the v ector space of formal quasi-lo cal p erturbations of this state vector { a Ω ϕ | a ∈ A } . W e then define an inner pro duct via ⟨ a Ω ϕ | b Ω ϕ ⟩ := φ ( a ∗ b ) . In general, this is not positive definite, so we ha ve to quotien t by the kernel and complete to obtain the Hilb ert space L 2 ( A, φ ), whic h has a natural action by A . Notice that φ is realized b y the v ector state Ω ϕ ∈ L 2 ( A, φ ), and in fact this is the universal realization of φ as a vector state. Recall φ is a pure state if and only if L 2 ( A, φ ) is an irreducible represen tation of A . F rom this p oin t of view, one wa y to sa y that φ and ψ lie in distinct sup erselection sectors is that L 2 ( A, φ ) and L 2 ( A, ψ ) are inequiv alent as represen tations of the algebra A . Conv ersely , t wo pure states are in the same sup erselection sector if they hav e (unitarily) isomorphic GNS representations, or equiv alen tly , one state can b e realized as a vector state in the GNS representation of the other. Imp ortan tly , the ab o ve discussion shows that w e can naturally lump quasi-lo cal p erturba- tions of a single state into one an ob ject in a category . This leads us to pass from the set of states to the c ate gory of Hilb ert space representations of A and b ounded intert winers, where we can utilize the mathematical machinery of category theory to study states and v arious notions of equiv alence that o ccur in the study of quan tum phases of matter. The reason that this story is often unfamiliar to the working physicist is that sectorization do es not usually o ccur in finite volume: the algebra of lo cal op erators M d ( C ) has a unique irreducible representation, hence all pure states lie in the same sup erselection sector. Indeed, the existence of distinct superselection sectors in the infinite volume limit (sectorization) lies at the heart of most rigorous explanations of macroscopic phases and emergent phenomenon, whic h can b e witnessed with recent progress in top ologically ordered phases of matter [ NO22 ]. If states are in different phases (whatever w e migh t mean by this term), they are necessarily in distinct superselection sectors though the con v erse is not usually the case. Nev ertheless, this principle tells us that whenever w e wan t to mathematically characterize phases, working with sectors (and hence working in the category of Hilb ert space represen tations of the quasi-local algebra). The key observ ation for us is that sectorization also o ccurs with non-lo c al op er ators . Sim- ilarly to the case of states, w e can consider quasi-local p erturbations of a cp-map via a GNS t yp e construction. How ever, the role of Hilb ert space representation of A that w e use in this picture m ust b e generalized to a c orr esp ondenc e b et ween C ⋆ -algebras. Definition 2.1. Let A and B b e C ⋆ -algebra. A c orr esp ondenc e is a linear A - B bimo dule X , together with a map X × X → B , written ( x, y ) 7→ ⟨ x | y ⟩ ∈ B , satisfying 1. ⟨· | ·⟩ is linear in the second v ariable and conjugate linear in the first v ariable. 2. ⟨ x | y ⟩ ∗ = ⟨ y | x ⟩ . 3. ⟨ x | x ⟩ ≥ 0 in B , and ⟨ x | x ⟩ = 0 implies x = 0. 4. The norm ∥ x ∥ = ||⟨ x | x ⟩|| 1 2 is a Banach norm. 2 w e are using this term in the more general sense of physicists, rather than the more specialized usages in algebraic quantum field theory , or more recen tly in top ologically ordered spin systems [ NO22 ]. 5 5. ⟨ x | y  b ⟩ = ⟨ x | y ⟩ b . 6. ⟨ a  x | y ⟩ = ⟨ x | ( a ∗ )  b ⟩ Notice that if B = C , then X with ⟨· | · ⟩ b ecomes a Hilb ert space, and the left action of A is simply a Hilb ert space representation. Let X b e an A - B corresp ondence, and x ∈ X . Then the matrix co efficien t asso ciated to x defines a cp map Φ x : A → B b y Φ x ( a ) = ⟨ x | a  x ⟩ . Con versely , starting from a cp map Φ : A → B , we start with a “state vector” Ω Φ , but now p erturb on the left b y elements of A and on the right b y elements of B span { a Ω Φ b : a ∈ A, b ∈ B } . Then w e define a B -v alued inner pro duct ⟨ a 1 Ω Φ b 1 | a 2 Ω Φ b 2 ⟩ := b ∗ 1 Φ( a ∗ 1 a 2 ) b 2 . Again, we quotient b y the kernel and complete to obtain an A - B corresp ondence, which we call L 2 (Φ). This is the univ ersal corresp ondence containing Φ as a “vector state”. F or the same reason as for states, it is extremely useful to consider non-lo cal op erators together with their quasi-lo cal p erturbations in a single ob ject, namely a corresp ondence. In particular, we can access categorical structure. Indeed, the collection of A - B corresp ondences forms a C ⋆ -category , where morphisms b et ween A - B corresp ondences X and Y are linear bi- mo dule maps f : X → Y such that ⟨ f ( x ) | y ⟩ Y = ⟨ x | f ∗ ( y ) ⟩ X for some f ∗ : Y → X . Suc h maps are automatically b ounded, and comp osition is the usual (for details, see [ CHPJP22 ]). Op erators, unlike states, ha ve a natural comp osition op erators, which is the crucial part of their existence. This is reflected at the correspondence lev el through the tensor pr o duct of corresp ondences. If X is an A - B corresp ondence and Y is a B - C corresp ondence, then X ⊠ Y is an A - C corresp ondence, defined with an inner pro duct on simple tensors ⟨ x 1 ⊗ y 1 | x 2 ⊗ y 2 ⟩ X ⊠ Y := ⟨ y 1 | ⟨ x 1 | x 2 ⟩ X  y 2 ⟩ Y . As usual w e quotien t b y the k ernel and complete, then equip the resulting space with the ob vious left A -action and righ t C -action. If x ∈ X and y ∈ Y , and Φ x , Φ y are the associated v ector state cp maps, then Φ x ◦ Φ y is a vector state in X ⊠ Y via the state v ector x ⊗ y . The collection of C ⋆ -algebras, corresp ondences, and intert winers assem bles into a C ⋆ 2- category [ CHPJP22 ]. F or a single algebra A , the collection of all A - A corresp ondences is a C ⋆ -tensor category , which w e denote Bim( A ). F or a quasi-lo cal algebra, this C ⋆ -tensor category is the “sectorization” of non-lo cal op erators. 