Dualities in one-dimensional quantum lattice models: topological sectors

It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin-$\frac{1}{2}$ Heisenberg XXZ model. One model has symmetry operators forming the fusion category $\mathsf{Rep}(\mathcal S_3)$ of representations of the group $\mathcal S_3$. We find that the mapping between its topological sectors and those of the XXZ model is associated with the non-trivial braided auto-equivalence of the Drinfel'd center of $\mathsf{Rep}(\mathcal S_3)$.


Sec. I | Introduction
Over the past few years tremendous progress has been achieved in our understanding of quantum theories by interpreting symmetries in terms of topological operators.More specifically, correlation functions of the theories including symmetry operators are insensitive to topologypreserving deformations of the submanifolds supporting the operators, unless they pass through charged operators [1].In this context, ordinary global symmetries are generated by codimension-one invertible operators, which together with the requirements that symmetry operators can be fused, implies that these furnish a representation of a group.This new approach has led to generalizations of the notion of symmetry, whereby operators are not necessarily supported on onecodimensional submanifolds and/or are not necessarily invertible.This manuscript is concerned with such generalized symmetries in the context of translation invariant one-dimensional quantum lattice models.
In virtue of their topological nature, any categorically symmetric model in (1+1)d can be lifted to a (gapped) boundary condition of the Turaev-Viro-Barrett-Westbury topological quantum field theory (TQFT) [40,41] with input datum the corresponding spherical fusion category [4,5,42].Mathematically, gapped boundary conditions admit a classification in terms of module categories over the input category [43][44][45][46][47], the case of pure gauge theories having received special attention [48][49][50][51][52][53][54][55].This holographic viewpoint on symmetries has also garnered a lot of interest [6-8, 56, 57].Crucially, the bulk TQFT can be reconstructed from any choice of gapped boundary condition, so that bulk topological lines are encoded into the Drinfel'd center of the corresponding spherical fusion category of (boundary) topological lines [47].This suggests a notion of duality between models canonically associated with distinct boundary conditions of the same bounding TQFT.
Inspired by these developments, we initiated in ref. [58] a systematic study of dualities in one-dimensional quantum lattice models from the viewpoint of their (categorical) symmetries.One merit of our approach is to make very concrete the concepts and results alluded to abovewhich are often formal and abstract otherwise-as well as demonstrate that this approach to dualities agrees and extends traditional ones [59].Within our framework, an equivalence class of dual models is given by a choice of input (spherical) fusion category together with an algebra of local operators.A representative of such a class then corresponds to a specific lattice realization of the underlying theory.Choosing a lattice realization loosely boils down to picking a collection of degrees of freedom, which happen to be encoded into a choice of module category over the input fusion category.This means that models that only differ in a choice of module category are dual to one another.Importantly, dualities thus defined are such that any symmetric local operator is mapped to a dual symmetric local operator, whereas non-symmetric local operators in one theory are mapped to non-local nonsymmetric operators in the dual theory.Generally speaking, we can interpret such dualities as arising from some generalized gauging procedure of the categorical symmetry [60][61][62][63][64][65][66][67][68][69][70].It follows from our construction that duality transformations are naturally associated with maps between module categories.
Practically, given a known one-dimensional quantum lattice model, a suitable choice of input fusion category can be extracted from a detailed understanding of its symmetries-which are typically generated by non-local operators.The Hamiltonian itself is then built from linear combinations of symmetric local operators obtained from the data of a module category over the input fusion category.The algebra entering the characterization of the equivalence class of dual models is that generated by the set of local operators.Keeping the same linear combination of symmetric operators, but choosing a different realization via a choice of module category, yields a dual model [58].
A key technical novelty of our approach is our ability to explicitly write down, in the form of matrix product operators [71][72][73], the non-local lattice operators generating the symmetries of a given family of dual models.Similarly, we are able to implement a duality relation via MPOs that transmute the local operators of a given Hamiltonian into those of one of its duals [58].The main teaching of the present manuscript is that these MPOs can be further exploited so as to construct isometries relating the full spectra of dual models.A crucial aspect of such a mapping is the delicate interplay between duality relations and sectors of the models that come into play.
Indeed, the study of sectors cannot be dissociated from the constructions of duality mappings.Consider for instance the Kramers-Wannier duality of the transverse field Ising model [29].Given a closed chain, one formulation of this duality identifies the simultaneous action of two Pauli Z operators on qubits located at neighbouring vertices with that of a single Pauli Z operator on a qubit located along the edge bounding these vertices, and vice versa for Pauli X operators.It turns out that such a duality mapping imposes kinematical constraints for both the original Hamiltonian and its dual.Indeed, it follows from the definition that acting simultaneously on all qubits with Pauli X or Pauli Z operators, respectively, must leave every state invariant.In other words, these kinematical constraints force both models into the even charge sectors of their respective Z 2 -symmetry.This means in particular that the symmetry cannot be spontaneously broken on either side of the duality [74,75].Accessing the odd charge sector of the original model requires locally altering the duality mapping, which in turn modifies the boundary condition of the dual model from periodic to antiperiodic.As such, it is not possible to define a mapping of local operators without addressing the mapping of sectors.
The purpose of the present manuscript is to completely address the fate of sectors upon dualizing for the case of closed boundary conditions.Here, closed boundary conditions include the familiar case of periodic boundary conditions, but more generally also contains symmetrytwisted boundary conditions [76,77].These boundary conditions are special in the sense that while they do break invariance with respect to the original translation operator, they do so in a way allowing us to define a twisted translation operator, together with corresponding twisted momentum, with respect to which invariance is preserved.For these kinds of boundary conditions, sectors are labelled by combinations of fluxes given by symmetry twists and symmetry charges that decompose the remaining twisted symmetry.Borrowing terminology from the study of topological order, we refer to these super-selection sectors labelled by symmetrytwisted boundary conditions and twisted symmetry sectors as topological sectors [27,28,42,78].
Recently there has been renewed interest in the physical realization of duality transformations and their applications in quantum technologies.In the context of quantum simulation, the fact that dualities typically change the phase of the states on which they act can be exploited to efficiently prepare states in a given phase [79][80][81][82].A prototypical example is the case of the Kramers-Wannier duality, where the duality MPO generates a long-range entangled GHZ state from a trivially entangled product state.The same duality transformations can also be used to generate quantum circuits that permute the anyons of a topologically ordered state [83], which has applications in the construction of topological quantum memories [84].Both of these applications ultimately rely on an understanding of the non-trivial action of a duality on the topological sectors of a model, as well as the explicit realization of operators that implement these transformations.The framework we present in this manuscript provides both of these features, and we expect the duality operators we construct here to guide the physical realization of more general duality transformations.
We will make use of several categorical concepts, which for convenience are summarized in tab.I together with their physical meaning in this work.Concretely, our construction goes as follows: We begin by choosing an input fusion category D, from which we can construct an (abstract) algebra of local operators referred to as the bond algebra [59,85].At this stage, these local operators do not yet admit an explicit matrix representation on a Hilbert space.Instead, they are written in terms of string diagrams, which allows us to compute their operator products using the diagrammatic manipulations of the input category.A particular equivalence class of dual models is then built by taking certain linear com-binations of such local operators.The choice of these operators completely determines the spectrum of these models, and in this sense captures the physical properties that can be directly inferred from the spectrum.The next step is to find explicit matrix representations of the local operators that build up these models, which then specify the Hilbert space and the Hamiltonian.These are classified by different choices of module categories M over the input category D. The choice of module category therefore provides a particular physical realization of the physical properties captured in the spectrum associated with an equivalence class of dual models.
Ignoring boundary conditions by considering infinite chains, one can construct operators that intertwine between dual representations of the local operators determined by different choices of (indecomposable) module categories M and N .Explicitly, such an operator can be written in terms of an MPO intertwiner that acts as a map between module categories; the consistency conditions on this MPO intertwiner are equivalent to those of a so-called D-module functor M → N in Fun D (M, N ).In the special case where the module categories are the same on both sides, we interpret the MPO as a symmetry, which is labeled by a D-module endofunctor in Fun D (M, M).Importantly, the composition of D-module endofunctors endows the category of endofunctors with the structure of a fusion category.This fusion category is denoted as D ⋆ M and referred to as the Morita dual of D with respect to M; it describes the symmetries of the Hamiltonian associated with the specific choice of module category M.
As alluded to above, in order to realize these dualities as explicit isometries one needs to carefully consider the boundary conditions.In our setup, the symmetrytwisted boundary conditions discussed above are given by an endofunctor of the module category, which tells us the way in which the degrees of freedom at either end of the chain have to be glued together.The additional condition that translation invariance is preserved up to a local unitary transformation is satisfied by requiring this endofunctor to possess a D-module structure.As expected , these are organized into the same fusion category D ⋆ M that describes the symmetries, hence the name.The symmetries of a model interact with these boundary conditions and typically one is left with less symmetry in the presence of a non-trivial symmetry twist.In general, this can be understood using 'tubes', which are modifications of the usual periodic MPOs to include the action on the symmetry twist.The topological sectors of the model are then characterized as irreducible representations of the category of tubes, which are well known to be in correspondence with simple objects in the Drinfel'd center Z(D ⋆ M ) [6,27,42,86,87].We confirm this directly on the lattice and explicitly compute the projectors onto the various topological sectors.
Importantly, the symmetry fusion categories D ⋆ M for different choices of M are all Morita equivalent [88,89].It is well known that this guarantees that the centers Z(D ⋆ M ) are equivalent as categories, which guarantees that the topological sectors of dual models can be mapped to one another.By generalizing the MPO intertwiners constructed for the infinite case to accommodate the twisted boundary conditions, we can explicitly compute these maps and construct the isometries that relate the full set of eigenvectors of one model to those of any of its dual models.
We illustrate our construction with a family of dual models whose fusion categories of symmetry operators are in the Morita class of the fusion category Rep(S 3 ) of representations of the symmetric group S 3 .We distinguish four models associated with the four indecomposable module categories over Rep(S 3 ).One of them is the spin- 1  2 XXZ Heisenberg model.In the infinite case, the Hamiltonian commutes in particular with non-local operators labelled by group elements in S 3 .Similarly, equivalence classes of boundary conditions that preserves the translation invariance of the model are shown to be labelled by conjugacy classes of S 3 .Given a conjugacy class and one of its representatives, the corresponding boundary condition is such that the resulting Hamiltonian only commutes with operators labelled by group elements in the centralizer of the representative.It follows that this Hamiltonian decomposes into twisted symmetry sectors indexed by irreducible representations of the centralizer.Putting everything together, we find eight topological sectors, which are in one-to-one correspondence with simple objects in the Drinfel'd center Z(Vec S3 ) of the fusion category Vec S3 of S 3 -graded vector spaces, or equivalently irreducible representations of the quantum double D(S 3 ).These simple objects also encode elementary anyonic excitations in Hamiltonian realizations of the S 3 Dijkgraaf-Witten theory.As evoked above, this is no mere coincidence and confirms the fact that the spin- By construction, Hamiltonian models that only differ from the spin- 1  2 XXZ Heisenberg model in the choice of module category over Rep(S 3 ) are dual to it, and also dual to one another.We explore these dual models within our framework.We highlight in particular a model whose Hamiltonian commutes in the infinite case with non-local symmetry operators labelled by irreducible representa-tion of S 3 , as opposed to group elements.Correspondingly, boundary conditions are also labelled by irreducible representations.We explicitly construct the boundary terms as well as the lattice symmetry operators that preserve the boundary conditions.The symmetry charge sectors decomposing the corresponding Hamiltonians are presented in detail.As for the XXZ Heisenberg model, we find a total of eight topological sectors, which are now in one-to-one correspondence with simple objects of the Drinfel'd center Z(Rep(S 3 )).Duality is then guaranteed by the Morita equivalence between Vec S3 and Rep(S 3 ).Furthermore, we study in detail how topological sectors of this model are mapped to those of the XXZ Heisenberg model upon duality.We show in particular how the two-dimensional charge sector of the XXZ Heisenberg model with periodic boundary conditions is mapped onto the trivial charge sector of the dual model with a nonabelian boundary condition.This is a concrete physical realization of the non-trivial permutation of anyons taking place at the only non-trivial invertible domain wall between topological orders described by Z(Vec S3 ) and Z(Rep(S 3 )), respectively.
The manuscript is organized as follows: We begin by reviewing in sec.II the case of infinite chains with an emphasis on the lattice duality operators transmuting local symmetric operators into one another.This section is also the opportunity to introduce the relevant technical preliminaries together with the graphical calculus that is used throughout our work.The characterization of topological sectors is presented in sec.III together with the operators mapping sectors of dual models onto one another, as well as the isometries realizing the dualities between the Hamiltonians.Finally, we present sec.IV several examples that illustrate the various results obtained in this work.Our manuscript is complemented by app.A and app.B that compile numerous results regarding categorical Morita equivalence and the quantum double of a finite group, respectively.

