Quantum Current and Holographic Categorical Symmetry

We establish the formulation for quantum current. Given a symmetry group $G$, let $\mathcal{C}:=\mathrm{Rep} G$ be its representation category. Physically, symmetry charges are objects of $\mathcal{C}$ and symmetric operators are morphisms in $\mathcal{C}$. The addition of charges is given by the tensor product of representations. For any symmetric operator $O$ crossing two subsystems, the exact symmetry charge transported by $O$ can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center $Z_1(\mathcal{C})$. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in $Z_1(\mathcal{C})$ and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.


Introduction
The electric charge is a conserved quantity. Classically, we think that the electric charge is a continuous quantity and we talk about the charge density ρ. The global charge Q = ρ is conserved. If we divide the whole system into two parts A and B, and denote Q A = A ρ and Q B = B ρ, then (1.1) The change of charge in one subsystem must compensate that in the other. When the charge in A increases, there must be charge flowing from B to A, i.e., a current. Again one can consider the current density j , and we have the famous differential equation for local conservation of charge However, we know in reality that electric charge is discrete instead of continuous. The above treatment, including the density of charge or current as well as the differential equation, is only an effective approximation at macroscopic or statistical level. The case becomes even worse when we consider other symmetries. The electric charge is just the conserved quantity of global U(1) symmetry, taking value in integers (with appropriate unit). The addition of electric charges is the usual addition of integers. We may switch to, for example, angular momentum, which is the conserved quantity of the rotation symmetry, SU (2). The quantum angular momentum takes value in, no longer ordinary numbers, but the representations of the symmetry group SU (2). The addition of angular momentum is also more complicated than the addition of numbers. For example, the total angular momentum of two spin 1/2's together is the "direct sum" of a spin 0 singlet and a spin 1 triplet: 1/2 ⊗ 1/2 = 0 ⊕ 1.
In this paper we give a serious treatment to the local conservation of quantum symmetry charge, which seems long missing in the literature, although the mathematics involved is quite fundamental. The essential technical step is to realize that in a quantum system with symmetry, the symmetric operators are just the morphisms in the tensor (or fusion when the group is finite) category of symmetry charges, or equivalently, symmetric (or invariant) tensor networks. With this in mind, we show that for any symmetric operator crossing two subsystems, a symmetry charge can be extracted which we interpret as the symmetry charge transported by the operator. Moreover, we define the notion of quantum current, which is the collection of symmetric operators that can transport a symmetry charge over a long distance.
It can be seen directly from our definition that a quantum current is an object in the Drinfeld center of the fusion category of symmetry charges. Our work is a generalization of the Noether theorem: when the symmetry, which can be discrete, is given, the possible current is fully determined, by computing the Drinfeld center. The local conservation is now written as where C is the category of symmetry charges and Z 1 (C) is the category of quantum currents. The functor on the left hand side Hom(− ⊗ X , Y ) compute the ways how X can be converted to Y . The internal hom [X , Y ] Z 1 (C) is an object in Z 1 (C) that represents the functor Hom(− ⊗ X , Y ). Physically, [X , Y ] Z 1 (C) is the quantum current providing the universal answer for the ways how X can be converted to Y . We have to use the internal hom to compute the quantum current, including the charge difference and how the charge flows, for discrete symmetry charges. On the other hand, quantum currents can be identified with excitations in a topological order in one higher dimension. In fact, our work provides a concrete physical interpretation to the categorical symmetry [1][2][3][4][5][6][7][8][9][10][11][12][13].
The term categorical symmetry was first proposed in Ref. [1], which aims at providing an invariant shared by gapped phases before and after a phase transition. For example, the 1+1D transverse field Ising model has two gapped phases, 2 symmetric and 2 symmetry breaking: One exhibits the 2 symmetry while the other exhibits the dual 2 symmetry. Putting the two 2 together, one says that there is a categorical symmetry of 2+1D 2 topological order (or toric code model [14]), which has both 2 charges and 2 fluxes. From a purely mathematical point of view, this categorical symmetry is the Drinfeld center of the fusion category Rep 2 of symmetry charges. Because of the boundary-bulk correspondence of topological ordered phases [15][16][17], one can say, virtually, that categorical symmetry is the topological order in one dimension higher. Because of this relation, in this paper we call such invariant (topological order in one dimension higher) as the holographic categorical symmetry, in order to avoid possible confusion with e.g., fusion category symmetry [18][19][20][21] and other similar notions which may also be referred to as categorical symmetry. 1 However, the physical meaning of holographic categorical symmetry remains mysterious. Ref. [4,5] proposed to view holographic categorical symmetry as the transparent patch operators. Based on the idea of topological Wick rotation [11], Refs. [3,13] proposed to view holographic categorical symmetry (or the background category of the enriched category description [6,9] of gapped phase) as the sectors of non-local symmetric operators. Moreover, in the context of topological order, one can condense the excitations in the bulk topological order to obtain a boundary theory, or fuse defects in the bulk with the boundary theory. Thus, by topological Wick rotation, the algebras and defects in the holographic categorical symmetry (corresponding to the topological order in the bulk) should play an important role in classifying phases and phase transitions (corresponding to the boundary theory). Such point of view has been emphasized in Refs. [2,5,10].
Our formulation of quantum currents clarifies the confusion around various concepts regarding holographic categorical symmetry. We give a rigorous definition, Definition 4. 13, for what operators qualify as quantum currents. We also give a definition, Definition 4. 19, for what quantum currents are condensed. These definitions are tested in concrete one dimensional fixed-point lattice models. We prove that the condensed quantum currents in the fixed-point model indeed form a Lagrangian algebra in the Drinfeld center, which also determines the fixed-point defects (or excitations) in the fixed-point model. Therefore, we established the correspondence between quantum currents and the holographic categorical symmetry; holographic categorical symmetry is about the transport property of the physical system. We also want to emphasize that the quantum currents can be measured, observed and understood, in the same dimension of the physical system; no fictional one dimensional higher bulk is required.
The paper is organized as follows: in Section 2 we give some necessary preliminary notions and fix our notations; in Section 3 we explain basic techniques on how to manipulate symmetric operators, viewing them as graphs or tensor networks; in Section 4 we motivate the definition for quantum current and introduce some related notions such as the charge transported by symmetric operators and the condensation of quantum currents; in Section 5 we give a rigorous treatment to the renormalization process in 1+1D lattice models; finally in Section 6 we show that in the fixedpoint lattice models condensed quantum currents form a Lagrangian algebra and give a series of examples. We summarize main contents in this paper in Table 1. Ground state subspace of two defects A ′′ B ′ A and A B A ′ :