2.2 Microscopic SymTFT picture F usion category symmetries of quantum field theories hav e emerged as an imp ortan t to ol in understanding quantum field theories and quantum man y-b ody systems, particularly in (1+1)d where the story is better understoo d. In this paper, w e will fo cus on the lattice setting. The usual picture for fusion categorical symmetry utilizes an algebra of “symmetry op erators”, enco ded either b y matrix pro duct op erators (MPOs) or (weak) Hopf C ⋆ -algebra ac ting on the lo cal Hilb ert spaces [ MYLG25 , IO26 , MdAGR + 22 ]. An alternativ e picture has emerged which pro vides p o werful to ols for analyzing the structure of symmetric phases (b oth gapp ed and gappless) called SymTFT [ KWZ17 , FMT24 , EJ25 , BPSNW25 ]. In the SymTFT picture for a spin system with lo cal Hamiltonian H = P H I , we hav e a de- comp osition of the system in to a one dimension higher TQFT (called the SymTFT) sandwic hed b et w een a top ological b oundary and a physical b oundary , as in the following picture: 6 A A I ≃ H I ∈ B I . The ph ysical boundary (also sometimes called the dynamical boundary) is usually v ery com- plicated, and should contain the lo cal terms H I of the Hamiltonian. The top ological b oundary , in contrast, is simple and witnesses the categorical symmetry via top ological defects. A k ey feature of this story is that the symmetry op erators are secondary , and the TQFT itself takes cen ter stage. This allo ws for the direct application of TQFT ideas to the study of symmetric phases. F rom the op erator algebra p erspective, it is natural to try to access a SymTFT decom- p osition by idenitifying the subalgebra B ⊆ A of lo cal op erators lo calized near the ph ysical b oundary , and then trying to reco ver the rest of the SymTFT structure from this. This leads to a natural question: when can we realize an abstract quasi-local algebra the lo cal op erators at a physical b oundary of a TQFT? This problem was addressed in [ EJ25 ]. The k ey idea is that the non-lo cal string op erators that create an yons in the bulk can be pushed in to the b oundary to define non-lo c al op erators on the b oundary theory . Due to the top ological nature of the bulk string op erators, we exp ect that after lo cal p erturbations (given b y p ost comp osing, so “right” p erturbations), we can obtain op erators that are lo c alize d in any other (sufficiently lar ge) r e gion we like . In particular, the corresp ondence asso ciated to a string op erator should con tain op erators lo calized in any sufficien tly large region, and should b e generated as a righ t mo dules by the pieces of the string op erator lo calized in any large enough in terv al. T o formalize this at the level of corresp ondences, let’s fo cus on the case where space is discrete and one-dimensional, i.e. Z . W e assume that the lo cal subalgebras A I are finite dimensional C ⋆ -algebras assigned to interv als I , and that we hav e Haag dualit y for interv als. F or any A - A corresp ondence X , set X I := { x ∈ X : ax = xa for all a ∈ A I c } . The v ectors x ∈ X I are precisely the elements that corresp ond to op erators lo calized in I , i.e. the asso ciated cp maps Φ x : A → A satisfy Φ x ( a ) = a Φ x (1) = Φ x (1) a for all a ∈ A I c . A DHR bimo dule will essen tially b e any corresp ondence generated under p ost-composition with quasi- lo cal p erturbations by op erators localized anywher e . Because of the finite scale of the lattice, w e will only assume that we can b e lo calized in any sufficien tly large interv al. W e hav e the follo wing formal definition. Definition 2.2. ( DHR bimo dules ). An A - A corresp ondence X is a DHR -bimo dule if for there is some R ≥ 0 so that for an y interv al with | I | ≥ R , dim( X I ) < ∞ and X = X I A . This definition lo oks slightly different than previous definitions in the literature [ Jon24 , EJ25 ]. W e say an A - B corresp ondence X is right finite if there exists a finite set { b i } ⊆ X suc h that for all x ∈ X , x = X i b i ⟨ b i | x ⟩ . W e call { b i } a finite pro jectiv e basis. No w we mak e contact with the usual definition of DHR bimo dule in the literature. Theorem 2.1. Supp ose A is a lo cally finite-dimensional net of algebras ov er Z satisfying weak algebraic Haag dualit y . An A - A corresp ondence X is a DHR bimodule if and only if there exists an S suc h that for all in terv als | I | ≥ S , there is a finite pro jective basis contained in X I . 7 Pr o of. Supp ose | I | ≥ R , so that X I A = X . By w eak Haag duality , there is some T such that A ′ I c ∩ A ⊆ A I + T . Th us if we tak e X I A I + T ⊆ X I + T , restricting the inner pro duct gives this subspace the structure of a righ t Hilbert mo dule ov er the finite dimensional C ⋆ -algebra A I + T . But Hilb ert mo dules o ver finite dimensional C ⋆ -algebras alwa ys admit pro jectiv e bases, and th us we can find a { b i } ⊆ X I + T with X i b i ⟨ b i | x ⟩ = x for all x ∈ X I A I + T . But by the DHR condition then X = X I A ⊆ X I A I + T A = X . Th us an y x ∈ X can b e written as P i x i a i , where x i ∈ X I A I + T and a i ∈ A . Th us X i b i ⟨ b i | x ⟩ = X i,j b i ⟨ b i | x j a j ⟩ = X j X i b i ⟨ b i | x j ⟩ ! a j = X j x j a i = x . Hence w e ha ve a pro jective basis lo calized in I + T , and th us setting S = R + T , w e hav e the desired result. No w, we connect abstract DHR bimo dules to the bulk SymTFT picture. Theorem 2.2 ([ Jon24 ]) . . The C ⋆ -tensor category of DHR( A ) consisting of DHR bimo dules has a natural unitary braiding. W e can now describ e the SymTFT picture. Definition 2.3. Let A b e a quasi-lo cal algebra ov er Z whic h is lo cally finite dimensional and satisfies weak algebraic Haag duality . A unital subalgebra B ⊆ A is called a ph ysical b oundary subalgebra if 1. S I ( B ∩ A I ) is norm dense in B . 2. There exists a conditional exp ectation E : A → B suc h that ( B ⊆ A, E ) is a Lagrangian Q-system in DHR( B ). Some immediate consequences of the definition: • E : A → B is lo cality preserving, i.e. there is some R such that E ( A I ) ⊆ B I + R = B ∩ A I + R . • The inclusion is irreducible, i.e. B ′ ∩ A = C . • DHR( A ) is trivial. • There is a fusion category C of A mo dules internal to DHR( B ), realized concretely as a class of corresp ondences ov er A , such that DHR( B ) ∼ = Z ( C ) [ EJ25 , HJJY25 ] • The fusion hypergroup of C acts b y unital completely p ositiv e maps on A , such that the in v arian t quasi-lo cal op erators are precisely B . The fusion category C ab ov e is called the symmetry c ate gory . Notice that from this p erspec- tiv e, the symmetry category C is secondary , and the symmetric op erators are what we fo cus on. The symmetric op erators can arise ho wev er you wish: for example, as the lo cal op erators that commute with an MPO or (weak)-Hopf algebra action. The category C and its action by quan tum channels (ucp maps) are c anonic al ly define d by the symmetric subalgebra. 