Sec. II | Infinite chains
After introducing some technical definitions, we review in this section our systematic and constructive approach to symmetry and duality operators for infinite onedimensional lattice models.

II.A. Technical preliminaries
We consider in this manuscript families of onedimensional quantum lattice models that are dual to one another.These dual models are characterized by distinct lattice realizations of a (categorical) symmetry generated by operators that are organized into a (spherical) fusion category.We like to think of choosing a fusion category as picking a backbone that supports the various theories.Lattice realizations are then obtained by choosing collections of physical degrees of freedom that are compatible with the backbone.Mathematically, a choice of lattice realization is associated with a module category over the input fusion category.In this context, a map between distinct module categories amounts to a duality operator, whereas a map from a module category to itself amounts to a symmetry operator.This is the program that was presented in ref. [58] in the case of infinite one-dimensional lattices.The purpose of the present manuscript is to continue this systematic study and explain the subtle interplay between duality relations, boundary conditions and topological sectors.Let us first briefly review the infinite case.
We set the stage by introducing the technical ingredients alluded to above.We encourage the reader to consult ref. [3] for details.Succinctly, a fusion category D encodes a collection of objects interpreted as (possibly non-elementary) topological charges that can be fused to one another.Throughout this manuscript, we notate via i⟩ for the vector space of maps from the (typically not simple) object It follows that the monoidal associator boils down to a collection of complex matrices whose actions can be conveniently depicted in terms of string diagrams as where the indices i, j, k, l label basis vectors in the homspaces H Y5 Y1Y2 , H Y4 Y1Y6 , H Y4 Y5Y3 and H Y6 Y2Y3 , respectively.Henceforth, we refer to the entries of these complex matrices as the F -symbols.
Given a fusion category D, a module category M over it is roughly speaking a collection of representatives of simple objects M 1 , M 2 , . . .∈ I M that are acted upon by the objects in D.More concretely, a right module category M over D is a triple (M, ◁, ◁ F ) that consists of a category M, an action ◁ : M × D → M and a (natural satisfying a 'pentagon axiom' involving the monoidal associator F of D. The isomorphism ◁ F shall be referred to as the (right) module associator and its components written as For instance, every fusion category D has the structure of a module category over itself via its monoidal structure known as the regular module category, whereby the module associator ◁ F is provided by the monoidal associator F .Introducing the notation , i⟩ for the vector space of maps from the (typically not simple) object It follows that the module associator boils down to a collection of complex matrices whose actions can be depicted in terms of string diagrams as where the indices i, j, k, l label basis vectors in the homspaces M3Y2 and H Y3 Y1Y2 , respectively.Henceforth, we refer to the entries of these complex matrices as ◁ F -symbols.
Mirroring the concepts above, we can define a notion of left module category (M, ▷, ▷ F ). Combining the notions of left and right module category then yields the concept of bimodule category.Concretely, a (C, D)bimodule category is a sextuple (M, ▷, ◁, ▷ F, ◁ F, ▷◁ F ) such that the triples (M, ▷, ▷ F ) and (M, ◁, ◁ F ) define left Cand right D-module categories, respectively, and where isomorphism satisfying two 'pentagon axioms' involving either ◁ F or ▷ F .The isomorphism ▷◁ F shall be referred to as the bimodule associator and its components written as As before, the bimodule associator boils down to a collection of complex matrices whose actions can be depicted in terms of string diagrams where the indices i, j, k, l label basis vectors in the homspaces V M3 XM1 , V M2 XM4 , V M2 M3Y and V M4 M1Y , respectively.Henceforth, we refer to the entries of these complex matrices as ▷◁ F -symbols.
The final ingredient we require is a notion of structurepreserving map between module categories over the same fusion category.Given two (right) module D-categories (M, ◁, ◁ F ) and (N , • ◁, • ◁ F ), we define a D-module functor between them as a pair (F, ω) that consists of a functor F : M → N and a (natural) isomorphism ω : ◁ − satisfying a 'pentagon axiom' involving both ◁ F and • ◁ F .Components of the isomorphism ω shall be written as ◁ Y for all Y ∈ I D and M 1 ∈ I M , and boil to a collection of complex matrices where V N M := Hom N (F(M ), N ).Such module functors form a category denoted by Fun D (M, N ).Throughout this manuscript, we notate via X 1 , X 2 , . . .representatives of isomorphism classes of simple objects in such categories of module functors.Moreover, we shall typically refer to a given module functor as an object X in the corresponding category, in which case the actual functor shall be denoted by X F and the module structure by X ω.In other words, we employ the shorthand notation X ≡ ( X F, X ω).Keeping this convention in mind, the action of the matrices (7) can also be depicted in terms of string diagrams as where the indices i, j, k, l label basis vectors in the homspaces V N1 M1 , V N2 M2 , V N2 N1Y , and V M2 M1Y , respectively.Henceforth, we refer to the entries of these complex matrices as X ω-symbols.Notice that we are performing a slight abuse of notation so as to have string diagrams akin to those associated with bimodule associators.The reason is the following: We shall often consider D-module endofunctors in categories D ⋆ M := Fun D (M, M).But every category D ⋆ M can be equipped with a monoidal structure via the composition of module functors.As a matter of fact, since we shall focus on cases where M is indecomposable, D ⋆ M even has the structure of a fusion category [3].Moreover, the category M is naturally endowed with the structure of a left module category over D ⋆ M .Indeed, notice that every D-module functor in  M, N ) can be composed so as to yield module functors in Fun D (M, O).Given representatives X 1 and X 2 of isomorphism classes of simple objects in Fun D (N , O) and Fun D (M, N ), respectively, composition yields a new functor X1 F( X2 F(−)) in Fun D (M, O) with the obvious module structure.Crucially, even though X 1 and X 2 are simple, the composite of the corresponding module functors is typically not a simple object in Fun D (M, O).Therefore, there exist a collection of complex matrices whose entries can be defined graphically following the convention of eq. ( 8) as where the indices i, j, k, l label basis vectors in the homspaces M and V N M , respectively.Henceforth, we refer to the entries of these complex matrices as F F -symbols.All the symbols introduced so far fulfill consistency conditions descending from the pentagon axioms satisfied by the corresponding isomorphisms.Writing an analogous consistency condition for the composition of module functors in terms of these F F -symbols would require the introduction of yet another set of symbols referred to as • F -symbols, thereby defining these new symbols in terms of F F -symbols.With our notations, we would have so that • F -symbols encode the associativity condition for the composition of module functors.But such a consistency condition may not exist in general.In other words, there may be an obstruction for the composition of module functors to be associative up to an isomorphism [90].That being said, such obstructions do not occur within our construction due to the specific compositions of module functors considered.In addition to this pentagon axiom defining the associativity of the the composition of module functors, note that the F F -symbols are involved in a coherence relation involving − ω-symbols (see eq. ( 19) below).
We defined in this section seven sets of symbols, namely F -, ▷ F -, ◁ F -, ▷◁ F -, − ω-, F F -and • F -symbols.Note that, in general, there are neither explicit formulas nor direct ways to compute these various symbols.Rather, they are obtained by solving the consistency conditions they are required to satisfy.This implies in particular that these symbols are typically defined up to basis choices for the various hom-spaces.The results presented in this manuscript hold regardless of these choices, and as such we shall implicitly choose them so as to simplify the values of the symbols considered. 1 Besides, it is not necessary to solve for all these symbols individually.Indeed, it is possible to deduce the ▷◁ F -, ▷ F -, − ω-, F F -and • Fsymbols from the knowledge of the F -and ◁ F -symbols for every possible choice of indecomposable (right) module category over D. We include this data for the example considered below where D = Rep(S 3 ) as supplementary material.

II.B. Tensor networks
A key aspect of our construction is the use of tensor networks [73], as a way to parameterize lattice models as well as their symmetry and duality operators.As we shall emphasize, this language is not only very natural but quickly becomes necessary as we consider non-elementary models, and translating the tensor networks into more explicit or familiar objects often turns out be a tedious exercise.We hope that the new results presented in sec.IV will convince the most skeptical readers of the benefits of this approach.
The types of tensor networks we consider build upon the graphical calculus of string diagrams, which was briefly employed above.First of all, the ◁ F -symbols associated with a (right) D-module category M can be de- 1 For instance, we work in a basis where evaluates to the identity matrix whenever Y 1 , Y 2 or Y 3 is the unit object; we make similar choices for the other symbols in this manuscript.

picted as
where we should think of the first diagram as the pasting of the string diagrams appearing on the l.h.s. and r.h.s. of eq. ( 8) into a tetrahedron.By convention, ◁ Fsymbols for which the fusion rules are not everywhere satisfied vanish.We can now construct tensors whose non-vanishing entries are provided by these ◁ F -symbols.We shall do so graphically.Let us first introduce a couple of graphical conventions.Unlabelled purple strings denote the following morphism: where the relevant module category will always be clear from the context.In the same spirit, unlabelled gray patches denote the following formal vectors: Putting these graphical conventions together, we define the following collection of tensors labelled by simple objects in D and a choice of basis vector: whose entries are provided by ◁ F -symbols.As we shall recall below, these tensors play a crucial role within our framework as they can be exploited to generate algebras of (categorically) symmetric operators.For this reason, we refer to them as symmetric tensors for the remainder of this manuscript.
Let us now introduce another type of tensors that evaluate to the module structure of functors between Dmodule categories.Let us first provide a graphical representation for the entries of the matrices specifying the module structure X ω of a D-module functor X F associated with an object X in Fun D (M, N ): , where we should think of the diagram as the pasting of the string diagrams appearing on the l.h.s. and r.h.s. of eq. ( 8).Adapting the conventions introduced above, we define the following collection of tensors labelled by simple objects in D and Fun D (M, N ): As we mentioned above, in the case of module endofunctors in D ⋆ M , we can think of the corresponding tensors as evaluating to the ▷◁ F -symbols of M as a (D ⋆ M , D)bimodule category.Contracting two such tensors whose labelling objects match is accomplished by concatenating them, horizontally or vertically, identifying the objects labelling the module strings, and tracing over the basis vectors along which the contraction takes place.Given our diagrammatic conventions, we have for instance .
Tensor networks of this kind are referred to as Matrix Product Operators (MPOs).Finally, we require another family of tensors whose non-vanishing entries evaluate to the F F -symbols associated with a triple (M, N , O) or right D-module categories.Specifically, we define , and the corresponding tensor is constructed following the same steps as for symmetric tensors.Tensors of this kind shall be referred to as fusion tensors, as they locally implement the fusion of MPOs.
We mentioned above that the various isomorphisms entering the definitions of module categories and module functors must satisfy some 'pentagon axioms' ensuring the self-consistency of the constructions.In terms of tensors defined in this section, these axioms translate into 'pulling-through conditions', whereby MPOs built out of tensors of the form ( 16) are pulled through symmetric tensors of the form (14). Concretely, the pentagon axiom fulfilled by the module structure of a D-module functor which is true for any labelling of the various strings and basis vectors.Notice that as we pull the MPO encoding the D-module functor X through the symmetric tensor that evaluates to the ◁ F -symbols of M, it transforms into a new symmetric tensor that evaluates to the • ◁ F -symbols of N .Making specific choices for the D-module categories M and N yields the pulling-through conditions associated with all the other pentagon axioms mentioned so far.Specializing to the case M = N , we find the pentagon axiom satisfied by the bimodule associator ▷◁ F and the (right) module associator ◁ F .Choosing M = D and N arbitrary yields the pentagon axiom fulfilled by the module associator ◁ F that involves the monoidal associator F .Finally, when M = N = D, the pulling-through condition amounts to the pentagon axiom satisfied by the monoidal associator F .As we shall review below, these various pulling-through conditions encode the action of symmetry and duality operators.
Another family of tensor network relations will play a crucial role in the following.These encode the composition rule of module functors.We mentioned earlier that given two representatives X 1 and X 2 of isomorphism classes of simple objects in categories Fun D (M, N ) and Fun D (N , O), respectively, these could be composed so as to yield a D-module functor between M and O that decomposes into simple objects of Fun D (M, O).Graphically, this translates into the composition of the corresponding MPO tensors by means the fusion tensors that evaluate to the F F -symbols introduced previously: which is true for any labeling of the various strings and basis vectors.Specializing to the case M = N , the composition of D-module functors provides the monoidal structure of D ⋆ M so that the fusion tensor evaluates to the ▷ F -symbols of the left D ⋆ M -module category M. The diagrammatic relation above then provides the fusion of the corresponding MPOs symmetries.