Condensed quantum currents
Internal hom [A, A] ∈ Z 1 (C), which is also a Lagrangian algebra in Z 1 (C) Holographic categorical symmetry Frobenius algebras in Rep G are classified by (H ⊂ G, ω 2 ) For trivial ω 2 , A := Fun(G/H),

Preliminaries and notations
We first review the group representation category and fix our notation for graphical calculus. ρ g is referred to as the group action or symmetry action. An object in Rep G is physically referred to as a (G-)symmetry charge.
We may use simply V to denote a representation (V, ρ), and write the group action ρ g (a) as g ⊲ a or ga, when the explicit form of ρ is not important for the discussion.
• A morphism between two representations (V, ρ) and (W, τ) is a linear map f : V → W that commutes with group actions, f ρ g = τ g f for all g ∈ G. The space of morphisms from (V, ρ) to (W, τ) is denoted by Hom((V, ρ), (W, τ)). In particular, the endomorphism space Hom((V, ρ), (V, ρ)) is exactly the subspace of symmetric operators on V , which satisfy ρ g f ρ g −1 = f . The following terms morphism in Rep G, -intertwiner, intertwining operator invariant tensor, -symmetric tensor, 2 -symmetric operator, will be used interchangeably.
Graphically, an object is represented by a line and a morphism is represented by a node between two lines Composition of morphisms is done from bottom to top 3) The representation category enjoys additional nice structures, one of which is the tensor product. Given two representations (V, ρ) and (W, τ), their tensor product is again a representation with action given by (ρ ⊗ τ) g = ρ g ⊗ τ g . (2.5) Tensor product is graphically represented by juxtaposing lines There is always the trivial representation 1 := ( , id), id g = id for all g ∈ G. 1 is the unit of tensor product. For any representation (V, ρ), there is a dual representation (V * , ρ ⋆ ) 3 where the underlying vector space is the dual vector space V * := Hom Vec (V, ) 4 , and the group action is graphically represented as (2.8) If one choose a basis of V and the corresponding dual basis of V * , then the matrix representation of g on V * is the transpose of the matrix representation of g −1 on V . We can check that the above defined (V * , ρ ⋆ ) is indeed the dual object of (V, ρ) in Rep G: The pairing between V * and V The copairing where {a} is a basis of V and {δ a } the corresponding dual basis δ a (b) = δ a,b , is also symmetric Therefore, the pairing and copairing both remain as morphisms in Rep G and exhibit (V * , ρ ⋆ ) as the dual object of (V, ρ). Another structure is the direct sum. Given two representations (V, ρ) and (W, τ), their direct sum (V, ρ) ⊕ (W, τ) := (V ⊕ W, ρ ⊕ τ) is again a representation with action given by (2.13) An isomorphism is a morphism invertible under composition. Two representations are isomorphic V ∼ = W when there is an isomorphism between them; in other words, V and W differ by a basis change which commutes with group actions. A nonzero representation is irreducible (or simple) if it is not isomorphic to a direct sum of two nonzero representations. For a compact group G, all finite dimensional representations are completely reducible, i.e. isomorphic to a direct sum of irreducible representations. For reader's convenience, we review the categorical formulation of direct sum here Definition 2.2. In a category whose hom-sets form abelian groups and composition is bilinear, the direct sum of two objects A, B, if exists, is an object Y together with two pairs of morphisms (2.14) We refer to p A , p B as projections 5 and q A , q B as embeddings.

Remark 2.3.