8 2.3 Charge categories F o cusing on the ph ysical b oundary algebra B itself, w e can ask a slightly differen t question: whic h abstract quasi-lo cal algebras ov er Z are suitable to b e ph ysical b oundaries of a TQFT in the first place? An ob vious answ er is that DHR( B ) should be the Drinfeld cen ter of a fusion category . In [ HJJY25 ], it w as sho wn that under some reasonable assumptions DHR( B ) is automatically the Drinfeld center of a category called the char ge c ate gory D , which is generally differen t from (or “dual to) the symmetry category C , the latter of which only arises when B is em b edded as a physical b oundary subalgebra of a quasi-lcoal algebra A . T o describe this category , consider the truncated quasi-lo cal algebra B − , whic h is the C ⋆ - subalgebra of B generated by B I with I ≤ 0. W e are interested in non-lo cal op erators (i.e. cp maps) defined only on B − that are lo calized around 0. F ollowing the translation of this idea in to bimo dules, we hav e the follo wing definition. Definition 2.4. A B − - B − corresp ondence X is a char ge bimo dule if there exists some I = [ − k , 0] suc h that X I B = X . W e denote the category of charge bimo dules by DHR − ( B ). DHR − ( B ) is a C ⋆ -tensor category with the relative tensor pro duct, how ever it do es not ha ve a braiding. F rom [ HJJY25 , JNP25 ], we do in general hav e a braided tensor functor DHR( B ) → Z (DHR( B − )). Under some reasonable assumptions, DHR − ( B ) is a fusion category , and the functor describ ed ab ov e is an equiv alence [ HJJY25 , JNP25 ]. In particular, for (multi)- fusion spin chains (see the next section), this is a braided equiv alence [ HJJY25 ]. The main class of abstract physical b oundary algebras that we consider are fusion spin c hains, built from a fusion category D , and a strongly tensor generating ob ject X . In this case, the charge category is D itself. Note that in the categorical symmetry context, the fusion spin c hains in question arise as the symmetric sub algebr as of a larger quasi-lo cal algebra A with symmetry category C whic h in general is distinct from D , but is alwa ys Morita equiv alent to it b y our ab ov e discussion. 3 Dualit y op erators W e will treat the ph ysical b oundary algebra B as fundamen tal, and consider the different top ological b oundaries of our SymTFT as v ariable. Given a fixed physical b oundary algebra B , the p ossible topological b oundaries in the SymTFT decomposition corresp ond to Lagrangian algebras A ∈ DHR( B ). Definition 3.1. Let B ⊆ A b e a SymTFT decomp osition with fusion category symmetry C . A duality op er ator consists of a completely p ositiv e map Ψ : A → A such that: 1. There exists an R ≥ 0 such that Ψ( A F ) ⊆ ( A ) F + R 2. Ψ restricts to a ∗ -automorphisms from B to B . W e note that one consequence of our assumption, is that B is in the multiplicativ e domain of the channel Ψ, so that in particular Ψ( b 1 ab 2 ) = Ψ( b 1 )Ψ( a )Ψ( b 2 ). W e also note that Ψ is automatically unital. If we take a duality channel Ψ : A 1 → A 2 , Ψ | B is a b ounded spread isomorphism, sometimes called a quantum cellular automata (QCA). Question 3.2. Given a bounded spread automorphism α : B → B , ho w do we parameterize extensions to duality channels Ψ : A 1 → A 2 suc h that Ψ | B = α ? The version of this question which asks when we can extend a b ounded spread isomorphism on B to an honest (inv ertible) QCA on A was answered in [ JSW26 ]. 9 Another imp ortan t reason for considering dualit y op erators is their role in the con text of generalized symmetry . It was argued in [ SSS24 , LOZ23 ] that the canonical example of a du- alit y op erators, the Kramers-W annier op erator (see the next section for a full explanation), implemen ts a form of generalized fusion category symmetry that go es b ey ond the usual internal story . This is witnessed by the fact that the Kramers-W annier op erator almost has the fusion rules of the Ising fusion category , but “mixes with” translation. The idea is that under an y RG flo w which is constrained by the internal Z 2 symmetry , this will give rise to an honest in ternal symmetry in the IR. This concept is sometimes called emmanent symmetry in the literature [ CS23 ]. P art of our motiv ation is to systematically study this type of symmetry using our analysis of dualit y op erators. In the next section, we will give a detailed description of an imp ortan t family of examples: the generalized Kramers-W annier c hannels in the Q-system mo del. W e will see in this class of examples, the UV-category of duality op erators can b e a Z -graded tensor category . 3.1 Example: Generalized Kramers-W annier W e briefly review the (1+1)d lattice mo del for top ological phases enriched with fusion category symmetries discussed in [ MYLG25 ]. This sp ecific construction provides a conv enient w ay to construct symmetric lo cal op erators on an H -symmetric quantum spin chain, where H is a Hopf C ⋆ -algebra. W e start with the Hopf C ⋆ -algebra H as the lo cal version of the fusion category symmetry and define the charge category as the represen tation category of the Hopf C ⋆ -algebra, isomorphic to C ∨ Hilb f := F un C ( Hilb f , Hilb f ), where f is a unitary fib er functor f : C → Hilb . A generic lattice mo del with fusion category symmetry ( C , f ) consists of: local Hilb ert spaces that carry representations of the Hopf algebra and lo cal interactions giv en by Hermitian intert winers b et w een represen tations on the lo cal Hilb ert space [ Ina22 ], whic h directly generalizes the construction with finite group symmetry in [ LZ24 ]. Note that for differen t fib er functors f , h of C , the asso ciated dual categories are not the same C ∨ Hilb f ≇ C ∨ Hilb h . Thus the c hoice of the c harge category is not unique. Examples include the isocategorical groups, Rep † ( D 4 n ) for n ∈ Z , n > 3, Rep † ( A ⋊ Z 2 ) × A for finite ab elian group A [ MYLG25 ]. W e will choose the c harge category as the one associated with the trivial fib er functor fgt : C → Hilb , and call it the reference charge category . Giv en a symmetry category C = Rep † ( H ∗ ), we c ho ose the reference c harge category D := C ∨ Hilb fgt in which ob jects are symmetry charges (lo calized excitations on the physical b oundary) and morphisms are interactions. W e use fusion c hannels to define the lo cal interactions. A lo cal C -symmetric tensor in region I is X 1 Y 1 f X 2 Y 2 · · · · · · X n Y m X ⊗ I ∈ A I := End D ( X ⊗ I ) F or I ⊆ J , w e define the natural inclusion A I  → A J , f 7→ 1 ⊗ J I X 10 whic h is graphically represen ted as f · · · X ⊗ I · · · 7→ f · · · X ⊗ J · · · . Then define A := colim I A I . Iden tifying each A I with its image in A yields an abstract spin system ov er Z . F or concrete calculations of the lo cal tensor, we