II.C. Symmetric Hamiltonians and dualities
Let us now put together the ingredients presented earlier into a recipe for constructing symmetric Hamiltonians on infinite one-dimensional lattices, and duality relations between them [58].First we need to pick a microscopic Hilbert space.Let D be a fusion category and M a (right) indecomposable module category over it.We consider the following C-linear span over {M ∈ I M }, {Y ∈ I D } and basis vectors {i} in the hom-spaces defined following the convention of eq. ( 13).
Notice that this Hilbert space is typically not a tensor product of local Hilbert spaces.We then define local operators acting on this microscopic Hilbert space via matrix multiplication of the form obtained by taking linear combinations of contractions of two symmetric tensors as defined in eq.(11).Any combinations of such local operators can then be organized into a local Hamiltonian Notice that the definition of the local operators fixes certain combinations of objects and morphisms in D, thereby imposing kinematical constraints on the genuine physical degrees of freedom of the model, which are provided by object and morphisms in M.This implies in particular that we can often consider a subspace of the microscopic Hilbert defined above when dealing with a specific Hamiltonian.Crucially, any such Hamiltonian H M remains invariant under the action of symmetry operators.Indeed, it follows immediately from the pulling-through condition eq. ( 18) that the MPO symmetry ) commutes with the Hamiltonian H M .Remark that this symmetry condition relies solely on the local operators being constructed out of symmetric tensors evaluating to the ◁ F -symbols of M, and is oblivious to the specific definitions of these local operators, i.e. choices of objects and morphisms in D as well as complex coefficients b n,i .Importantly, for any D-module category M that we consider in this manuscript, the fusion category D ⋆ M is found to be Morita equivalent to D (see app.A).The physical implications of this mathematical result are discussed in the next section.
If symmetry operators are labelled by objects in D ⋆ M , duality operators are labelled by objects in Fun D (M, N ) with N a D-module category distinct from M. These duality operators have exactly the same form as the symmetry operators (22) with the difference that the top purple strings are now labelled by objects in N and the tensor evaluates to X ω-symbols, i.e. entries of the matrices specifying the module structure X ω of the module functor X F corresponding to the simple object X in Fun D (M, N ).Denoting by T X M|N such an intertwining MPO, from the pulling-through conditions (18) follows the commutation relation where and b N i,n defined exactly as in eq. ( 21) but with respect to N .It follows immediately from our construction that the Hamiltonian H N thus constructed remains invariant under the action of symmetry operators labelled by objects in the Morita dual D ⋆ N of D with respect to N .
The duality operator performs the transmutations of the local operators defining the Hamiltonians H M and H N .However the knowledge of this operator is not quite sufficient to write down a set of isometric transformations mapping models to one another.Obtaining such transformations indeed requires an analysis of the topological sectors of the corresponding models.This is the main teaching of this manuscript and the purpose of the following sections.

II.D. Illustration
Before concluding this section, let us consider an illustrative example, namely the transverse-field Ising model.We encourage the reader to consult ref. [58] for additional examples.Let D be the fusion category Vec Z2 of Z 2 -graded vector spaces.This fusion category has two simple objects, which we denote by 1 and m.The fusion rules read Given any Vec Z2 -module category M, we consider the Hamiltonian with local operators given by By definition, only hom-spaces for which the corresponding objects satisfy the fusion rules are non-vanishing, which is the case of all the hom-spaces appearing above.Since the outcome of the fusion of two objects is uniquely determined, hom-spaces are necessarily one-dimensional and we labelled by 1 the corresponding unique basis vectors.Let us now choose specific Vec Z2 -module categories.Let M = Vec Z2 be the regular module category.Recall that in this case the module associator ◁ F boils down to the monoidal associator F of Vec Z2 , which happens to be trivial.It means that the ◁ F -symbols equal 1 whenever all the fusion rules are satisfied, and 0 otherwise.Consider the microscopic Hilbert space (20).It follows from the fusion rules that objects in I D are fully determined by a choice of objects in I M .This means that the physical degrees of freedom are labelled by objects in I M and located in the 'middles' of the corresponding strings so that the effective microscopic Hilbert spaces is isomorphic to The operator b M i,1 acts on this Hilbert space as |1/m⟩ → |m/1⟩ on the site i, where we identify |1⟩ and |m⟩ with the +1 and −1 eigenvectors of the Pauli S z operator, respectively.The operator b M i,2 acts as the identity operator whenever the degrees of freedom at sites i and i + 1 agree, and minus the identity operator otherwise.Putting everything together, we find that the Hamiltonian (24) boils down to which we recognize as the transverse-field Ising model.This model has a (global) Z 2 symmetry generated by ś i S x i .The fusion category Vec Z2 admits another (indecomposable) module category over it, namely the category M = Vec of vector spaces.This category has a unique object, which we denote by 1, and the module structure is provided by 1 ◁ 1 ≃ 1 ≃ 1 ◁ m.As for the previous case, the module associator is trivial.For this example, the physical degrees of freedom are identified with the unique basis vectors of the hom-spaces Hom M (1 , m}, and thus the effective Hilbert space is still isomorphic to Â i C 2 .It readily follows from the definition of the local operators that the Hamiltonian (24) now boils down to which we recognize as the Kramers-Wannier dual of the transverse-field Ising model.This model also has a Z 2 symmetry, which is now generated by . Within our framework, obtaining two models that are dual to one another is guaranteed by the fact that they only differ by a choice of Vec Z2 -module category.This implies that altering the definitions of the local operators b M i,1 and b M i,2 would still yield two dual models, and the resulting models would always be invariant under the action of operators labelled by objects in the fusion categories (Vec Z2 ) ⋆ respectively, which are Morita equivalent [58].
Let us now explicitly construct the duality operator transmuting the Hamiltonians H Vec Z 2 and H Vec into one another.Applying the recipe presented above, this duality operator can be written as an MPO labelled by the unique simple object in Fun Vec Z 2 (Vec Z2 , Vec) ∼ = Vec such that the individual tensors evaluate to the module structure of the corresponding functor Vec Z2 → Vec, which happens to be trivial.Graphically, it reads where dotted strings are labelled by the unique simple object Vec that we have been notating via 1.This duality operator should be interpreted as a map from any symmetric operator associated with Vec Z2 to a symmetric operator associated with Vec.Concretely, it follows from the fusion rules in Vec Z2 that this operator maps states , as expected.

Sec. III | Topological sectors and dualities
Building upon the constructions above, we introduce in this section the notion of twisted boundary condition and present a characterization of topological sectors.We subsequently explain a method to compute the mapping of topological sectors under a duality relation.

III.A. Boundary conditions and tube category
Given an input fusion category D, we reviewed in the previous section a recipe to construct dual local operators associated with choices of module categories over D. These operators are invariant under the action of symmetry operators labelled by objects in fusion categories that are Morita equivalent.Duality operators can then be constructed from the data of module functors between module categories [58].However, this is not enough in order to fully establish a duality relation between Hamiltonian models.Indeed, it is further required to establish how the mappings of symmetric operators interact with the topological sectors of the models.Let us consider a (finite) spin chain of length L + 1 with total Hilbert space H M given by over {M ∈ I M }, {Y ∈ I D } and basis vectors {i} in the hom-spaces defined following the convention of eq. ( 13).
Notice that we have left the boundary condition unspecified.Loosely speaking, choosing a boundary condition amounts to picking a relation, or a map, between degrees of freedom at sites L + 1 and 1 so as to close the chain.For instance, periodic boundary conditions would be obtained by enforcing M L+1 = M 1 .Within our formalism, such a map between degrees of freedom is provided by an endofunctor of M in Fun(M, M).But the corresponding (possibly twisted) boundary conditions should not break translation invariance of a Hamiltonian acting on this Hilbert space.In other words, given a choice of boundary condition, there should still be an isomorphism between vector spaces related by a translation by one site.This requires the action of the endofunctor to commute with the module action of D on M up to a natural isomorphism, i.e. it must be equipped with a D-module structure.Therefore, we consider boundary conditions classified by D ⋆ M = Fun D (M, M).This is also the fusion category that describes the symmetry of the model, and as such these boundary conditions are referred to as symmetry-twisted boundary conditions.
As mentioned in sec.II A, the category D ⋆ M is equipped with a monoidal structure provided by the composition of D-module endofunctors.We further commented that M is naturally endowed with the structure of a left module category over D ⋆ M , and as such we can employ the same graphical calculus for objects in D ⋆ M as that for objects in D. Specifically, we consider microscopic Hilbert spaces over {M ∈ I M }, {Y ∈ I D } and basis vectors {i} in the hom-spaces defined following the convention of eq. ( 13), and where {A ∈ I D ⋆ M } are representatives of isomorphism classes of simple objects in D ⋆ M encoding choices of boundary conditions.A couple of important remarks: Firstly, boundary conditions are promoted within our approach to genuine degrees of freedom of the model, which implies in particular that they can be acted upon.Secondly, given an arbitrary boundary condition, the effective number of sites may not be L-as would be the case for periodic boundary conditions for instance-but rather L + 1.This may seem somewhat paradoxical, but as we shall see this is a characteristic feature of non-abelian boundary conditions.
Given a choice of D-module category M, let us study the decomposition of the Hilbert space H M into superselection sectors of the symmetry.In order to perform such a decomposition, we consider tensor networks T A,A ′ ,X,X ′ ,k,k ′ M|M that describe the action of the symmetry in the presence of twisted boundary conditions.These are of the form where A, A ′ , X, X ′ ∈ I D ⋆ M , while k, k ′ label basis vectors in the hom-spaces Hom D ⋆ M (X ′ , A ′ bX) and Hom D ⋆ M (X b A, X ′ ), respectively.We distinguish two types of tensors in this expression.On the one hand, we have the same MPO tensors as in eq. ( 22), which evaluates to the ▷◁ Fsymbols of M as a (D ⋆ M , D)-bimodule category.On the other hand we have two fusion tensors of the form (11) that evaluates to the ▷ F -symbols of M as left module category over D ⋆ M .Given the geometry of such tensor networks, we will henceforth refer to them as 'tubes'.It follows from our graphical calculus that these symmetry tubes can be interpreted as linear maps H M → H M .Let us now demonstrate that the subspace of linear maps spanned by the symmetry tubes is closed under multiplication.Graphically, multiplication of symmetry tubes is obtained by stacking them on top of one another and contracting the indices along which the stacking takes place, i.e.
, where we are using the convention defined in eq. ( 12).Notice that this contraction is accompanied with the identifications of objects in D as well as in D ⋆ M .At this point, we can use the fusion of MPO tensors defined in eq. ( 19) together with the recoupling theory of fusion tensors to express this stacking as a complex linear combination of tubes.Recoupling fusion tensors amounts to changing the contraction patterns of a given collection of tensors, which is rendered possible due to the coherence relations satisfied by the ◁ F -symbols these tensors evaluate to [22].Concretely, the fusion tensors satisfy graphical identities of the form of eq. ( 2), where the Fsymbols would be that of D ⋆ M .These recoupling moves can be explicitly found in ref. [58].Putting everything together, we obtain the following multiplication rule: where it follows from the definition of the F -symbols that in particular the first sum is over simple objects X 3 appearing in the decomposition of the monoidal product We exploit the results obtained above to introduce the tube category Tube(D ⋆ M ), whose objects are objects A, A ′ in D ⋆ M and hom-spaces Hom Tube(D ⋆ M ) (A, A ′ ) are vector spaces spanned by tubes T A,A ′ ,−,−,−,− M|M as defined previously [91,92].The composition rule is then provided by the multiplication rule (32).It follows that the Hilbert space H M defines a representation in Fun(Tube(D ⋆ M ), Vec), and thus admits a decomposition into superselection sectors labelled by irreducible representations V of Tube(D ⋆ M ).Henceforth, we refer to these superselection sectors as topological sectors.Crucially, there is a well-known equivalence between the category of representations of the tube category and the Drinfel'd center Z(D ⋆ M ) [88,89,[93][94][95] (see app.A 2),3 so that topological sectors of H M can be labelled by simple objects Z in Z(D ⋆ M ).As a matter of fact simple objects in Z(D ⋆ M ) are often obtained by computing the minimal idempotent tubes w.r.t. the multiplication defined in eq.(32).Such a simple object encodes a (possibly not simple) twisted boundary condition as well as a symmetry charge that decomposes the action of the tubes leaving the boundary condition invariant.
By considering the space of all tubes and introduce the convention that tubes with mismatching objects multiply to zero, we can consider the tube algebra of all tubes.In addition to being closed under multiplication, the tube algebra is closed under Hermitian conjugation according to This closure under Hermitian conjugation equips the tube algebra with the structure of a * -algebra.In virtue of its finiteness, it follows that the tube algebra is block diagonal in the topological sectors so we can define a new basis where Z labels a simple object of Z(D ⋆ M ), and A i ≡ (A, i) runs over all simple objects A of D ⋆ M decomposing Z as an object of D ⋆ M as well as the corresponding degeneracy labels i.The normalization factor is chosen to be # Z := ℓ(Z)/FPdim(D), where ℓ(Z) corresponds to the number of simple objects appearing in the decomposition of Z and FPdim(D) is the Frobenius-Perron dimension of D [3].These new basis elements behave like matrix units under the tube multiplication, i.e. they diagonalize the multiplication: where 1 M denotes the identity operator on H M .This means that for A i = A ′ j they define a complete set of idempotents that project onto states within a certain topological sector Z with boundary condition A and degeneracy i.Consequently, for A i ̸ = A ′ j they define isometries, which, within a topological sector Z, map between states with boundary condition A with degeneracy i and states with boundary condition A ′ and degeneracy j.
where in addition to symmetric tensors evaluating to the ◁ F -symbols of M, we now require an MPO tensor evaluating to the ▷◁ F -symbols of M as a (D ⋆ M , D)-bimodule category.Notice that such a definition allows for very general types of boundary conditions.It immediately follows from the pulling-through conditions translating the pentagon axioms of the bimodule associator ▷◁ F of M involving the right module associator ◁ F of M on the one hand, and that involving the left module associator ▷ F of M as a module category over D ⋆ M on the other hand, that these local operators can be pulled through the tubes (31) defined previously.By considering arbitrary linear combinations of local operators ( 21) and ( 37), we construct families of Hamiltonians associated with boundary condition A: We have already established that away from the boundary, the local operators commute with the tubes.An arbitrary tube would however modify the boundary condition provided by A: This means that only a subset of tubes of the form T A,A,−,−,−,− M|M would leave the boundary condition invariant and as such commute with the Hamiltonian H M,A .Practically, this means that in general the symmetry operators leaving H M,A invariant are not organized into D ⋆ M , as is the case in the infinite chain scenario.The symmetry charge sectors decomposing the action of these tubes commuting with H M,A then provide topological sectors that are in one-to-one correspondence with simple objects of Z(D ⋆ M ).Note that in general a given simple object of Z(D ⋆ M ) is not necessarily associated with a simple boundary condition, and conversely, the same topological sector can be found in the decomposition of Hamiltonians with boundary conditions provided by different simple objects in D ⋆ M .As alluded to above, this is the statement that, as an object in D ⋆ M , a simple object Z in Z(D ⋆ M ) decomposes over simple objects in D ⋆ M .We shall provide concrete examples of these statements in sec.IV.
Let us refine the statements above using the matrix unit basis considered above.A given Hamiltonian H M,A can be decomposed into topological sectors as where the sum is over all topological sectors Z, such that A appears in the decomposition of Z as an object of D ⋆ M , and degeneracy labels i.The Hamiltonians H M,A Z,i can be thought of as the elementary building blocks of a Hamiltonian with a given boundary condition.Importantly, all these elementary Hamiltonians within a given topological sector Z have the same spectrum since they can be related by the isometry As pointed out above, this implies that a given topological sector Z can be found in Hamiltonians H M,A for different choices of A, as long as the boundary condition A appears in the decomposition of Z into simple objects of D ⋆ M .