Such an object Y is simultaneously a product and coproduct of A and B, and by the universal property of limit, is unique up to unique isomorphism. Thus it is fine to talk about the direct sum and denote it by A ⊕ B.
Rep G is also a unitary category with unitary structure given by the usual Hermitian conjugate.

Remark 2.4.
In a unitary category such as Rep G, it is always possible to choose q i = p † i .

Now suppose that
where X i are irreps (irreducible representations). In more elementary words, the above means that after a change of basis of V ⊗ W , the group actions all become block-diagonal. We depict the projection map p i from V ⊗ W to X i by (2.16) and the embedding map by (2.17) The normalization is taken to be Two fundamental constructions will be useful later and we fix the notation here: Definition 2.5. Given a set S, the vector space of finite formal linear combinations of elements in S, is called the free vector space on S, and denoted by C(S). Definition 2.6. Given a subset T of a vector space V , the subspace of all linear combinations of vectors in T , is called the space spanned by T , and denoted by 〈T 〉.
Remark 2.7. S is automatically a basis of C(S). 〈T 〉 is the smallest subspace of V containing T , and T is not necessarily a basis of 〈T 〉.

Submission
The following notion is also useful in later analysis: Definition 2.8. Given two finite dimensional Hilbert spaces V and W , a linear map U : which is equivalent to U † U = id W . On the other hand, U : V → W is called partially isometric if any one of the following equivalent conditions holds (denote by ker U ⊥ the orthogonal complement ker U in V and by Im U the image of U): 1. The restriction U : ker U ⊥ → W is isometric; 2. The restriction U † : Im U → V is isometric; 3. The restriction U : ker U ⊥ → Im U is unitary; Remark 2.9. For technical simplicity, in this paper we mainly work with the example Rep G whose objects and morphisms have underlying vector spaces and linear maps that can be calculated concretely. We would like to emphasize that the graphical calculus techniques here and below remain valid even if we consider a more general unitary fusion category (UFC). For a general fusion category, we continue to interpret objects as symmetry charges and morphisms as symmetric operators; however, we lose access to underlying vectors and linear maps, unless we are willing to deal with (weak) Hopf algebras and their (co-)modules. 1. An object of 3. The tensor product of (X , β X ,− ) and Example 2.11 (Permutation group S 3 ). S 3 is the simplest non-Abelian group with 6 elements: where a, b are called two generators of S 3 . S 3 has three irreducible representations λ 0 , λ 1 , τ listed below: where ω = e 2πi/3 . Let Irr(C) denote for the set of isomorphism classes of simple objects in semisimple category C. We have Irr(Rep Denote the basis of Λ as {0, 1}. Correspondingly Λ * has dual basis {δ 0 , δ 1 }. Then Λ * is isomorphic to Λ through the isomorphism 0 → δ 1 , 1 → δ 0 . We also list the data in Z 1 (Rep S 3 ): Therefore, there are 8 simple objects in Z 1 (Rep S 3 ) labelled as: Example 2.12 (Quaternion group Q 8 ). Q 8 is a non-Abelian group with 8 elements: (2.25) Q 8 has five irreducible representations γ 0 , γ 1 , γ 2 , γ 3 , Γ listed below: 6 We follow the usual convention and abuse the same notation β X ,− and β Y,− for different pairs (X , β X ,− ) and (Y, β Y,− ). The reader is reminded that the pairs should be understood as a whole; the half-braiding β X ,− depends on the pair (X , β) instead of only the object X . Indeed, the same object X may be equipped with different half-braidings to form different objects in the Drinfeld center.