first pro ject { Y } and { X } in to simple ob jects f = X { i,α } , { j,β }  ι Y 1 j 1 ,β 1 ⊗ · · · ⊗ ι Y m j m ,β m  ◦ M { j,β } { i,α } ◦  p i 1 ,α 1 X 1 ⊗ · · · ⊗ p i n ,α n X n  , where i, j, · · · ∈ Irr( D ) and ι S i,β : i → S and p i,α S : S → i are direct sum decomp osition of S such that M { j,β } { i,α } is a matrix blo c k with row indexed by { j , β } and column indexed b y { i, α } M { j,β } { i,α } : i 1 ⊗ · · · ⊗ i n → j 1 ⊗ · · · ⊗ j m as morphisms in fusion spaces. In particular,  ι S i,β  † = p i,α S . T o further decomp ose M { j,β } { i,α } , w e choose a trivialization T j i of the lo cal fusion junction D ( i 1 ⊗ · · · ⊗ i n , j 1 ⊗ · · · ⊗ j m ) expanded by the set of bases { e l } : e l · · · · · · Note that changing the tree graph T to T ′ yields a different choice of bases { e ′ l } , which are related to { e l } b y a series of F-mov es. The p en tagon axiom ensures the path indep endence of basis transformations. Com bining the ab ov e tw o steps, f is decomp osed under a c hoice of T as X 1 Y 1 f X 2 Y 2 · · · · · · X n Y m = X l ∈ T j i X { i,α } , { j,β } M [ l ] { j,β } { i,α } ·  ι Y 1 j 1 ,β 1 ⊗ · · · ⊗ ι Y m j m ,β m  ◦ e l ◦  p i 1 ,α 1 X 1 ⊗ · · · ⊗ p i n ,α n X n  . Equiv alently , any linear map f : H → K b etw een Hilb ert spaces can b e factorized through its image via f = H V † ↠ Im( f ) Σ − → Im( f ) U  − → K where U , V are isometries and Σ ∈ End(Im( f )) is a p ositiv e-definite Hermitian map. This is the basis-indep enden t formulation of the singular v alue decomp osition (SVD) [ LZ24 ]. After the direction sum decomp osition of { X } and { Y } , w e p erform the usual SVD on eac h blo ck M [ l ] { j,β } { i,α } . 11 Example 1 . When m = n = 2, f : X 1 ⊗ X 2 → Y 1 ⊗ Y 2 can b e decomp osed as f = X i 1 α 1 ,i 2 α 2 X j 1 β 1 ,j 2 β 2 X l,γ ,δ M [ l ] γ ,j 1 β 1 ,j 2 β 2 δ,i 1 α 1 ,i 2 α 2 · j 1 β 1 j 2 β 2 k γ α 1 i 1 δ α 2 i 2 Y 1 Y 2 X 1 X 2 , where SVD is p erformed on each matrix blo c k f [ l ]. T o do SVD in the horizontal direction, w e transform the tree graph T → H by F-mov es. 3.1.1 Q-system model As discussed ab o ve, giv en a Hopf C ⋆ -algebra H describing the lo cal version of a fusion category symmetry , the lo cal Hilb ert space of an H -symmetric quantum spin chain is Irr( Rep † ( H )). This is a large-enough lo cal Hilbert space, in which differen t H -symmetric phases (SPTs or SSBs) are embedded. H -symmetric phases are in one-to-one corresp ondence with the Morita classes of unitary separable F rob enius algebras (Q-systems) in C ∨ Hilb f . On an infinite chain, the Q-system is lab eled b y ( Z , H , A, 1 − m † m ), where the each site of the spin c hain is indexed by i ∈ Z , H is the Hopf C ⋆ -algebra describing the symmetry of the spin chain, Ψ( { i } ) = A is the lo cal Hilb ert space at site i and Φ( { i, i + 1 } ) = 1 − m † m giv es the nearest-neighbor interaction. In short, the total Hilb ert space is giv en by H = N i ∈ Z A and the Hamiltonian is H = X i ∈ Z 1 − ( m † m ) i,i +1 = X i ∈ Z 1 − m A A m † A A . W e summarize the notations in the table b elo w (dual) Hopf C ⋆ -algebra ( H ∗ ) H Symmetry category C Rep † ( H ∗ ) Charge category D := C ∨ Hilb f Rep † ( H ) Comm uting pro jector fixed p oin t mo del ( Z , H , A, 1 − m † m ) A is Q-system in C ∨ Hilb f , lo cal Hilb ert space H i = A ∀ i ∈ Z , Hamiltonian H = P i 1 − ( m † m ) i,i +1 . Th us giv en a c harge category , w e can construct a (1+1)d lattice mo del realizing all H -symmetric phases via the pro jection P k : Irr( Rep † ( H )) → A k . The Hamiltonian is written as H = − X k L X i =1 λ k  m i,i +1 k P i k P i +1 k  †  m i,i +1 k P i k P i +1 k  for all A k in C ∨ Hilb f with the asso ciated tuning parameter λ k . 12 3.1.2 Dualit y op erator W e generalize the Kramers-W annier dualit y op erator to an y finite ab elian group and further pro vide the (1+1)d quantum mo del at the self-dual p oint. 3 Similar work has b een done in the con text of Z X -calculus [ TVV23 , TTVV24 , GST24 , LXY25 ]. Let A b e a finite ab elian group and χ : A × A → U(1) b e a symmetric non-degenerate bicharacter (SNB). W e consider the F ourier transform of Fun ( A ): F : Fun ( A ) → Fun ( ˆ A ) , where ˆ A : A → U(1) is the group of characters. Cho ose the sets of bases { δ a } a ∈ A of Fun ( A ) and { δ χ } χ ∈ ˆ A of F un ( ˆ A ), F ( δ a ) = 1 p | A | X χ ∈ ˆ A χ ( a ) δ χ = 1 p | A | X b ∈ A χ b ( a ) δ b , where the c haracter χ b ( a ) := χ ( b, a ) is induced by the SNB. W e define the following t wo op er- ators: F = , F − 1 = . Define the symmetric map f : F un ( A ) → Fun ( A ), i.e. the symmetric lo cal op erator f ∈ End( F un ( A )), as: f ( δ a ) = X b ∈ A K ( b, a ) δ b = X b ∈ A K e ( a − 1 b ) δ b , where the matrix co efficien t K ( b, a ) dep ends on the difference a − 1 b only as f is an A -equiv ariant map (see App endix. A for details). Its F ourier transform ˆ f : F un ( ˆ A ) → F un ( ˆ A ) is given by ˆ f ( δ a ) = 1 | A | X b,c ∈ A χ a − 1 b ( c ) K ( c ) δ b , where the complex conjugate is χ • ( a ) := χ • ( a − 1 ) = χ − 1 • ( a ). Denote the symmetric lo cal op erator as O ( f ) := m † m f , w e can show the follo wing prop ert y: f = ˆ f . The con volution is given by f ∗ g := m ◦ ( f ⊗ g ) ◦ m † whic h maps to m ultiplication under the F ourier transformation: [ f ∗ g = b f b g . 3 This construction is based on discussions with Chenqi Meng. 13 Using this prop ert y , we define the KW op erator as D = · · · m † m i − 3 2 i − 2 m † m i − 1 2 i − 1 m † m i + 1 2 i + 0 m † m i + 3 2 i + 1 m † m i + 5 2 i + 2 · · · . W e hav e D η a = η a D = D , ∀ a ∈ A. The KW duality op erator maps symmetric op erators O ( f ) on the sites to F ouier-transformed ˆ f op erators on the links, i.e. D O ( f ) i,i +1 = ˆ f i + 1 2 D . (1) W e can chec k the fusion rule D ⊗ D ∗ = M a ∈ A η a , where D ∗ := D ⊗ T + as discussed in [ SSS24 ]. 