III.C. Intertwining tubes and dualities
Given an input fusion category D and a pair (M, N ) of (right) D-module categories, we explained in sec.II C how MPOs In order to account for boundary conditions, the intertwining MPOs (22) need to be promoted to intertwining where and X, X ′ represent isomorphism classes of simple objects in Fun D (M, N ), while k, k ′ label basis vectors in the hom-spaces Hom Fun D (M,N ) (X ′ , B ▷ X) and Hom Fun D (M,N ) (X ◁ A, X ′ ).
As before, we distinguish two types of tensors entering the definition of these intertwining tubes.One the one hand, we have MPO tensors evaluating to X ω-symbols.On the other hand, we have two fusion tensors of the form (17) that evaluates to the F F -symbols associated with the triples (M, M, N ) and (M, N , N ) of right D-module categories, respectively.Note that in virtue of the composition of D-module functors, Fun D (M, N ) is equipped with a (D ⋆ N , D ⋆ M )-bimodule structure, hence the definition of the hom-spaces above.
It follows from the various pulling-through conditions that descend from the coherence axioms involving module associators, bimodule associators, module functors and composition of bimodule functors that these intertwining tubes can be pulled through local operators of the form (37).This operation gives rise to commutation relations of the form Note however that there is no guarantee that there will exist non-vanishing intertwining tubes associated with any pair (A, B) of boundary conditions.Indeed, not every boundary condition of a model associated with the D-module category N is compatible with a given boundary condition of a dual model associated with the Dmodule category M. In order to obtain the mapping of topological sectors associated with the duality provided by D-module functors in Fun D (M, N ) further require to project H M,A and H N ,B onto specific symmetry charges.This is done via the symmetry tube idempotents with respect to the multiplication rule (32).Given an intertwining tube associated with a pair (A, B) of boundary conditions, if acting on both side with such charge projectors yields a non-trivial tensor, then the corresponding topological sectors are mapped onto one another by the duality.Repeating this process for every combination of boundary conditions and the corresponding symmetry charges provides the mapping of all the topological sectors realizing an equivalence . Mathematically, the existence of such an equivalence is guaranteed by the fact that D ⋆ M and D ⋆ N are Morita equivalent (see app.A).
Similar to the symmetry tubes, one can define the multiplication of intertwining tubes via stacking, and use the recoupling theory of the fusion tensors given by the • Fsymbols to express the result as a linear combination of new intertwining tubes. 4The computation parallels that of the multiplication of symmetry tubes, yielding Additionally, the Hermitian conjugate of an intertwining tube T M|N can be expressed as a linear combination of the opposite intertwining tubes T N |M : Putting together symmetry tubes and intertwining tubes, we can construct a finite * -algebra spanned by tubes {T M|M , T M|N , T N |M , T N |N } where incompatible tube multiplications such as T M|M • T N |N are defined to be zero [83,96].As before, this * -algebra is block diagonal in the topological sectors and can be decomposed into matrix units.The decomposition of the * -subalgebras spanned by the symmetry tubes T M|M and T N |N is unchanged, while we also have where Z, A i are defined as before, Z denotes a simple object of Z(D ⋆ N ) in the image of the topological sector Z under the duality, and B j ≡ (B, j) refers to the simple object B of D ⋆ N appearing in the decomposition into simple objects of Z as an object of D ⋆ N with degeneracy label i.The normalization factor is now chosen to be # Z, Z := (ℓ(Z)ℓ( Z)) Using these matrix units, one can construct explicit isometries that relate a given Hamiltonian H M,A Z,i in the topological sector Z with degeneracy i to a dual Hamiltonian H N ,B Z,j in the dual topological sector Z with degeneracy j: thereby demonstrating that duality transformations do preserve the spectrum.

III.D. Illustration
Let us illustrate the concepts presented in this section with the case of the transverse-field Ising model.Starting from the fusion category D = Vec Z2 , we constructed in sec.II D local symmetric operators associated with a choice of module category M over Vec Z2 .We identified the model associated with M = Vec Z2 as the transversefield Ising model and that associated with M = Vec as its Kramers-Wannier dual.Moreover, we provided an explicit lattice operator that transforms symmetric operators of one model into symmetric operators of the other.
Let us now examine the topological sectors of these two models and their mappings under the duality.
Let us first consider the case M = Vec Z2 .By construction, boundary conditions are labelled by simple objects in Fun Vec Z 2 (Vec Z2 , Vec Z2 ) ∼ = Vec Z2 and by convention the site L+2 is defined to be the site 1.In order to construct the boundary operators as per eq.( 37), we require the MPO tensors that evaluate to the F -symbols of Vec Z2 : Applied to configurations where M L+1 = M 1 , this unitary transformation amounts to fixing M L+1 = 1, leaving an effective total Hilbert space with L degrees of freedom.
It follows that the boundary terms associated with 1 are given by leading to the Hamiltonian which is the original untwisted Hamiltonian on periodic boundary conditions.This model enjoys a Z 2 -symmetry generated by ś L i=1 S x i and as such decomposes into two charge sectors which are even and odd with respect to this symmetry operator, respectively.Let us notate via ([1], 0) and ([1], 1) the corresponding topological sectors.
Considering now the Hamiltonian twisted by the simple object m in Vec Z2 , the Hilbert space consists of L + 1 degrees of freedom subject to the constraint that m b M L+1 = M 1 .Applying the unitary (50) to such configurations amounts to fixing M L+1 = m, again leaving an effective total Hilbert space with L degrees of freedom.Importantly however, the 1 and m-twisted Hilbert spaces are only effectively the same, but are in fact orthogonal.The boundary terms associated with m are given by leading to the Hamiltonian referred to as the antiperiodic transverse-field Ising model.The model retains the same Z 2 -symmetry as in the periodic case and as such also decomposes into even and odd charge sectors.We notate via ([m], 0) and ([m], 1) the corresponding topological sectors.
Let us now consider the case M = Vec.As per our construction, boundary conditions for this dual model are labelled by simple objects in Fun Vec Z 2 (Vec, Vec) ∼ = Rep(Z 2 ) (see app.A) and by convention the site 1  2 is defined to be the site L + 1 2 .The Hamiltonian boundary terms defined as per eq.( 37) requires MPO tensors evaluating to ▷◁ Fsymbols of Vec as a (Rep(Z 2 ), Vec Z2 )-bimodule category: where V ≡ (V, ρ : Z 2 → End(V )) is an irreducible representation of Z 2 .Choosing V to be the trivial representation 0 immediately yields i.e. the Kramers-Wannier dual of the transverse-field Ising model with periodic boundary conditions.The Z 2symmetry is now generated by and the topological sectors associated with the even and odd charge sectors are notated via ([1], 0) and ([1], 1), respectively.Similarly, choosing V to be the sign representation 1 yields Let us now establish the mapping of topological sectors under the duality associated with the unique simple object in Fun Vec Z 2 (Vec Z2 , Vec) ∼ = Vec.We already presented in sec.II D the MPO intertwiner that performs the mapping of the local symmetric operators.In order to account for boundary conditions it needs to be promoted to intertwining tubes of the form eq. ( 42), which in turn requires the introduction of the following tensors: for any (V, ρ) ∈ I Rep(Z2) .Let us consider for instance the topological sector ([1], 1), i.e. the odd sector of the periodic transverse-field Ising model H Vec Z 2 ,1 .Selecting this topological sector amounts to acting with the projector e ).It follows from eq. ( 58) that when pulling this projector through the intertwining tube T 1,0,1,1,1,1 ). Performing analogous computations for the other sectors confirms that under this duality, the topological sectors ([1], 1) and ([m], 0) are swapped, while the topological sectors ([1], 0) and ([m], 1) remain unchanged.