Example 2.13 (Special unitary group SU
where α denotes the complex conjugate of α. SU (2) has infinite numbers of irreps labelled by l, which are non-negative integers and half-integers. The dimension of irrep labelled by l is 2l + 1. We list generators for irreps l = 0, 1/2, 1 below: For each l, group elements in SU (2) represented by l are {e iθ (n x J x +n y J y +n z J z ) |θ , n x , n y , n z ∈ , n 2 x + n 2 y + n 2 z = 1}. The dual representation l = 1 1. For any two disjoint subsets K 1 and K 2 , the representation associated to their disjoint union is the tensor product of those associated to K 1 and K 2 The total Hamiltonian has the form We denote such a quantum system by (L, H K , H).
Remark 2.15. The empty set ⊂ L is necessarily associated with the trivial group representation. For the singleton subset {i} ⊂ L (i is just a lattice site), we will simply write the associated representation as (H i , ρ i ). It is clear that When no confusion arises, we will abuse notation and do not distinguish O K fromÕ K .

Symmetric operators as graphs
In this section we set up the formulation that any symmetric operator in a system with onsite symmetry G can be represented by graphical calculus in Rep G. In the language of tensor network, any symmetric operator can be represented by a G-invariant tensor network. Each local Hilbert space can be decomposed to irreducible representations. 2. For each local Hilbert space or more generally each representation V , the projection maps to and the embedding maps from the representative irreps: p a;n V : V → X a , q V a;n : X a → V where n goes from 1 to the multiplicity of X a in V .
3. In particular, for each tensor product of two representative irreps, the projection maps to and the embedding maps from the representative irreps: Tree graphs decorated by X a and p, q's in a charge decomposition serve as bases of intertwiner spaces. We may draw any graph in Rep G and interpret it as an intertwiner between several representations. More specifically, let V 1 , . . . , V n and W 1 , . . . , W m be representations, an intertwiner in f ∈ Hom(V 1 ⊗ . . . ⊗ V n , W 1 ⊗ . . . ⊗ W m ) can be represented by the graph: With a chosen charge decomposition, the intertwiner above can be expanded in terms of "basis" graphs. As an illustrative example, consider the intertwiner space Hom( One first decompose all external legs to irreps, and then fuse the irreps one by one in a chosen order. One choice of basis graphs is where a, b, t, d, e, i, j, k, l, m, n runs over all possible values. Another choice is where a, b, c, d, e, i, j, k, l, m, n runs over all possible values. The only remaining third choice is left as an exercise for the reader. In general, a basis of the intertwiner space Hom(V 1 ⊗ . . . ⊗ V n , W 1 ⊗ . . . ⊗ W m ) can be obtained by decorating a tree graph, which is bivalent between external and internal legs and trivalent between internal legs: • The external legs are fixed and decorated by V 1 , . . . , V n , W 1 , . . . , W m ; • The internal legs are decorated by representative irreps; • The vertices are decorated by projection and embedding maps from a chosen charge decomposition.
Different orders to fuse internal legs lead to different tree graphs and thus different sets of basis intertwiners. Basis intertwiners on different tree graphs are related to each other by the F-moves. After fixing basis graphs, every symmetric operator has a unique linear expansion and concrete calculation is possible. However, for the purpose to read out the total symmetry charge being transported by symmetric operators, we are going to use a slightly more effective method to rewrite the symmetric operators.