3.1.3 UV-category of duality op erators F rom the duality transformation on lo cal op erators ( 1 ), we define the following self-dual lattice mo del: H = O i ∈ Z F un ( A ) , H = − X i ∈ Z O ( f ) i,i +1 − X i ∈ Z ( ˆ f ) i + 1 2 , whic h in the IR realizes fusion category symmetry TY χ,ϵ A . How ever, in the UV-category of dualit y channels is a differen t from the IR symmetry category . F rom the Q-system ( Q, m, ι ) mo del in a unitary fusion category C ∨ Hilb fgt , the following system is at the self-dual p oin t: H = O i ∈ Z Q, H = X i ∈ Z 1 − ( m † m ) i,i +1 − 1 d Q X i ∈ Z ( ιι † ) i , where ι : F un ( e )  → Fun ( A ). Example 2 . • When A = Z n , the Hamiltonian is H = − X i 1 2 (1 + Z ( n ) i Z ( n ) i +1 ) − X i 1 2 (1 + X ( n ) i ) . When n = 2, it recov ers the transverse field Ising chain. In this case, χ is exactly the Hadamard op erator as discussed in [ GST24 ]. In this case, the b ounded spread isomor- phism α on the Z n -symmetric subalgebra leads to a Z -graded extension of Z n as the dualit y c hannel restricting to α or say QCA acting on Fun ( Z n ) has an infinite order in the group QCA( B ). • When A = Z 2 × Z 2 , Q ∈ Rep ( Z 2 × Z 2 ), one self-dual Hamiltonian is H = − X i ∈ 2 Z ( Z i − 1 X i Z i +1 + Z i X i +1 Z i +2 ) . The duality channels assembly into Z 2 -graded extension of Z 2 × Z 2 , i.e. TY χ,ϵ Z 2 × Z 2 isomor- phic to Rep † ( D 8 ), Rep † ( Q 8 ) and Rep † ( H 8 ) for differen t ( χ,  )’s. Note that TY χ diag ,ϵ = − 1 Z 2 × Z 2 14 is isomorphic to an anomalous integral fusion category . The UV categorical structure of dualit y c hannels is determined b y the order of QCA acting on Z 2 × Z 2 -symmetric subal- gebra in the group QCA( B ). F or a single dualit y , the general situation giv es a Z -graded extension of Z 2 × Z 2 , while a finite Z n -graded quotient can happen on the lattice if the QCA has finite order. Up on fixing the lattice realization on tensor-pro duct Hilb ert space, the fact that the fusion rule inv olving the non-inv ertible ob ject D is mixed with translation suggests that the lattice categorical symmetry structure is not exactly TY χ,ϵ Z 2 , but a Z -graded extension of Z 2 , hence a new set of F-symbols. W e follow the conv en tion in [ SSS24 ] D ⊗ D = ( 1 ⊕ η ) T ± where T ± is a righ t(left) full translation arising from the identification of lattices before and after the duality transformation. The Z -graded tensor category C Z is defined as: Irr( C 0 ) = { 1 , η } , Irr( C 1 ) = { D 1 2 } , Irr( C − 1 ) = { D − 1 2 } , Irr( C 2 ) = { T + , η T + } , Irr( C − 2 ) = { T − , η T − } , Irr( C 3 ) = { D 1 2 T + } , Irr( C − 3 ) = { D − 1 2 T − } , Irr( C 4 ) = { T + T + , η T + T + } , Irr( C − 4 ) = { T − T − , η T − T − } , · · · where the ob jects are defined as T + := T − := η T + := X η T − := X D 1 2 := D − 1 2 := D 1 2 T + := D − 1 2 T − := 15 ( T + ) 2 := ( T − ) 2 := · · · W e used the MPO’s defined ab o ve to compute F -symbols. W e start with determining the in tertwiners ιρ a = ρ b ⊗ c ι for each b ⊗ c = a , where a, b, c are simple ob jects in C Z 4 . Note that as η T • is defined as an ob ject, w e tak e the intert winer η T • → T • ⊗ η in to accoun t: T + ⊗ η = X X X = X = η T + In tertwiner X ; T − ⊗ η = X X X = X = η T − In tertwiner X . (2) See App endix. A for a detailed F -sym b ol computation. Note that in this mo del-dep enden t construction, w e can only compute F -symbols for finitely many simple ob jects and it is uncertain whether such a Z -graded extension exists with solutions to the p en tagon equations. Thus case- b y-case study is limited. In the rest of the pap er we consider multiple duality op erators acting on an abstract spin chain without referring to a sp ecific internal symmetry C to start with and sho w that the UV-category of duality op erators is in fact an F n -graded extension of C . 3.2 The general picture In this section, we b egin b y answering the question 3.2 . 3.2.1 Bimo dule channels and top ological b oundary defects Supp ose we hav e a physical b oundary algebra B and tw o top ological b oundary extensions B ⊆ A 1 and B ⊆ A 2 . W e w ant to capture top ological defects b etwe en these b oundary conditions in terms of quan tum channels betw een A 1 and A 2 . In the SymTFT picture, this is obtained b y “sw eeping out” the defect. But this pro cedure acts trivially on the ph ysical b oundary . In pictures, this lo oks as follo ws. 4 An intert winer is a rank-3 tensor to reduce the comp osed lo cal tensors. 16 Ψ ∈ Ch B ( A 1 , A 2 ) := B ( A 1 , A 2 ) B B A 1 A 2 id B − → time T o capture this idea in op erator algebra language, we hav e the following definition. Definition 3.3. Let B b e an abstract spin system and B ⊆ A 1 and B ⊆ A 2 lo cal extensions. A bimo dule channel b et ween these extensions is a completely p ositiv e map Ψ : A 1 → A 2 suc h that Ψ | B = Id B . W e denote the set of bimo dule c hannels Ch B ( A 1 , A 2 ). W e note that as b efore, this implies Ψ is B - B bimo dular, and automatically unital. The collection of top ological b oundary extensions and bimo dule channels extends to a category , defined as follows. Definition 3.4. Fix a physical b oundary algebra B , and define category T ∂ ( B ) as follows: • Ob jects are all top ological b oundary extensions of B . • The set of morphisms b et ween extensions A 1 and A 2 is given b y the set Ch B ( A 1 , A 2 ), and comp osition is just comp osition of channels. W e note that T ∂ ( B ) can b e view ed as enriched ov er the appropriate category of conv ex spaces. Indeed, there is an ob vious conv ex structure on Ch B ( A 1 , A 2 ), and this comp osition is “bilinear”. Our goal is to show that this formally defined category of cp maps “matches” with the in tuitive picture of top ological defects b et ween top ological b oundary conditions. Notice that a bimo dule channel is an actual morphism Ψ : A 1 → A 2 in the line ar c ate gory DHR( B ). The question is, whic h abstract morphisms ψ : A 1 → A 2 in DHR( B ) corresp ond to bimo dule channels? Let L 1 , L 2 b e comm utative Q-systems in a unitary braided tensor category . W e recall the c onvolution algebr a associated to these, as in [ HBJP23 ], see also [ BJ21 , HJL W25 ]. Consider the finite dimensional vector space Hom C ( L 1 , L 2 ), with conv olution pro duct f ∗ g := m ◦ ( f ⊗ g ) ◦ m † ∈ Hom C ( L 1 , L 2 ) . This is a commutativ e C ⋆ -algebra with ∗ -op eration denote b y # and defined as f # := f : f # := f † W e denote this C ⋆ -algebra as H ( L 1 , L 2 ). Prop osition 3.1. Let B b e a physical b oundary algebra with top ological b oundary extensions B ⊆ A 1 , A 2 and let Ψ : A 1 → A 2 b e a B - B bimo dule in tertwiner. Then Ψ is completely p ositive if and only if Ψ is a p ositiv e elemen t in the C ⋆ -algebra H ( L 1 , L 2 ). 