Sec. IV | Examples with Rep(S 3 ) symmetry
We present in this section a series of examples that are mathematically non-trivial and physically relevant, thereby showcasing the potential and merits of our approach.The examples we consider all have symmetry operators organized into fusion categories that are in the Morita class of Rep(S 3 ), namely the category of (finitedimensional) representations of the symmetric group S 3 .

IV.A. Foreword
Before presenting the examples, let us mention a couple of different ways the constructions presented in this manuscript can be used in practice.
On the one hand, we can pick a module category M over a given fusion category D, make a choice of boundary condition, and construct a lattice Hamiltonian by considering any linear combination of local operators as defined in eq. ( 21).The resulting Hamiltonian is guaranteed to be symmetric with respect to operators organized into D ⋆ M , which is in the Morita class of D by definition.The same linear combinations of local operators but for a different choice of module category yields a dual lattice Hamiltonian.The corresponding duality operator is then provided by the intertwining tubes given in eq. ( 42).This is the recipe we would typically follow when wishing to define a new family of lattice Hamiltonians satisfying certain symmetry conditions.
On the other hand, our method can be employed in order to investigate new properties and duality relations of a known lattice model.This first requires to rewrite the Hamiltonian of interest within our framework.To do so, we must identify a suitable choice of input fusion category D. Such an input fusion category can be chosen to be any Morita dual of a subcategory of symmetry operators of the starting Hamiltonian.We then need to pick the appropriate D-module category so we can find a linear combination of local operators (21) so as to recover the Hamiltonian.Concretely, let us consider for instance a lattice Hamiltonian with a G-symmetry so that the category of symmetry operators is the category Vec G of G-graded vector spaces.For any subgroup H of G, any Morita dual of the fusion category Vec H is a valid choice of input fusion category D, even though Vec H does not capture the whole symmetry of the Hamiltonian.There will then be a choice of module category over D such that there exists a linear combination of local operators (21) producing the desired Hamiltonian.Module categories over D then classify duals of this Hamiltonian with respect to the (sub)symmetry encoded into Vec H . Indeed, any duality relation between two models is always with respect to a given symmetry.This implies in particular that choosing a small input fusion category reduces the number of dual models that can be considered.The same reasoning applies for any one-dimensional lattice model satisfying some categorical symmetry.This latter scenario is the one we explore in this section.Throughout, the input fusion category is chosen to be the category D = Rep(S 3 ) of finite-dimensional representations of the symmetric group S 3 .For a specific choice of Rep(S 3 )-module category, we show how to recover the Heisenberg XXZ model within our framework.After studying its topological sectors, we explore dual models associated with various choices of module categories over Rep(S 3 ), and construct the explicit lattice operators performing the corresponding duality transformations.

IV.B. Local operators
The input fusion category D of the models we consider is the category Rep(S 3 ) of finite-dimensional representations of the symmetric group S 3 = ⟨r, s | r 2 = 1 = s 3 , rsr = s 2 ⟩.Recall that this fusion category has three simple objects provided by the irreducible representations of the group.We denote these simple objects as 0, 1, 2 and refer to them as the trivial, sign and two-dimensional irreducible representations, respectively.The fusion rules read 0 b Y ≃ Y , with Y any simple object in Rep(S 3 ), 1 b 1 ≃ 0, 1 b 2 ≃ 2 and 2 b 2 ≃ 0 ' 1 ' 2 (see app.B 4 for a brief review of the element structure and representation theory of S 3 ).
For every A ⊆ S 3 subgroup, the category Rep(A) is a module category over Rep(S 3 ) via the restriction functor Res S3 A : A (Y ) for every M ∈ Rep(A) and Y ∈ Rep(S 3 ).Every Rep(S 3 )-module category is of this form.Given a Rep(S 3 )-module category M, we consider the Hamiltonian with local operators which are obtained by contracting symmetric tensors evaluating to ◁ F -symbols of M. By construction, the Hamiltonian H M is left invariant by the action of MPOs (22) labelled by objects in D ⋆ M .In order to write down the Hamiltonian and the symmetry operators more explicitly, we now need to consider the different module categories M separately.

IV.C. M = Vec
The first Rep(S 3 )-module category we consider is M = Vec, which amounts to choosing the trivial subgroup of S 3 .We notate the unique simple object in Vec via 1 ≃ C such that 1 ◁ Y ≃ 1 for any Y ∈ I D .For this example, the ◁ F -symbols reduce to Clebsch-Gordan coefficients of S 3 , i.e. basis vectors of hom-spaces of the form Hom where i, j, k label basis vectors in the vectors spaces Y 1 , Y 2 and Y 3 underlying the corresponding representations, respectively.We review the definition of these Clebsch-Gordan coefficients in app.B 4. Given the definition of the local operators (61), we only require the following non-vanishing ◁ F -symbols: Let us consider the Hilbert space (30) with no regard for the boundary conditions at the moment.Since the module category Vec has a unique simple object and the local operators b M i,Y constrain the D-strings to be labelled by 2, the only physical degrees of freedom are basis vectors in hom-spaces Hom M (1 ◁ 2, 1) ≃ C 2 ≃ C[|1⟩, |2⟩].Therefore, the effective microscopic Hilbert space is isomorphic to Â i C 2 .In other words, we are dealing with a Hamiltonian acting on spin- 1  2 particles located at half-integer sites of the lattice.How does the local operator b M i,2 act on this effective Hilbert space?Notice first that in virtue of the contraction of the symmetric tensors, the local operator b M i,2 is a sum of two terms.It then follows from the definition of the ◁ F -symbols that it acts by projecting out states for which the hom-space basis vectors at sites i − 1 2 and i + 1 2 agree, and acts as |1, 2⟩ → |2, 1⟩ or |2, 1⟩ → |1, 2⟩ otherwise.Similarly, the operator b M i,1 acts as the identity operator when the hom-space basis vectors at sites i − 1 2 and i + 1 2 agree, and minus the identity operator otherwise.Putting everything together, we find that the Hamiltonian (60) in the infinite chain case boils down to where we introduced the notation S ± := 1 2 (S x ± iS y ).This is the spin- 1  2 Heisenberg XXZ model.What is the symmetry of the Hamiltonian H Vec ?On the one hand, there is a Z 2 symmetry generated by , which acts in particular as S ± → S ∓ .On the other hand, there is a Z 3 symmetry generated by ś i ( ω 0 0 ω ) i+ 1 2 with ω = e 2iπ/3 , which acts in particular as S ± → ω ±1 S ± .Crucially, these symmetry operators do not commute so that we have an overall Z 2 ⋉ Z 3 ≃ S 3 symmetry, whereby Z 2 acts on Z 3 by complex conjugation.Let us confirm that this is indeed what our general theory predicts.By construction, we know that the Hamiltonian H Vec is left invariant by symmetry operators organized into the fusion category (Rep(S 3 )) ⋆ Vec .But as a fusion category (Rep(S 3 )) ⋆ Vec is equivalent to Vec S3 (see app.A), which means that H Vec is left invariant by symmetry operators labelled by group variables g ∈ S 3 .Our construction further provides an explicit expression for these symmetry operators in the form of MPOs via where ρ : S 3 → End( 2) is the representation matrix of 2. It is immediate to confirm that the Z 2 symmetry is generated by the MPO labelled by the order 2 group element r ∈ S 3 , whereas the Z 3 symmetry is generated by the MPO labelled by the order 3 group element s ∈ S 3 .
Let us now examine the topological sectors.Consider a spin chain of length L + 1. Generally speaking, boundary conditions of the Hamiltonian H Vec are of the form with S = S x , S y , S z and K ∈ End(C 2 ) a unitary matrix.In particular, we can choose K = ρ(g) with ρ : S 3 → End(2) for any g ∈ S 3 .These boundary conditions, which are labelled by simple objects in (Rep(S 3 )) ⋆ Vec ∼ = Vec S3 , do correspond to those predicted by our approach. 5Recall that in general, given a simple object in Vec S3 , the corresponding boundary condition is provided by the local operators defined in eq.(37), where the MPO tensors appearing in these local operators are rotated versions of the symmetry MPO tensor (63).It follows that boundary conditions are provided by the local operators where σ : S 3 → End(1) is the character of the sign representation 1.We can now explicitly check that this definition agrees with that proposed above.For instance, for which readily agrees with eq. ( 65).
Now that we have confirmed that the boundary conditions prescribed by our tensor network approach agree with the usual definitions, let us analyze these boundary conditions in more detail.Naturally, the boundary condition labelled by g = 1 corresponds to a periodic spin chain.In this case, the Hamiltonian has the same S 3 symmetry as in the infinite chain case, and therefore the Hilbert space decomposes into charge sectors labelled by the three irreducible representations of S 3 .The boundary condition labelled by g = r corresponds to anti-periodic spin chain.Interestingly, it is clear from eq. ( 67) that the resulting Hamiltonian is no longer S 3 symmetric but merely Z 2 symmetric, and as such decomposes into charge sectors labelled by irreducible representations of Z 2 .This is true for any (twisted) antiperiodic boundary condition labelled by representatives of the conjugacy class [r] = {r, rs, rs 2 }.Finally, the boundary condition labelled by g = s corresponds to a twisted periodic spin chain.We can readily check that the resulting Hamiltonian retains the Z 3 symmetry of the periodic case, whereas the Z 2 symmetry is lost.Therefore, the model decomposes into charge sectors labelled by irreducible representations of Z 3 .Akin to the previous scenario, these statements are valid for any twisted periodic boundary condition labelled by representatives of the conjugacy class [s] = {s, s 2 }.Overall, we find eight topological sectors that are in one-to-one correspondence with simple objects in Mod(D(S 3 )), i.e. simple modules over the Drinfel'd double of S 3 .This is in agreement with the general results presented in the previous section.Indeed, our approach predicts that topological sectors are organized into the Drinfel'd center Z(Vec S3 ), which is equivalent to Mod(D(S 3 )) as we review in app.B 2 in the case of an arbitrary finite group G.