Definition 3.2.
Consider an intertwiner f ∈ Hom(V, W ). An image decomposition (l,r) of f is a factorization of f through its image where Im f is the image of f (Definition B.8). Graphically, In more concrete words, we can think the image decomposition as a singular value decomposition performed on symmetric tensors, which we call symmetric singular value decomposition (SSVD). Explicitly, there are two steps in an SSVD of f : 2. The usual SVD performed on the m a ×n a matrices M a mn for all representative irreps X a , leads to a SSVD of f : where v a and w a are n a × n a and m a × m a unitary matrices, and Σ a is a m a × n a rectangular diagonal matrix with non-negative real entries.
where Here it is easy to see that the ambiguity ofl andr arises from choices of block diagonal unitary matrices in the SSVD and how one separate the non-zero singular values of M a .

Remark 3.3.
In the factorization f =rl,r is a monomorphism. And in any abelian category (in this paper all categories used are moreover semisimple),l is an epimorphism, proof of which is omitted.
Remark 3.4. The pair (l,r) is clearly not unique. Such ambiguity is related to the sectors of quantum currents and morphisms between quantum currents. We will come to this point later.

Remark 3.5.
We introduce a trial-and-error method to compute the image, which is more straightforward (if succeed). Given two representations (V, ρ), (W, τ), note that we can endow the operator space Hom Vec (V, W ) with a natural group action by post-composing τ g . For an intertwiner f : V → W , we can then consider the cyclic sub-representation of Hom Vec (V, W ) generated by f , C(G) f := 〈τ g f , g ∈ G〉. Pick any v ∈ V , there is an intertwiner: with which it is easy to check that ξ v is symmetric. If C(G) f happens to be isomorphic to Im f for some v, the computation is completed. For this method to work, the necessary and sufficient conditions are 1. Im f ∼ = C(G) f is a cyclic sub-representation of W . v should be chosen as a preimage of a cyclic vector in Im f . In this case ξ v automatically maps onto Im f .

2.
ξ v must also be injective, which means that ker There are simple cases that these conditions are satisfied, for example, V is the regular representation C(G) of Abelian group G, where one can take v = e the identity element of G. However, in practice, the second condition is difficult to check, and also we can not know in advance whether Im f is a cyclic representation or not. Such method can only be used with trial-and-error. Consider an intertwiner f ∈ Hom(Λ ⊗ Λ, Λ ⊗ Λ) taking the following form: where the similarity transformation can be thought as changing to the basis with fixed symmetry charge, i.e., changing f ∈ End(Λ ⊗ Λ) to a M a ∈ End(Λ ⊕ λ 0 ⊕ λ 1 ). Since M Λ = 1, there is only one non-zero singular value in M Λ , 7 we see Im f ∼ = Λ. Graphically, where by requiringl andr to be intertwiners, the image decomposition is solved as where c 1 , c 2 are non-zero complex numbers such that c 1 c 2 = 1. A different choice of (l,r) correspond to changing c 1 and c 2 to c ′ 1 and c ′ 2 such that c ′ 1 c ′ 2 = 1. Here since dim Hom(Λ ⊗ Λ, Λ) = 1, choices of (l,r) only differ by a scalar.
Similarly, a general intertwiner f ∈ Hom(Λ ⊗ Λ, Λ ⊗ Λ) reads Consider an intertwiner f ∈ Hom(Γ * ⊗ Γ , Γ ⊗ Γ * ) taking the following form: where the similarity transformation can be thought as changing to the basis with fixed symmetry charge in Hom( Since there is only one non-zero singular value in M γ 1 , we see Im f ∼ = γ 1 . Graphically, (3.20) where the image decomposition is solved as where c 1 , c 2 are non-zero complex numbers such that c 1 c 2 = 1.