17 Pr o of. Let M 1 and M 2 b e the unitary categories of B - A 1 and B - A 2 corresp ondences that forget to DHR( B ) as B - B corresp ondences. Then B - B bimo dular cp-maps Ψ : A 1 → A 2 are in bijectiv e corresp ondence with cp-multipliers b et ween M 1 and M 2 in the sense of [ JP17 ], which b y [ HP23 , HJL W25 , V er22 ] corresp ond to p ositiv e elements in the C ⋆ -algebra H ( L 1 , L 2 ). In the ab o v e statement, we gav e a criteria for determining if a morphism is cp, whereas the morphisms in T ∂ ( B ) are unital cp maps. How ev er, ev ery non-zero B -bimodular cp-m ultiplier Ψ is almost unital, in the sense that Ψ(1 A 1 ) = λ 1 A 2 for some strictly p ositiv e scalar λ . Indeed, the irreducibility of the inclusions ( B ′ ∩ A i ) = C 1 A i implies b oth A 1 and A 2 as B - B bimodules con tain B with m ultiplicity 1. Since Ψ is B -bimo dular, this implies Ψ(1 A 1 ) = λ 1 A 2 for some scalar λ . T o see this is non-zero, note that since Ψ is cp, Ψ(1 A 1 ) = p ∈ A 2 is a non-zero p ositive elemen t of the C ⋆ -algebra A 2 if and only if Ψ is non-zero. No w, since finite-dimensional commutativ e C ⋆ -algebras are isomorphic to C n , there exists a basis of orthogonal minimal pro jections p i ∈ H ( L 1 , L 2 ) suc h that p i ∗ p j = δ i,j p i p # i = p i i ◦ i † = X p i The equation follows from that fact that i ◦ i † is the unit for H ( L 1 , L 2 ). F or each i , there is some non-zero, p ositiv e scalar λ i suc h that p i (1) = λ i 1. W e then define the c hannel Ψ i := p i λ i . The next prop osition shows that the conv ex space Ch B ( A 1 , A 2 ) is actually a simplex, mean- ing that there are finitely many extreme p oin ts and every elemen t is a uniquely sp ecifiable as a con vex combination of extreme p oints. Theorem 3.2. If B ⊆ A 1 and B ⊆ A 2 are local extensions of an abstract spin system, then Ch B ( A 1 , A 2 ) is a simplex with extreme p oints given by { Ψ i } describ ed ab o ve. Pr o of. Since the p i are bases for the whole conv olution C ⋆ -algebra H ( A 1 , A 2 ), the B-bimo dular ucp maps Ψ : A 1 → A 2 can b e uniquely written written as Ψ = P i β i Ψ i , but since the Ψ i are unital, Ψ(1) = 1 implies P i β i = 1, hence Ψ is a conv ex combination. No w we will demonstrate how to upgrade results from [ BJ21 ], to allow us to completely understand cp maps b etw een Lagrangian algebras. Recall the canonical corresp ondence b e- t ween Lagrangian algebras L ∈ Z ( D ) and indecomp osable right D -module categories. Giv en a Lagrangian algebra, we write the corresp onding D -module category as M L . The collection of right mo dule categories of D assembles into a unitary 2-category , whose 1-morphisms are  -mo dule functors, and morphisms are natural transformations. W e will turn this in to a unitary multi-fusion category as follows: pick a representativ e of eac h indecomp os- able mo dule category M i and consider the righ t unitary mo dule category M := L i M i . F or con venience we set M 1 = D as a right D -mo dule category . Then define the indecomp osable unitary m ulti-fusion category e D := End D ( M ) . e D is an indecomp osable unitary m ultifusion category , where the blo c ks are indexed by the mo dule categories and the blo c ks are giv en by D ij := End D ( M j , M i ) . 18 The D ij are in vertible D ii − D j j bimo dule categories. F urthermore, W e ha ve Z ( D ) ∼ = Z ( e D ) ∼ = Z ( D ii ) . No w let F i : Z ( D ) → Z ( D ii ) b e the comp osition of the equiv alence Z ( D ) = Z ( D 11 ) ∼ = Z ( D ii ) with the forgetful functor to D ii . Let I i : D ii → Z ( D ) b e its (bi)-adjoint. Then there are exactly n isomorphism classes of Lagrangian algebras in Z ( D ), given by { A i := I i ( 1 D ii ) } n i =1 , with A 1 b eing the canonical Lagrangian algebra in Z ( D ). W e can express A i as the pair ( L Y ∈D 1 j Y ⊗ Y , ψ A i ) where ψ A i ,W := M Y ,Z ∈ Irr( D 1 i ) X i √ d Z √ d Y Y Z W ¯ Y ¯ Z W i α α • . Here i is lab elling the region, meaning we pro ject onto the i th summand of 1 in D (we assume a blank lab el indicates the region is lab elled by 1). { α } is a basis for Hom D 1 i ( Y , W ⊗ Z ) with β † ◦ α = δ α,β 1 Y , and { α • } ⊆ D i 1 ( ¯ Y ⊗ W, ¯ Z ) α • := (ev Y ⊗ 1 Z ) ◦ α † ◦ (1 Y ⊗ 1 W ⊗ co ev Z ) . Eac h ob ject A i is canonically endow ed with the structure of a Q-system in Z ( D ), with structure maps L i L i L i = M X ∈ Irr( D 1 i ) 1 √ d X ¯ X X X X ¯ X ¯ X i , L i = M X ∈ Irr( D 1 i ) p d X ¯ X X i The comultiplication and counit are giv en b y the reflected diagrams of the multiplication and unit maps resp ectiv ely , with the same normalizing co efficien ts. Th us A i is a connected Q-system in Z ( D ). F rom ab o ve we see Z ( D )( A i , A j ) ∼ = M X ∈ Irr( D ij ) D ij ( 1 , X ⊗ ¯ X ) . No w for Y ∈ Irr( D ij ), define Φ Y = M X ∈ Irr( D 1 j ) ,Z ∈ Irr( D 1 i ) X i √ d X √ d Z d Y X Z Y ¯ X ¯ Z α • α (3) where α ranges ov er a basis for e D ( Y ⊗ ¯ X , ¯ Z ). It is easy to v erify that Φ Y ∈ Z ( D )( A j , A i ). Comparing this with [ BJ21 ], w e note that our Φ Y = 1 d 2 Y e Y in their notation. With this dictionary in mind, then following the calculations in [ BJ21 , Section 3.1] w e hav e the following. 19 Theorem 3.3. 1. { Φ Y } Y ∈ Irr( D ij ) forms a linear basis for Z ( D )( L j , L i ). 2. F or Y , Z ∈ Irr( D ij ), Φ Y ∗ Φ Z = δ Y ,Z d 2 X d 2 Y d 2 Z Φ Y . 3. F or Y ∈ Irr( D ij ) and Z ∈ Irr( D j k ) Φ Y ◦ Φ Z = P X ∈ Irr( D ik ) d X d Y d Z N X Y Z Φ X . Prop osition 3.4. Let B ⊆ A 1 and B ⊆ A 2 b e top ological b ounda y extensions. 1. There is a canonical bijectiv e assignment F un C ( M A 1 , M A 2 ) → Ch B ( A 1 , A 2 ), X 7→ Φ X b e- t ween the extreme p oints Ch B ( A 1 , A 2 ) and equiv alence classes of simple C -mo dule functors in F un C ( M A 1 , M A 2 ). 2. If A 1 , A 2 and A 3 are Lagrangian algebras, let X ∈ F un C ( M A 1 , M A 2 ) , Y ∈ F un C ( M A 2 , M A 3 ) and Φ X ∈ Ch B ( A 1 , A 2 ) , Φ Y ∈ Ch B ( A 2 , A 3 ) the asso ciated channels. Then Φ X ◦ Φ Y = X Z ∈ Irr ( F un C ( M A 1 , M A 3 ) ) d Z d X d Y N Z X,Y Φ Z where N Z X,Y denotes the fusion rule for mo dule functors. If w e take a mo dule fucn tor X : M 1 → M 2 , the cp map b et ween the asso ciated Lagrangian algebras A 1 and A 2 is denoted by Φ X as ab o ve. A 1 A 2 Φ X ↔ M A 1 M A 2 X . Giv en that A is a comm utative Q-system in DHR( B ) ∼ = br Z ( D ), for X ∈ D , we can write do wn the lo cal op erators explicitly (for example, see [ JSW26 ]): f · · · · · · X ⊗ I symmetric lo cal op erators in B ⊆ f σ · · · · · · X ⊗ I A ∈ Z ( D ) lo cal op erators in A , where σ A,X : A ⊗ X ≃ − → X ⊗ A is half-braiding in Z ( D ). Combining the bimo dule channels with the lo cal op erators, we hav e O I ⊆ Z M d ( C ) ∼ = f · · · · · · X ⊗ I A 7→ f · · · · · · X ⊗ I A Φ X . See ( 3 ) for the explicit formula of the basis Φ X . 