IV.D. M = Rep(Z2)
The second Rep(S 3 )-module category we consider is M = Rep(Z 2 ), whose simple objects are denoted by 0 Z2 and 1 Z2 .As mentioned in sec.IV B, the module action is given by M ◁ Y = M b Res S3 Z2 (Y ), for every M ∈ Rep(Z 2 ) and Y ∈ Rep(S 3 ), with the restriction functor fully specified by its action on the simple objects of Rep(S 3 ), namely Res S3 Z2 (0) ≃ 0 Z2 , Res S3 Z2 (1) ≃ 1 Z2 and Res S3 Z2 (2) ≃ 0 Z2 ' 1 Z2 .The non-vanishing ◁ F -symbols that are relevant for our construction are given by and , As for the previous choice of module category, let us begin by writing down the Hamiltonian in the infinite spin chain case.The parametrization of the microscopic Hilbert space significantly differs from that of the previous scenario.The local operators b M i,Y always constrain the D-strings to be labelled by 2.Moreover, it follows from the definition of the module action that hom-spaces in M are all one-dimensional.The only fluctuating physical degrees of freedom thus correspond to the M-strings.We thus find an effective microscopic Hilbert space that is still isomorphic to Â i C 2 , where the spin-1 2 particles are now located at integer sites of the lattice.How does the local operator b M i,2 act on this effective Hilbert space?Identifying |0 Z2 ⟩ and |1 Z2 ⟩ with the +1 and −1 eigenvectors of the Pauli S z operator, respectively, it follows from the definitions of the ◁ F -symbols that b M i,2 acts as a projector onto the state |+⟩ at site i if the degrees of freedom at sites i − 1 and i + 1 agree, and minus this projector otherwise.The local b M i,1 simply acts as the Pauli S x operator at site i.Putting everything together, we find that the Hamiltonian (60) in the infinite chain case boils down to What is the symmetry of this Hamiltonian?Unlike the Hamiltonian H Vec , it is very difficult to identify the symmetry without relying on the fact that we obtained this Hamiltonian using tensor networks satisfying certain pulling-through conditions.Specifically, we know by construction that the Hamiltonian H Rep(Z2) is left invariant by symmetry operators organized into the fusion category (Rep(S 3 )) ⋆ Rep(Z2) , which happens to be equivalent to Rep(S 3 ).Given the representative X of an isomorphism class of simple objects in Rep(S 3 ), the corresponding symmetry operator is provided by an MPO generated by tensors that evaluate to ▷◁ F -symbols of Rep(Z 2 ) as a (Rep(S 3 ), Rep(S 3 ))-bimodule category.Naturally, the symmetry MPO associated with the trivial representation 0 acts trivially, whereas the symmetry MPO labelled by 1 can readily be checked to act as We notice immediately that the first set of ▷◁ F -symbols satisfy a symmetry condition whereby the entries only depend on the number of times the representation 1 Z2 occurs in the symbol.This symmetry condition can be exploited in order to rewrite the symmetry MPO associated with the above symbols as an MPO of the form (68) with building blocks This MPO should be interpreted as an operator acting on the integer sites of the effective Hilbert space.Let us now use this alternative form of the symmetry MPO labelled by 2 to explicitly check its commutation relation with H Rep(Z2) .The building block defined above verifies the following symmetry conditions: where O := ).It then follows from the three symmetry conditions above that the MPO commutes with S z i−1 (1+S x ) i S z i+1 , for any i, thus it commutes with H Rep(Z2) , thereby confirming that the MPO labelled by 2 is indeed a symmetry operator.Putting everything together, this confirms the Rep(S 3 ) symmetry of H Rep(Z2) .The exercise we just carried out exemplifies how non-trivial it may be to explicitly confirm certain properties of a given Hamiltonian, although these properties are immediate once the model has been recast within our framework.
Let us now examine the topological sectors.Consider a spin chain of length L + 1.We know from our general construction that topological sectors are in one-to-one correspondence with simple objects of Z(Rep(S 3 )), which is equivalent to Z(Vec S3 ) considered in the previous scenario in virtue of the Morita equivalence between Rep(S 3 ) and Vec S3 (see app.A).Therefore, we must find eight topological sectors corresponding to the simple objects Mod(D(S 3 )) as for M = Vec.The Hamiltonian H Rep(Z2) admits three types of boundary conditions labelled by simple objects in Rep(S 3 ).Naturally, choosing 0 identifies the degrees of freedom at sites L + 1 and 1 and thus corresponds to a periodic chain, in which case the Hamiltonian has the same Rep(S 3 ) symmetry as in the infinite case.We can then check that the Hilbert space decomposes into charge sectors indexed by conjugacy classes of the group, providing the simple objects labelled by ([1], 0), ([r], 0 Z2 ) and ([s], 0 Z3 ) of Z(Rep(S 3 )) in the notation of app.B 1.More generally, given a boundary condition labelled by a simple object A in Rep(S 3 ), the corresponding Hilbert space decomposes in charge sectors indexed by the simple objects of Z(Rep(S 3 )), which, when treated as (typically not simple) objects in Rep(S 3 ), contain A as subrepresentation (see app.B 4).We can obtain the explicit boundary conditions associated with the other simple objects in Rep(S 3 ) applying our general construction.Recall that a boundary condition is provided by the local operators defined in eq.(37), where the MPO tensors appearing in these local operators are rotated versions of the symmetry MPOs tensor evaluating to the ▷◁ F -symbols defined previously.Concretely, the boundary condition labelled by 1 identifies the degree of freedom at site L + 1 and the image of that at site 1 under the Pauli S x operator, so we still have an effective Hilbert space with L degrees of freedom, and ) are all required to be distinct from one another, i.e. each irreducible representation of S 3 can only appear once in {Y 1 , Y 2 , Y 3 }.
As usual, we begin by writing down the Hamiltonian in the infinite chain case.The microscopic Hilbert space underlying this model differs from that of the previous scenarios.First of all, the local operators still constrain the D-strings to be labelled by 2 and the hom-spaces in M are all one-dimensional.The only fluctuating physical degrees of freedom thus correspond to the M-strings.Since the category M counts three isomorphism classes of simple objects, we are now dealing with a system of spin-1 particles, in sharp contrast with the previous cases.Moreover, given an object M in Rep(Z 3 ), it follows from the fusion rules in Rep(Z 3 ) that M does not appear in the decomposition of ).As such, strands i and i + 1 cannot be labelled by the same object M -this is confirmed by the definition of the ◁ F -symbols above.We thus find an effective microscopic Hilbert space of spin-1 particles located at integer sites of the lattice, which is not a tensor product of local Hilbert spaces given the kinematical constraint we just described.We notate this microscopic Hilbert space via H Rep(Z3) .Within this microscopic Hilbert space, it follows from the definition of the ◁ F -symbols that b M i,2 modifies the label of the strand i to whichever other label-if any-is allowed by the kinematical constraint, and acts as the zero operator if this is not possible.Similarly, b M i,1 acts on the strands i − 1, i and i + 1 as the identity operator whenever the labels of the strands i − 1 and i + 1 are identical and minus the identity operator otherwise.Putting everything together, the Hamiltonian is given by with so that S x S z = ωS z S x .Notice that given certain configurations of spin-1 variables, acting with S x i + (S x i ) † may bring the corresponding state outside of H Rep(Z2) , in which case it is projected out by definition of the ◁ Fsymbols.
Our construction predicts that the Hamiltonian H Rep(Z3) is left invariant by operators organized into the fusion category ( has an S 3 symmetry.This symmetry works as follows: Firstly, notice that Rep(Z 3 ) and Vec Z3 are equivalent as fusion categories.Secondly, identifying the simple objects C g in Vec Z3 with the corresponding left cosets M = gZ 2 , the left Vec S3 -module structure of Vec Z3 is given by C g ▷ M := (gr(M ))Z 2 for any g ∈ S 3 and M ∈ S 3 /Z 2 , where r(M ) denotes the representative of M .Therefore, the symmetry operator labelled by g ∈ S 3 simply acts on the local Hilbert space C 3 associated with every spin-1 particle by permutation of the coordinates.The fact that H Rep(Z3) commutes with these symmetry operators finally follows from S z S z = ωS z S x , C(S x + (S x ) † ) = (S x + (S x ) † )C and CS z = (S z ) † C, where This can be readily confirmed using the symmetry MPOs whose building blocks evaluate to ▷◁ F -symbols of Rep(Z 3 ) as a (Vec S3 , Rep(S 3 ))-bimodule category.Boundary conditions are labelled by simple objects in Vec S3 and can be implemented using our general recipe.The analysis of the topological sectors then parallel that of the model H Vec , that is, conjugacy classes of S 3 define equivalence classes of boundary conditions and the corresponding charge sectors are labelled by irreducible representations of the centralizer of the conjugacy class.Topological sectors are found to correspond to simple objects of Z(Vec S3 ) as expected.
The remaining Rep(S 3 )-module category is Rep(S 3 ) itself.This model is both the simplest model to define-since all the data we need is provided by the monoidal structure of Rep(S 3 )-and the most difficult model to analyse explicitly.Indeed, contrary to the previous scenarios, we do not know how to rewrite the resulting Hamiltonian in terms of spin operators so we can hardly make the local operators and the symmetry operators as explicit.On the bright side, this illustrates the need for a systematic category theoretical approach in general.
By definition, the ◁ F -symbols of Rep(S 3 ) as a Rep(S 3 )module category coinciding with the F -symbols of Rep(S 3 ).The latter are typically referred to as 6jsymbols and are obtained by contracting Clebsch-Gordan coefficients.It was shown in ref. [58] that in the case where we choose module categories over themselves, the lattice models constructed following our approach boils down to so-called anyonic chains [30][31][32][33][34][35].In particular, the Rep(S 3 ) anyonic chain associated with our choice of local operator (61) was studied in [36,37].As such, we shall not review this model explicitly here and encourage the reader to consult the above references for detail.Let us merely stress the fact that since Fun Rep(S3) (Rep(S 3 ), Rep(S 3 )) ∼ = Rep(S 3 ) the model is left invariant by symmetry operators encoded into Rep(S 3 ) as for H Rep(Z2) .

IV.G. Duality Rep(Z2) → Vec
By definition, the Hamiltonians H Vec , H Rep(Z2) , H Rep(Z3)  and H Rep(S3) constructed above are all dual to one another.Duality relies upon the fact that they only differ by a choice of module category over Rep(S 3 ).Crucially, symmetry operators of these various models are encoded into categories that are Morita equivalent, although they are not necessarily equivalent as fusion categories.Morita equivalence ensures that the center of the categories of symmetry operators are equivalent as braided fusion categories, which is crucial to being able to map the topological sectors of one model onto those of another.We explained in sec.III C how to explicitly perform such a mapping via intertwining tubes.We illustrate in this section our construction with the duality between the models H Vec and H Rep(Z2) .
Let us first compute the mapping of local operators.Note that this mapping immediately provides the duality relation in the infinite chain case.Inspecting the local operators, we notice that H Vec can be obtained from H Z2 via the mapping Indeed, due to S y = iS x S z , we have for instance Let us now prove that this mapping is provided by a Rep(S 3 )-module functor in Fun Rep(S3) (Rep(Z 2 ), Vec), as predicted by the general construction.First of all, we can show that Fun Rep(S3) (Rep(Z 2 ), Vec) ∼ = Rep(Z 3 ), so that we distinguish three duality maps between H Vec and H Rep(Z2) labelled by simple objects in Rep(Z 3 ).These duality maps cannot be distinguished locally in the sense that they all perform the same transformations of local operators.Locally then, without loss of generality, we can focus on the duality operator labelled by the simple object 0 Z3 in Rep(Z 3 ).Globally however, they may perform different mappings of the states within topological sectors, and one needs to consider all the duality operators.
Constructing the MPO intertwiners that perform the transformation of the local operators only requires one type of tensors, namely those that evaluate to the 0 Z 3 ω-symbols of the Rep(S 3 )-module functor in Fun Rep(S3) (Rep(Z 2 ), Vec) associated with 0 Z3 ∈ Rep(Z 3 ).However, directly computing these tensors may prove challenging.Conveniently, these can be obtained via the following composition of module functors: On the one hand, the MPO intertwiner labelled by the unique object in Fun Rep(S3) (Rep(S 3 ), Vec) ∼ = Vec evaluates to Clebsch-Gordan coefficients of Rep(S 3 ), i.e.
On the other hand, simple objects X in Fun Rep(S3) (Rep(Z 2 ), Rep(S 3 )) ∼ = Rep(Z 2 ) op label MPO intertwiners whose building blocks read where ◁ F here refers to the Rep(S 3 )-module associator of Rep(Z 2 ).We provided the relevant entries for these tensors in sec.IV D. Importantly, composing module functors labelled by simple objects in the relevant categories do not yield simple objects in Fun Rep(S3) (Rep(Z 2 ), Vec).This means that after contracting the MPO intertwiners presented above, we must decompose the resulting MPO into simple blocks.Doing so, we are able to find the MPO intertwiner labelled by 0 Z3 whose building block is of the form for any M 1 , M 2 ∈ I Rep(Z2) and basis vector |i = 1, 2⟩ in Hom Vec (C ◁ 2, C) ≃ C 2 .The non-vanishing 0 Z 3 ω-symbols that are relevant to our computations are found to be Let us check that this MPO intertwiner indeed performs the mappings given in eq. ( 78).Given the specific form of ( 83) and the values of the 0 Z 3 ω-symbols, one can rewrite the resulting MPO intertwiner in a more traditional form as (84) with building blocks It is now immediate that the MPO tensor verifies the following symmetry conditions which in turn provide the mappings given in eq. ( 78), as expected.
We checked how the MPO intertwiners labelled by simple objects in Fun Rep(S3) (Rep(Z 2 ), Vec) transform the local symmetric operators entering the definition of H Rep(Z2) into those entering the definition of H Vec .In order to fully establish the duality relations between these models, we must further explain how topological sectors are mapped under these transformations.We know from the general construction that topological sectors of H Vec and H Rep(Z2) are associated with isomorphism classes of simple objects in Z(Vec S3 ) and Z(Rep(S 3 )), respectively.Naturally, since Vec S3 and Rep(S 3 ) are Morita equivalent, we have Z(Vec S3 ) ∼ = Z(Rep(S 3 )), which in turns guarantees the correspondence of topological sectors.However, it does not mean that under duality the topological sectors are mapped identically.It is indeed possible for permutations of sectors to take place, as already illustrated for the Kramers-Wannier duality.Let us investigate in some detail such a scenario.
Let us consider the boundary condition of H Rep(Z2) labelled by the simple object 2 in Rep(S 3 ).Concretely, we found that the local operators associated with this boundary condition are given by eq. ( 73).We further explained in sec.IV D that, given this boundary condition, the model decomposes into five charge sectors.In order to consider a specific topological sector of the model, it is thus required to project the model onto the corresponding charge sector.This can be performed via the projectors described in app.B 3, which are constructed from the half-braiding tensors associated with the simple objects in Z(Rep(S 3 )) labelling the topological sectors of interest.Concretely, let us consider the topological sector labelled by the simple object ([1], 2).As an object of Z(Rep(S 3 )), ( [1], 2) is also an object of Rep(S 3 ), namely 2 itself.We wish to compute what this topological operator is mapped to under the duality transformation labelled by 0 Z3 .
Carrying out this computation amounts to deriving the intertwining tubes for all boundary conditions defined in sec.III C, applying the projectors associated with all topological sectors on both side of the intertwining tube, and identifying which topological sectors that have non-vanishing overlap with ([1], 2).The MPO intertwiners provided by eq. ( 83) are only one component of the intertwining tubes (42).Indeed, we further require three-valent tensors that evaluate to F F -symbols capturing the composition Fun Rep(S3) (Vec, of module functors: for any X, X ′ ∈ I Rep(Z3) , A ∈ I Rep(S3) , B ∈ I Vec S 3 and M, M 1 , M 2 ∈ I Rep(Z2) .Since we are interested in the fate of the topological sector ( [1], 2) of H Rep(Z2) under the duality operator labelled by 0 Z3 , we fix A = 2 and X = 0 Z3 , which in turn constrains X ′ to be 1 Z3 ' 1 * Z3 .We then apply apply the projector associated with the simple object ( [1], 2) of Z(Rep(S 3 )) so as to select the corresponding topological sector amongst all possible charge sectors compatible with the boundary condition labelled by 2. The question is then, for which B ∈ I Vec S 3 and charge sectors of H Vec associated with the boundary condition labelled by B acting with the corresponding projector on the intertwining tube provides a non-vanishing operator?We find that B must be equal to C s ' C s 2 , where we should think of C as the vector space underlying 0 Z3 , yielding the topological sector associated with the simple object ([s], 0 Z3 ) of Z(Vec S3 ).The F F -symbols entering the definition of the intertwining tube that are relevant to this mapping are given by and In short, this shows that under the duality operator labelled by the Rep(S 3 )-module functor in Fun Rep(S3) (Rep(Z 2 ), Vec) ∼ = Rep(Z 3 ) identified with the simple object 0 Z3 , the topological sector ( [1], 2) of H Rep(Z2) is mapped to the topological sector ([s], 0 Z3 ) of H Vec .Conversely, we can show that the topological sector ([s], 0 Z3 ) is mapped to ([1], 2).We can similarly show that the remaining topological sectors are mapped identically.This permutation of topological sector corresponds to the non-trivial braided autoequivalence of Z(Rep(S 3 )) in BrEq(Z(Rep(S 3 ))) ≃ Z 2 (see app.A).