Quantum current
In this section we motivate the definition of quantum current. We would like to define a quantum current as a collection of symmetric operators that can transport symmetry charge all over the space. This physical idea will be put into precise mathematical definition later. For concreteness we will first focus on lattice system in one spatial dimension with onsite symmetry. We will show that a quantum current carries two important quantities: one is the symmetry charge being transported, and the other is the half-braiding that determines how the current extends over the space without leaving things behind along its path. Together, a quantum current is identified with an object in the Drinfeld center Z 1 (C).

Every symmetric operator carries a symmetry charge
To begin with, let's consider a bipartite system, A symmetric operator acting on the total space, is by definition an operator O ∈ Hom Vec (H, H) that commutes with symmetry actions In general, O does not commute with "partial" group actions ρ A or ρ B . Indeed, it can transport symmetry charge between A and B. We are tempted to represent O by a trivalent tree graph Then we interpret X as the symmetry charge transported by O from subsystem A to subsystem B and the intertwiners l, r as describing how the charge X leaves A and arrives as B. However, a large enough representation X can always do the job to represent O. We need to find the smallest X .
Note that there is a natural isomorphism ε by "bending legs": The smallest X is nothing but ImŌ.
In short, bending the legs on the graph corresponds to rotating the positions of indices of the tensors (in the corresponding bases and dual bases). Readers familiar with tensor network may find that the above is just doing SSVD for the tensor O while viewing two H A legs a ′ , a as input and two H B legs b ′ , b as output.
A more physical way to understand the above is to note that the dual group representation is the anti symmetry charge. Thus, H * A ⊗ H A may be thought as calculating the initial charge minus the final charge in A, i.e., the charge decrease in A, while H B ⊗ H * B may be thought as calculating the final charge minus the initial charge in B, i.e., the charge increase in B. The result ImŌ is the transported charge. We conclude the above discussion in the following definition.

Definition 4.2.
Given a symmetric operator O acting on a bipartite system H A ⊗H B , the symmetry charge transported by O from A to B is ImŌ, denoted by O↑ B A := ImŌ, whereŌ is given by the "bending leg" isomorphism (4.4).

Now we have
where ε is the natural isomorphism (4.4), and there are further natural isomorphism giving a bijection betweenl and l, and giving a bijection betweenr and r. These three steps are 1. Natural isomorphism ε: We apply the rotation of symmetric tensorŌ 2. Image decompositionŌ =rl: SSVD in Definition 3.2; 3. Natural isomorphism ε −1 : We apply the inverse rotation of tensors l ac b =l c a b and r b ca =r ba c .

Remark 4.3. The symmetry charge O↑
the total symmetry charge is conserved.

and the basis change between them is
where we find O↑ B A = ImŌ = Λ. And we solvel andr as the following form: where c 1 and c 2 are non-zero complex numbers such that c 1 c 2 = 1.
2. Consider the direct sum decomposition 1 , and the basis change between them is (4.14)

Submission
Then we havē where we find O↑ B A = ImŌ = 1.
At this stage, we find that O transports an angular momentum of spin 1.

Proposition 4.7.
When the conditions in Remark 3.5 are satisfied, we can simply define the space O↑ B A to be the operator space spanned by ρ B g Oρ B g −1 , g ∈ G: In other words, O↑ B A is the closure of O under the "partial" conjugation group actions on operators. Note that 4. The group action on O↑ B A is the "partial" conjugation by ρ B , with respect to which the tensors l, r in Eq. (4.5) are intertwiners. More precisely, one can associate the group representation In other words, as a group representation, But, it is easy to see that one can also associate this operator space with the group representation It is easy to see that A are dual to each other; the total symmetry charge of A, B together is conserved.
Proof. When the conditions in Remark 3.5 are satisfied byŌ, Then graphically, Remark 4.8. For the method in Remark 3.5 to work, the image ofŌ, i.e., the transported symmetry charge, is necessarily a cyclic representation. In Proposition 4.7, given an arbitrary symmetric operator O ∈ End(H), we actually do not know in advance whether the transported symmetry charge O↑ B A is a cyclic representation or not. It is just a convenient method that we can try at first. If O↑ B A is not cyclic, we will fail to obtain the correct result, i.e., there is no solution for l and r in Eq.
In this situation, we should extract the transported symmetry charge by the original method in Definition 4.2.
where α + is the trivial representation and α − is the sign representation. Consider the regular representation on a qubit R := C(|1〉, |ζ〉), where we identify |1〉 with the logical 0 and |ζ〉 with the logical 1, and write the nontrivial action ρ ζ = σ x . Then we have Consider two regions A and B both carrying the regular representation R, and a symmetric operator Let us try the method in Proposition 4.7 to extract the symmetry charge transported by O: We find no contradiction in solving l, r. Therefore, where one possible choice of l, r is 26) or in more compact tensor notation: .
Denote the basis of Λ as {0, 1} and the basis of λ 1 as {o}. One possible choice of l, r is Here since the intertwiner spaces Hom(Λ ⊗ λ 1 , Λ), Hom(λ 1 ⊗ Λ, Λ) are all of dimension 1, other choices of l and r are only up to a scalar.
Denote the basis of Γ as {|0〉, |1〉} and the basis of γ 2 as {|x 2 〉}. One possible choice of l, r is Other choices of l and r are only up to a scalar. And Denote the basis of γ 3 as {|x 3 〉}. One possible choice of l, r is Other choices of l and r are only up to a scalar.