20 3.2.2 P arameterizing dualit y op erators Let α ∈ QCA( B ). Then by [ Jon24 ], α acts by braided auto equiv alences on DHR by t wisting. Giv en X ∈ DHR( B ), define X α := X as a vector space with a  x  b := α − 1 ( a ) xα − 1 ( b ) ⟨ x | y ⟩ α := α ( ⟨ x | y ⟩ ) . A t the lev el of morphisms, if f : X → Y is an in tert winer, then e α ( f ) := f as a set fucntion, whic h is easily seen to b e an intert winer b et ween the twisted bimo dules. By [ Jon24 ], this is again a DHR bimo dule, and defines a braided autoequiv alence of DHR( B ), whic h we denote e α . No w, let Ch α B ( A 1 , A 2 ) denote the cp maps Ψ : A → A with Ψ | B = α . W e hav e the following result (c.f. [ JSW26 ]). Theorem 3.5. Let B b e a ph ysical b oundary algebra, and let α ∈ QCA( B ). Suppose B ⊆ A 1 , B ⊆ A 2 , B ⊆ A 3 are top ological b oundaries, so that A 1 , A 2 , A 3 are Lagrangian algebras in DHR( B ). 1. Then there is a canonical (conv ex) isomorphism λ : Ch α B ( A 1 , A 2 ) ∼ = Ch B ( e α ( A 1 ) , A 2 ). 2. If Ψ ′ ∈ Ch α B ( A 1 , A 2 ) and Ψ ∈ Ch β B ( A 2 , A 3 ), then Ψ ◦ Ψ ′ = λ − 1 (( λ (Ψ) ◦ e β ( λ (Ψ ′ ))) ∈ Ch β ◦ α B ( A 1 , A 3 ) . Pr o of. Let Ψ ∈ Ch α B ( A 1 , A 2 ). Then considering A 1 as a DHR bimo dule of B , we claim the map of v ector spaces Ψ : A 1 → A 2 actually defines a B - B bimo dule map e α ( A 1 ) → A 2 , which we will denote λ (Ψ). W e compute Ψ( b 1  a  b 2 ) = Ψ( α − 1 ( b 1 ) aα − 1 ( b 2 )) = b 1 Ψ( a ) b 2 . Th us Ψ ∈ Ch B ( e α ( A 1 ) , A 2 ). Conv ersely the same computation shows that for any Ψ ∈ Ch B ( e α ( A 1 ) , A 2 ), the underlying map of vector spaces Ψ : A 1 = e α ( A 1 ) → A 2 is in Ch B ( e α ( A 1 ) , A 2 ). The statemen t concerning comp osition follows trivially . The ab o v e theorem can b e represented graphically b y asserting that we alw ays hav e the decomp osition Ch α B ( A 1 , A 2 ) : B B A 1 A 2 α − → time = B B B A 1 e α ( A 1 ) A 2 α id B − → time . Theorem 1.1 from the introduction now follows. 21 Pr o of of The or em 1.1 . F or any α ∈ QCA( B ), we hav e an auto equiv alence e α ∈ Aut br ( Z ( C )). By [ ENO10 ], this defines an inv ertible C - C bimo dule category , which as a right C mo dule cat- egory corresp onds to e α − 1 ( I ( 1 )). The s imple ob jects of this bimo dule category are in bijective corresp ondence with the minimal con volution idemp oten ts of the algebra H ( I ( 1 ) , α − 1 ( I ( 1 )) ∼ = H ( α ( I ( 1 )) , I ( 1 )). Thus the result follows. 3.2.3 Univ ersal tensor categories and emanan t symmetries W e turn our attention to characterizing emanant symmetries of duality op erators. W e assume w e hav e a physical b oundary subalgebra B ⊆ A for a fixed b oundary , with symmetry category C . W e w an t to understand UV-categories of duality op erators on A , whic h will flo w to in ter- nal symmetries under any C -constained R G. In general these categories will not b e fusion, in particular if the restriction of Ψ | B ∈ QCA( B ) do es not ha ve finite order. W e hav e already c haracterized the dualit y operators that sit o ver a fixed QCA, α ∈ QCA( B ), and w e ha ve a form ula to compute how these comp ose, given also by the previous comp osition. If Ψ ′ ∈ Ch α B ( A, A ) and Ψ ∈ Ch β B ( A, A ), then Ψ ′ ◦ Ψ ∈ Ch α ◦ β B ( A, A ), so it will decompose as a con vex combination, and thus this lo oks like a kind of (normalized) fusion rule. W e will see there is a universal unitary tensor category here, though it is not necessarily fusion. Theorem 3.6. Let B ⊆ A b e a ph ysical b oundary subalgebra, and let { α j } n j =1 ⊆ QCA( B ). Let π : F n → QCA( B ) b e the homomorphsim of groups sending the jth generator to α j . 1. There is a canonical F n -graded extension of C , denoted C # F n . 2. F or each element g ∈ F n , there is a bijection X ↔ Φ X b et w een the simple ob jects X ∈ ( C # F n ) g of the g -graded comp onen t and the extreme p oin ts of the simplex Ch π ( g ) B ( A, A ). 3. If X ∈ ( C # F n ) g and Y ∈ ( C # F n ) h , then Φ X ◦ Φ Y := P Z ∈ ( C # F n ) gh d Z d X d Y N Z X Y Φ Z . Pr o of. W e note that since QCA( B ) ≤ Aut( B ), the action on bimo dules is just b y conjugation, and the action of QCA on DHR bimo dules is just the restriction of this general action, w e get a 2-group homomorphism QCA( B ) → Aut br ( Z ( C )). Composing π yields a 2-group homomor- phism e π : F n → Aut br ( Z ( C )). By the standard extension theory of fusion categories [ ENO10 ], the existence of an F n -graded extension is obstructed b y a class o 4 ( e π ) ∈ H 4 ( F n , U(1)), and if this v anishes, the solutions to the p en tagon equation for the asso ciator form a torsor ov er H 3 ( F n , U(1)). Both of these cohomology groups are trivial for free groups, and th us there exists a unique F n -graded extension. W e will call this category C # F n No w, given an α ∈ QCA( A, A ), we hav e an isomorphism Ch α B ( A, A ) ∼ = Ch B ( α ( A ) , A ), and by [ BJ21 , Section 3.2] the extreme p oints corresp ond to the simple ob jects in α graded comp onen t (or more precisely , the g graded comp onen t for any g ∈ π − 1 ( α ). Then the c laim ab out comp osition follo ws from part 2 of Theorem 3.5 . This giv es us Theorem 1.2 Pr o of of Cor ol lary 1.3 . An y category X obtain under RG from an emanant symmetry will necessarily b e a quotien t of C # F n , and th us will b e a G-graded quotient of a quotient of C , where G is some quotient of F n . But b y [ EJ25 ], C must b e integral, hence an y quotien t of C is integral [ EGNO15 , Lemma 3.5.6]. Th us X is a G -graded extension of an integral category , whic h is neccessarily weakly in tegral. Indeed, if X ∈ X is a simple ob ject, then d ( X ) 2 = d ( X ⊗ X ∗ ), but X ⊗ X ∗ is in the quotien t of C , which is integral. 22 A An algebraic construction of generalized KW dualit y op era- tors W e generalize the Kramers-W annier duality MPOs to an y finite ab elian group and further pro vide the (1+1)d quantum mo del at the self-dual p oint. Similar w ork has b een done in the con text of Z X -calculus [ TVV23 , TTVV24 , GST24 , LXY25 ]. Let A b e a finite ab elian group and χ : A × A → U(1) b e a symmetric non-degenerate bic haracter (SNB). W e consider the F ourier transform of F un ( A ): F : Fun ( A ) → Fun ( ˆ A ) , where ˆ A : A → U(1) is the group of characters. Cho ose the sets of bases { δ a } a ∈ A of Fun ( A ) and { δ χ } χ ∈ ˆ A of F un ( ˆ A ), then F ( δ a ) = 1 p | A | X χ ∈ ˆ A χ ( a ) δ χ , where F ( δ a )( χ ) := χ ( a ). F or a finite ab elian group A , there exists an isomorphism ι : A ≃ − → ˆ A , th us F ( δ a ) = 1 p | A | X χ ∈ ˆ A χ ( a ) δ χ = 1 p | A | X b ∈ A χ b ( a ) δ b , where the c haracter χ b ( a ) := χ ( b, a ) is induced by the SNB. W e define the following t wo op er- ators: F = , F − 1 = . Define the symmetric map f : F un ( A ) → Fun ( A ), i.e. the symmetric lo cal op erator f ∈ End( F un ( A )), as: f ( δ a ) = X b ∈ A K ( b, a ) δ b , where the matrix co efficien t K ( b, a ) is giv en by K ( b, a ) := f ( δ a )( b ) ∈ C . f being a symmetric map means that f is an A -equiv ariant map. Explicitly , for any elemen t φ of Fun ( A ), A acts on φ by a left translation: ( L a φ )( x ) := φ ( a − 1 x ) , for x ∈ A satisfying L a ◦ L b = L ab . It is easy to show that f comm uting with L a requires K ( b, a ) dep ends on the difference a − 1 b only , thus f ( δ a ) = X b ∈ A K e ( a − 1 b ) δ b , where K : A → U(1) can b e conv eniently c hosen to b e the character of A . Its F ourier transform ˆ f : F un ( ˆ A ) → F un ( ˆ A ) is given by ˆ f ( δ a ) = 1 | A | X b,c ∈ A χ a − 1 b ( c ) K ( c ) δ b , where the complex conjugate is χ • ( a ) := χ • ( a − 1 ) = χ − 1 • ( a ). W e can show the follo wing prop- ert y: 5 f = ˆ f . 