Sec. V | Discussion
We conclude our manuscript with a discussion of concrete applications of some of the results presented in this manuscript and comments on possible generalizations and extensions.

V.A. Application: symmetric tensor networks
Our study of the interplay between duality transformations and closed boundary conditions can be directly applied to the numerical diagonalization of symmetric Hamiltonians within the variational class of symmetrypreserving tensor networks.One standard approach for this problem involves decomposing the Hamiltonian into symmetric tensors and working in a fusion basis for the Hilbert space [98][99][100].In this basis, the matrix elements of the Hamiltonian can be obtained by invoking the recoupling theory of the symmetric tensors associated with the fusion category D, and the setting effectively becomes equivalent to that of one-dimensional models covered in this manuscript.For closed boundary conditions, the sectors are organized into Z(D), and one requires the half-braiding tensors as discussed for instance in app.B to thread the corresponding flux through the closed loop [13].For modular tensor categories, there is an equivalence Z(D) ∼ = D b D op so that half-braidings can be obtained from the braiding of D; this case was studied in detail in [101][102][103].
In the setup described above, the original Hamiltonian is given in terms of symmetric tensors associated with a choice of right D-module category M.However, when working in the usual fusion basis, we are effectively replacing the Hamiltonian by a dual Hamiltonian obtained by choosing M = D.This means that, when performing symmetric tensor network computations, one is generically simulating a dual model that has the same spectrum but typically much smaller degeneracies for given sectors.This last property is responsible for the computational advantage gained by working with symmetry preserving tensor networks.In order to simulate the full model with all possible boundary conditions, it is clear that a detailed understanding of how sectors are mapped into one another under duality is required, which we obtained in this manuscript.We showed in this manuscript that given a fusion category D and a choice of D-module category M we can construct local operators that commute with symmetric operators organized into the fusion category D ⋆ M := Fun D (M, M).In this context, we loosely define open boundary conditions as equivalence classes of extensions of the one-dimensional system to its boundary components in a way that is compatible with the 'bulk' D ⋆ M -symmetry.More precisely, we require open boundary conditions to be organized into categories that are equipped with a D ⋆ M -action, i.e.D ⋆ M -module categories.Given a pair (P, Q) of D ⋆ M -module categories, we would then consider microscopic Hilbert spaces of the form As discussed in the introduction, mathematically, spherical fusion categories serve as input data for topological quantum field theories via the Turaev-Viro-Barrett-Westbury construction [40,41].Crucially, these topological quantum field theories are fully-extended in the sense that they capture locality all the way down to the point.In this context, the Drinfel'd center Z(C) of C corresponds to the quantum invariant assigned to the circle by the state-sum [104], whereas Mod(C) is identified with the quantum invariant assigned to the point.Crucially, these invariants are related via a socalled 'crossing with the circle' condition stipulating that Dim Mod(C) ∼ = Z(C), where Dim is an appropriate categorification of the notion of dimension of a vector space suited to bicategories (see ref. [105,106] for the case of C = Vec G ).This relation formalizes the process whereby identifying the endpoints of a model with open boundary conditions yields a model with closed boundary conditions.We postpone a systematic study of dualities in onedimensional quantum lattice models with open boundary conditions to another manuscript.

V.C. Higher dimensions
The results presented in this manuscript can be largely extended to two-dimensional quantum lattice models following the ethos of categorification.Loosely speaking, it amounts to replacing fusion (1-)categories and module (1-)categories in our exposition by fusion 2-categories and module 2-categories.More specifically, given an input fusion 2-category and a choice of module 2-category over it, local symmetric operators akin to those considered in this manuscript can be constructed [61].These in turn define Hamiltonians that commute with operators organized into the (higher) Morita dual of the input fusion 2-category with respect to the chosen module 2-category.Similarly, duality relations are also encoded into module 2-functors between distinct module 2-categories.Boundary conditions and topological sectors will then be related to representations of higher-dimensional tube algebras as considered in ref. [55,[107][108][109] and to the highercategorical center of the symmetry fusion 2-category.
As an example, given a two-dimensional model with a G-symmetry, we can construct a tensor network operator performing the gauging of the symmetry and show that the resulting dual model commutes with projected entangled pair operators forming the fusion 2-category 2Rep(G) of '2-representations' of the group G [61,110,111], which is Morita equivalent to the fusion 2-category 2Vec G of Ggraded 2-vector spaces [61].
Recently, a closely related point of view has been embraced by the high-energy community under the name of 'sandwich construction' or 'symmetry topological field theory' [112].So far, this approach has been mostly employed in the continuum, although lattice versions of these ideas have appeared in the past [9,27,28,[113][114][115].Succinctly, this approach amounts to realising the partition function of a d-dimensional theory as the interval compactification of a (d+1)-dimensional topological field theory with two types of boundaries: one hosting a gapped boundary condition and another hosting a 'physical' typically non-topological boundary condition.In two spacetime dimensions on the lattice, the choice of three-dimensional topological field theory and its gapped boundary conditions amounts to our choices of input fusion category and module category over it, whereas the data of the physical boundary condition is encodes our choice of abstract algebra of operators.Explicitly performing the interval compactification, this would immediately recover our construction, or its completely analogue formulation in terms of anyonic chains presented in ref. [58].We note that in principle the sandwich construction produces a partition function, but this can be related to our Hamiltonian construction via the transfer matrix and the standard quantum to classical mapping.

App. A | Morita equivalence
In this appendix, we collect a few results about (categorical) Morita equivalence of fusion categories.

A.1. Motivating examples
Consider the fusion category Vec G of G-graded vector spaces.Recall that it is the C-linear category with simple objects the one-dimensional vectors spaces C g , g ∈ G, such that C g bC h ≃ C gh and Hom(C g , C h ) = δ g,h C for every g, h ∈ G. (Indecomposable) module categories over Vec G are labelled by pairs (A, ψ) with A ⊆ G a subgroup and ψ a representative of a cohomology class in H 2 (A, U(1)), such that the collection of simple objects is provided by G/A [116].Choosing A = G, we find that Vec is a (right) Vec G -module category via the forgetful functors Vec G → Vec.It is a well-known result that the category (Vec G ) ⋆ Vec := Fun Vec G (Vec, Vec) is equivalent to Rep(G) [3].Let us briefly review this derivation.A functor F : Vec → Vec is fully specified by the vector space V := F (C) it assigns to the unique simple object C in Vec.The Vec G -module functor structure in turn provides natural isomorphisms prescribed by for every g ∈ G.It follows from the defining coherence relation of module functors that (V, ω : g → ω g ) defines a representation of G. Similarly, we can show that module natural transformations between Vec G -module endofunctors of Vec correspond to intertwiners of G. Putting everything together, we have (Vec G ) ⋆ Vec ∼ = Rep(G).As we shall see below, this derivation demonstrates that Vec G and Rep(G) are Morita equivalent.A consequence of this result is that indecomposable module categories over Rep(G) are also parametrized by pairs (A, ψ).Indeed, for any (A, ψ), the category Rep ψ (A) of projective representations of A can be endowed with the structure of a (right) Rep(G)-module category via the restriction functor Res G A : Rep(G) → Rep(A).As one would expect, we have (Rep(G)) ⋆ Vec := Fun Rep(G) (Vec, Vec) ∼ = Vec G .The derivation of this result is more subtle than the previous one, as such we shall merely sketch here and refer the reader to ref. [3] for details.First, remember that Rep(G) can be equivalently defined as the category Mod(C[G]) of modules over the group algebra, whereas Vec G is equivalent to the category Mod(C G ) of module over the algebra of functions on G.As already mentioned, a functor F : Vec → Vec is fully specified by a vector space V := F (C).A Rep(G)-module structure on F then provides natural isomorphisms prescribed by Consider for instance a fusion category C as a (left) module category over itself.The functor X → (− b X) : C is a monoidal equivalence, whereby the module structure is provided by for every Y ∈ Ob(C).This is the statement that C and C op are Morita equivalent.Notice that the monoidal structure in C ⋆ C is provided by the composition of Cmodule functors, which explains why the monoidal equivalence is with C op instead of C.
Let us consider an interesting application of the derivation above.Given a fusion category C, we can construct another fusion category Z(C) known as the center of C. Objects in Z(C) consists of pairs (X, R −,X ), where X is an object of C and R −,X : − b X ∼ − → X b − is a collection of natural isomorphisms known as half-braidings.
These half-braidings are required to satisfy an 'hexagon axiom' involving the monoidal associator of C. It turns out that the center Z(C) can be identified with the category Fun C|C (C, C) of (C, C)-bimodule functors.Indeed, any functor in Fun C|C (C, C) is in particular a functor in Fun C (C, C) ∼ = C op and hence is of the form − b X with X ∈ Ob(C).The right C-module structure then imposes the existence of natural isomorphisms It follows from the defining coherence relation of module functors that these half-braiding isomorphisms fulfill the expected 'hexagon axioms'.Monoidal structure in Z(C) finally corresponds to the composition of (C, C)bimodule functors.Notably, it follows immediately from We review this important result below.
The endofunctor X ▷ − of M is equipped with a left Cand right D-module structures via respectively, for any X ′ ∈ Ob(C) and Y ∈ Ob(D).The 'hexagon axioms' satisfied by the half-braiding isomorphisms R −,X as well as the various 'pentagon axioms' fulfilled by the module associators ensure that the defining coherence relations of right and left module functors are satisfied.Conversely, let us consider an object Fun C|D (M, M).By definition, it is in particular a right D-module functor in Fun D op (M, M).In virtue of Fun D op (M, M) ∼ = C, it is of the form X ▷ − with X ∈ Ob(C).The (C, D)-bimodule structure is provided by natural isomorphisms for any X ′ ∈ Ob(C), Y ∈ Ob(D).These yield the following collection of isomorphisms of right D-module endofunctors:  such that there is an isomorphism X ▷ − ≃ − ◁ F (X) of (C, D)-bimodule functors for every X ∈ Ob(C).This in turn motivates the definition of our tube intertwiners as maps between the tubes associated with Z(C) and Z(D) that 'commute' with M (see eq. ( 42)).
A particularly compact way of stating the result presented in this section is that the centre construction defines a 2-functor Z : BrPic → EqBr, where BrPic refer to the Brauer-Picard 2-groupoid of fusion categories and invertible bimodule categories, whereas EqBr denotes the 2groupoid of braided fusion categories and braided equivalences [90].This implies in particular the group isomorphism BrPic(C) ∼ = EqBr(Z(C)) between the Brauer-Picard group of invertible (C, C)-bimodules and braided auto-equivalences of Z(C).

App. B | Quantum double
In this appendix, we illustrate with a detailed example the equivalence between the category of representations of the tube category and the Drinfel'd center of the underlying fusion category.| such that c 1 = q −1 ci c i q ci and q c1 = 1, the action of D(G) on V is provided by (a δ g ) ▷ |c j , v⟩ = δ g,acj a −1 ac j a −1 ⟩ b (q −1 acj a −1 aq j ) ▷ |v , for any a, g ∈ G.We shall often implicitly make use of the equivalence between simple objects V labeled by ([c 1 ], V ) in Mod(D(G)) and irreducible representations (V, ρ : D(G) → End(V)) such that (a δ g ) ▷ |c i , v⟩ = ρ(a δ g )|c i , v⟩ and ρ(a δ g ) iṽ jv ≡ ⟨c i , ṽ|ρ(a δ g )|c j , v⟩ = δ g,ci δ g,acj a −1 ρ(q −1 ci aq cj ) ṽ v , where ( V , ρ : Z Representations of the tube category defined as objects in Fun(Tube(Vec G ), Vec) then correspond to the modules over D(G) as defined in this section.

B.2. Drinfel'd center Z(VecG)
There is a famous braided monoidal equivalence between Mod(D(G)) and the Drinfel'd center Z(Vec G ) of the category of G-graded vector spaces.Given an object (V, ρ) in Mod(D(G)), we shall now review how to obtain the corresponding object in Z(Vec G ) (see [117] for more details).Since 1 δ g ⋆ 1 δ g = 1 δ g , the matrix ρ(1 δ g ) is a projector and a fortiori it is diagonalizable.Furthermore, since 1 δ g ⋆ 1 δ h = 1 δ h ⋆ 1 δ g , the matrices ρ(1 δ g ) and ρ(1 δ h ) commute for every g, h ∈ G, and thus the set {ρ(1 δ g )} g∈G is simultaneously diagonalizable w.r.t. a basis notated via {|ν⟩} ν=1,...,dim(V) .We denote by V g the subspace of V given by Im ρ(1 δ g ).Due to the unit element of D(G) being provided by ř g∈G 1 δ g , given ν ∈ {1, . . ., dim(V)}, there is a unique g ∈ G such that |ν⟩ ∈ Im ρ(1 δ g ).Consequently, V decomposes as We deduce that the left D(G)-module V has the structure of a G-graded vector space.Given two objects (V, ρ) and depicted as ).Let us now compute the half-braiding tensors associated with each simple object.Since simple objects in Mod(D(S 3 )) labeled by [1] correspond to irreducible representations of S 3 , it follows immediately from the definition of the Clebsch-Gordan coefficients that for every simple object (W, σ) in Rep(S 3 ). .
The remaining cases are slightly more involved.We begin with the simple object labeled by ([r], 1 Z2 ).What is the decomposition of Ind S3 Z2 (1 Z2 )?Let us consider the restrictions of the irreducible representations of S 3 to Z 2 , and check the multiplicity of 1 Z2 .We know that 1 Z2 does not appear in 0 since the odd element a acts trivially on it.On the other hand, we have the obvious fact that 1 restricts to 1 Z2 .Since Ind S3 Z2 (1 Z2 ) is three-dimensional as an irreducible representation of D(S 3 ), we must have Ind S3 Z2 (1 Z2 ) ≃ 1 ' 2. Invoking eq.(B3), the basis vectors are found to be .
This concludes the computation of the half-braiding tensor for Z(Rep(S 3 )).

1 2
XXZ Heisenberg model can indeed arise as effective theory on the boundary of a (2+1)d topological model with Z(Vec S3 ) topological order.
Y 1 , Y 2 , . . .∈ I D representatives of isomorphism classes of simple objects in D and their respective quantum dimensions via d Y1 , d Y2 , . . .∈ C. The fusion of objects is encoded into the monoidal structure (b, 1, F ) of D consisting a product rule b : D × D → D, a distinguished object 1 referred to as the trivial charge or vacuum, and a (natural) isomorphism F : (− b −) b − ∼ − → − b (− b −) satisfying a coherence relation known as a 'pentagon axiom'.The isomorphism F shall be referred to as the monoidal associator and its components written as Fun D (D, M) is of the form (− ◁ M, ◁ F M −− ) with M any object in M, establishing the equivalence M ∼ = Fun D (D, M).But Fun D (D, M) has the structure of a left D ⋆ M -module category via the composition D ⋆ M × Fun D (D, M) → Fun D (D, M) of D-module functors.It follows that M has the structure of a (D ⋆ M , D)bimodule category.In this context, the module functor structure X ω −− of an object X in D ⋆ M coincides with the bimodule associator ▷◁ F X−− of M as an invertible (D ⋆ M , D)-bimodule category.Given three (right) module D-categories M, N and O, D-module functors in Fun D (N , O) and Fun D ( III.B. Symmetric operatorsGiven a D-module category M and a choice of boundary condition A ∈ I D ⋆ M , let us now construct local operators that are invariant under the action of symmetry operators.We already explained in sec.II C how to construct D ⋆ M -symmetric operators away from the boundary in terms of tensors evaluating to the ◁ F -symbols of M. Local operators involving the boundary conditions are defined as b

( 22 )
labelled by simple objects in Fun D (M, N ) perform the transmutations of local operators b M i,n into b N i,n , which only differ by the choice of D-module categories.Let us now explain how these mappings of local operators interact with the topological sectors characterized above.

V
.B. Open boundary conditions The results presented in this manuscript focus on closed boundary conditions, raising the question, what about open boundary conditions?Accommodating open boundary conditions requires an extension of the framework employed in this manuscript.
and basis vectors {i} in the appropriate hom-spaces.The resulting model would have symmetry operators organized into Fun D ⋆ M (P, Q).The same way topological sectors correspond to simple objects in the center Z(D ⋆ M ) in the closed case, we would find that topological sectors in the open case define the bicategory Mod(D ⋆ M ) of D ⋆ M -module categories, D ⋆ M -module functors and D ⋆ M -module natural transformations.
for every U ∈ Rep(G), satisfying the defining coherence relation.Equivalently, the module structure of F prescribes a homomorphism V → V b Nat(Res, Res), where Nat(Res) denotes here the vector space of natural endotransformations of Res : Rep(G) → Vec.By definition, a vector in Nat(Res, Res) assigns to every object U in Rep(G) a morphism Res(U ) → Res(U ) in Vec satisfying a naturality condition with respect to any intertwiner of G.It turns out that this vector space Nat(Res, Res) can be equipped with a canonical Hopf algebraic structure.It follows from the defining axioms of module functors that the map V → V b Nat(Res, Res) endows V with a right comodule structure over Nat(Res, Res).We can show that a natural transformation Res → Res is fully specified by its component on the regular representation C[G].But endomorphic intertwiners of the regular representation are given by right multiplication by an element in C[G], and thus the component of the natural transformation on the regular representation must amount to left multiplication by some element in C[G]. 6So we have a right comodule structure over Nat(Res, Res) ≃ C[G], which is the same thing as a left module structure over the dual of the Hopf algebra C[G], namely C G .Putting everything together we find (Rep(G)) ⋆ Vec ∼ = Mod(C G ) ∼ = Vec G , as expected.We shall now put these results in the context of Morita equivalence.A.2. Definition Given two fusion categories C and D, they are said to be (categorically) Morita equivalent if there exists a left C-module category M such that D ⋆ M ∼ = C, or equivalently if there exists a right D-module category such that C ⋆ M ∼ = D op , or still equivalently if there exists an invertible (C, D)-bimodule category M [3, 90].

Finally, the induced
braided monoidal equivalence Z(C) ∼ − → Z(D) is provided by a functor F : Z(C) → Z(D)

B. 1 .
Hopf algebraLet G be a finite group with identity element 1.We denote by C[G] the group ring of G and C G the space of functions on G.The quantum double D(G) ≡ (D(G), ⋆, η, ∆, ϵ, S, R) of G is a quasi-triangular Hopf algebra whose underlying vector space is isomorphic to C[G] b C G .Denoting by a b δ g ≡ a δ g , with a, g ∈ G, basis elements of D(G), the Hopf algebraic structure is defined by(a δ g ) ⋆ (b δ h ) := δ a −1 ga,h (ab δ g ) b a δ h −1 g (comultiplication) ϵ(a δ g ) := δ g,1(counit)S(a δ g ) := a −1 δ a −1 g −1 a(antipode) for all a, b, g, h ∈ G and z ∈ C. The definitions above imply that both C[G] and C G are equipped with the structures of Hopf subalgebras of D(G) and we denote by ι C[G] and ι C G the corresponding embedding maps.The quasitriangularity is then provided by the invertible element R ≡ (ι C[G] b ι C G )( R) with R := ÿ g∈G g δ g .(B1) Let us consider the braided monoidal category Mod(D(G)) of left modules over the quantum double D(G).It follows from C[G] and C G forming subalgebras of D(G) as well as the relation a δ g = (1 δ g ) ⋆ ÿ h∈G a δ h = ÿ h∈G a δ h ⋆ (1 δ a −1 ga ) that an object in Mod(D(G)) is a vector space V equipped with left C[G]-and C G -module structures satisfying the straightening formula δ g ▷(a ▷ |ν⟩) = a ▷(δ a −1 ga ▷ |ν⟩) , for any a, g ∈ and |ν⟩ ∈ V.In particular, simple modules of D(G) are labeled by pairs ([c 1 ], V ) with [c 1 ] a conjugacy class of G with representative c 1 and V a simple left module over the group ring C[Z [c1] ] of the centralizer Z [c1] = {g ∈ G | gc 1 = c 1 g} of [c 1 ] in G. Denoting the constituents of [c 1 ] by {c i } i=1,...,|[c1]| , we have V = Span C { |c i , v⟩} ∀ i=1,...,|[c1]| ∀ v=1,...,dim( V ) so that dim(V) = |[c 1 ]| • dim( V ).Introducing the set Q [c1] = {q ci ∈ G} i=1,...,|[c1] [c1] → End( V )) is an irreducible representation of Z [c1] ⊂ G.The monoidal structure in Mod(D(G)) can then be conveniently defined in terms of the tensor product (V b W, (ρ b σ) • ∆) of representations (V, ρ) and (W, σ), whereas the braiding is provided by the isomorphismR = (σ b ρ)(R) • swap : V b W ∼ − → W b V ,where 'swap' simply permutes the order of vector spaces in the tensor product.In the context of our work, the quantum double D(G) of a finite group G arises-up to a normalization factor-as the groupoid algebra of the tube category Tube(Vec G ).By definition Tube(Vec G ) is the category with objectset G and morphisms of the form a : g → a −1 ga.Its groupoid algebra is then defined as the algebra with underlying vector space C[ |g a − → a −1 ga⟩ ], over a, g ∈ G, and algebra product |g a − → a −1 ga⟩ ⋆ |h b − → b −1 hb⟩ = δ a −1 ga,h |g ab − → (ab) −1 g(ab)⟩ .