5 W e use the conv ention that lo cal tensors contract from righ t to left and comp ose from b ottom to up. 23 Pr o of. LHS = δ a ⊗ δ b 7→ K ( b − 1 a ) δ a ⊗ δ b = 1 | A | X a ′ K ( b − 1 a ) χ ( b − 1 a, a ′ ) δ a ′ = 1 | A | X a ′ ,c ˆ K ( c ) χ ( b − 1 a, c ) χ ( b − 1 a, a ′ ) δ a ′ = 1 | A | X a ′ ,c ˆ K ( c ) χ ( b − 1 a, a ′ ) δ ca ′ = RHS . The con volution is f ∗ g := m ◦ ( f ⊗ g ) ◦ m † whic h maps to m ultiplication under the F ourier transformation: [ f ∗ g = b f b g . A.1 Kramers-W annier dualit y op erator The KW op erator is defined to b e D = · · · m † m i − 3 2 i − 2 m † m i − 1 2 i − 1 m † m i + 1 2 i + 0 m † m i + 3 2 i + 1 m † m i + 5 2 i + 2 · · · Denote the symmetric lo cal op erator as O ( f ) := m † m f , satisfying D η a = η a D = D , ∀ a ∈ A. By the prop erty prov ed in the last section, the KW dualit y op erator maps symmetric op er- ators O ( f ) on the sites to F ouier-transformed ˆ f op erators on the links, i.e. D O ( f ) i,i +1 = ˆ f i + 1 2 D . (4) No w we chec k the following fusion rule: D ⊗ D ∗ = · · · · · · 24 = · · · · · · . A t each lo cal tensor of the MPO, we hav e T = m m † m m † , T : Fun ( A ) ⊗ F un ( A ) → Fun ( A ) ⊗ Fun ( A ) . Explicitly: δ a ⊗ ˆ δ b = 1 p | A | X c χ ( c, b ) δ a ⊗ δ c 7→ 1 | A | 5 2 X c χ ( c, b ) χ ( a, c ) X c ′ ,a ′ χ ( a ′ , a ) χ ( c ′ , c ) χ ( a ′ , c ′ ) δ a ′ ⊗ δ c ′ = 1 | A | 5 2 X c, a ′ ,c ′ χ ( c, b ) χ ( c ′− 1 a, c ) χ ( a ′ , a − 1 c ′ ) δ a ′ ⊗ δ c ′ = 1 | A | 3 2 X a ′ χ ( a ′ , b ) δ a ′ ⊗ δ ab = 1 | A | ˆ δ b ⊗ δ ab , Th us D ⊗ D ∗ = M a ∈ A η a , where D ∗ := D ⊗ T + as discussed in [ SSS24 ]. A.2 Z -graded tensor category from duality operators Up on fixing the tensor-product Hilbert space lattice realization, the fact that the fusion rules in volving the non-in vertible ob ject D is mixed with translation suggests that the lattice categor- ical symmetry structure is not exactly TY χ,ϵ Z 2 , but a Z -graded extension of Z 2 . The Z -grading is incorp orated to encompass translation as an ob ject in this tensor category , in whic h there are infinitely-many simple ob jects, hence a new set of F-symbols. W e follo w the con ven tion in [ SSS24 ] D ⊗ D = ( 1 ⊕ η ) T ± where T ± is right(left) translation arising from the identification of lattices b efore and after the dualit y transformation. The Z -graded tensor category is defined as: Irr( C 0 ) = { 1 , η } , Irr( C 1 ) = { D 1 2 } , Irr( C − 1 ) = { D − 1 2 } , Irr( C 2 ) = { T + , η T + } , Irr( C − 2 ) = { T − , η T − } , Irr( C 3 ) = { D 1 2 T + } , Irr( C − 3 ) = { D − 1 2 T − } , 25 Irr( C 4 ) = { T + T + , η T + T + } , Irr( C − 4 ) = { T − T − , η T − T − } , · · · where the ob jects are T + := T − := η T + := X η T − := X D 1 2 := D − 1 2 := D 1 2 T + := D − 1 2 T − := ( T + ) 2 := ( T − ) 2 := · · · F η D η D W e start with the simplest non-trivial F-sym b ols inv olving only one non-inv ertible ob ject, for example, F η D 1 2 η D 1 2 . 26 D 1 2 X = X → Z Z := D 1 2 Z Z In tertwiner: Z D 1 2 X = X → X X := D 1 2 X X In tertwiner: X . (5) Th us F η D 1 2 η D 1 2 = − 1. F DD • • W e first consider F-sym b ols in volving tw o non-in vertible ob jects D ± 1 2 . W e start with deter- mining the intert winers ιρ a = ρ b ⊗ c ι for each b ⊗ c = a , where a, b, c are simple ob jects in C Z 6 . D − 1 2 ⊗ D − 1 2 = = 1 √ 2 1 ⊕ η = 1 √ 2 ( 1 ⊕ η ) T − denoted as D − 1 2 D − 1 2 In tertwiner: 2 1 4 | a ⟩ (6) where | a ⟩ conditions differen t fusion c hannels, the blue color indicates the added blo cks such that the D − 1 2 ⊗ D − 1 2 fusion rule would b e satisfied. Before pro ceeding with other F -sym b ols, w e first consider the in tert winers for η T + → T + ⊗ η 6 An intert winer is a rank-3 tensor to reduce the comp osed lo cal tensors. 27 and η T − → T − ⊗ η T + ⊗ η = X X X = X = η T + In tertwiner X ; T − ⊗ η = X X X = X = η T − In tertwiner X . (7) Then w e hav e D 1 2 ⊗ D 1 2 = 1 √ 2 T + ( 1 ⊕ η ) : D 1 2 D 1 2 = 1 √ 2 1 ⊕ η In tertwiner: 2 1 4 id | 1 ⟩ , 2 1 4 X | η ⟩ ; D − 1 2 ⊗ D 1 2 = 1 ⊕ η : D − 1 2 D 1 2 = 1 ⊕ η In tertwiner: | a ⟩ ; D 1 2 ⊗ D − 1 2 = 1 ⊕ η : D 1 2 D − 1 2 = 1 2 1 ⊕ η In tertwiner: √ 2 | a ⟩ Using the ab o ve intert winers ( 5 ),( 6 ), w e compute the F-sym b ols F D − 1 2 η D − 1 2 T − , F D − 1 2 η D − 1 2 η T − 28 X | a ⟩ = χ ( η , a ) Z | a ⟩ . Computations for F D 1 2 η D 1 2 T + , F D 1 2 η D 1 2 η T + , F D − 1 2 η D 1 2 1 , F D − 1 2 η D 1 2 η , F D 1 2 η D − 1 2 1 , F D 1 2 η D − 1 2 η are straightfor- w ard. F or F D − 1 2 D − 1 2 η T − , F D − 1 2 D − 1 2 η η T − , using ( 6 ) ( 7 ) w e hav e X | a ⟩ = X | a ′ ⟩ . F or F η D − 1 2 D − 1 2 T − , F η D − 1 2 D − 1 2 η T − , ( 6 ) we hav e Z | a ⟩ = id | a ′ ⟩ . F D T D 1 ,η Next w e compute the F-symbols inv olving translation and the non-inv ertible op erator. W e need to determine the intert winers for D 1 2 → T + ⊗ D − 1 2 , D 1 2 → D − 1 2 ⊗ T + , D − 1 2 → T − ⊗ D 1 2 and D − 1 2 → D 1 2 ⊗ T − . T + ⊗ D − 1 2 = = T + D − 1 2 = √ 2 D 1 2 In tertwiner: 2 − 1 4 ; D − 1 2 ⊗ T + = = T + D − 1 2 = √ 2 D 1 2 In tertwiner: 2 − 1 4 . 29 Belo w we list all other in tertwiners: D 1 2 → T − ⊗ D − 1 2 : in tertwiner from T − D 1 2 ; D − 1 2 → D 1 2 ⊗ T − : in tertwiner 2 − 1 4 from T − D 1 2 . W e hav e F D − 1 2 T + D − 1 2 1 = F D − 1 2 T + D − 1 2 η = 1. F DDD D T Next we consider F-symbols inv olving three non-inv ertible ob jects. W e will consider F D − 1 2 D − 1 2 D − 1 2 D − 1 2 T − , all other cases are similar. Notice that D − 1 2 ⊗ D − 1 2 = ( 1 ⊕ η ) T − . In order to obtain D − 1 2 T − after fusing with the third D − 1 2 , w e need to consider the following intert winer: T − ⊗ D − 1 2 = := 1 √ 2 D − 1 2 T − D − 1 2 T − → T − ⊗ D − 1 2 : In tertwiner 2 1 4 . 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