Symmetry charge flows via half-braiding
Now consider a tripartite subsystem where for simplicity we omitted the subscript of β in the graph which can be unambiguously read out from the decorations on the legs. We also draw the β node intuitively as a crossing-over. β must be compatible with the arbitrary choice of M, which turns out to be the following conditions: (a) β commutes with local symmetric operators (naturality): for any f : Remark 4.14. Such β is exactly the half-braiding. The above two conditions are exactly Eq. (2.20) and (2.21). Our convention for β is also justified.
It is then clear that a quantum current is associated with the pair (Q, β Q,− ) which is an object in the Drinfeld center Z 1 (C). Conversely, given an object (X , β X ,− ) ∈ Z 1 (C) we can construct quantum current operators by pasting β X ,− in the intermediate sites and fusing X to the source ant target sites with some choice of source and target intertwiners.  [4,5] and unconfined (non-local) symmetric operator [3] are defined elsewhere. Our definition here contrast with them in two aspects: (1) our definition is model independent; it only cares about the symmetry, or group representations, but not specific Hamiltonians. The condition (4.38), which is mathematically the naturality of half-braiding (2.20), requires a quantum current to commute with any possible local symmetric operators (including local Hamiltonians) that does not overlap with the source and the target. (2) We point out the condition (4.39), which is mathematically the hexagon equation for half-braiding (2.21), and physically the requirement for a quantum current to be able to consistently extend over an arbitrary long distance. This condition is overlooked in previous works [3][4][5] By Proposition 4.7 and ρ ζ = σ z in basis {|+〉, |−〉}, P + ⊗ P + and P − ⊗ P − both transport charge α + . However, if one adds them up, We extract the transported charge by O by going through the process in Diagram (4.6). We have , and the basis change between them is . This is what we mean by not closed under addition. 11 Denote the dual basis of Then we havē The explicit forms of l, r are easy to compute and we omit them. Therefore, we conclude that symmetric operators P + ⊗ P + , P − ⊗ P − transport the symmetry charge α + , different from α + ⊕ α + transported by their addition, i.e., (α + , β) is not closed under addition.
Because of this subtlety, we introduce a finer notion, that instead of allowing arbitrary source and target intertwiners, we restrict to bimodules of End(H s ) ⊗ End(H t ). (4.45) We call the set of realizations with s, M, t, l ∈ L, r ∈ R, a sector of (Q, β), denoted by (Q, β) L,R forms a vector space, and moreover an End(H s ) ⊗ End(H t )-bimodule. We call a sector simple if this bimodule is simple.
terms whose support is within M. Therefore, O can only cost energy, or create/annihilate excitations around s or t. We can also consider whether a simple sector (Q, β) L,R and that O ′ is condensed. By simpleness, the difference between O and O ′ must be local symmetric operators on s and t. Thus, we know that excitations created/annihilated by O around s or t must be of the trivial type. Therefore, a simple sector is condensed if its realizations create/annihilate only excitations of the trivial type.
For non-simple quantum current, e.g., (Q, β) = (Q 1 , β 1 ) ⊕ (Q 2 , β 2 ), by definition, if any of (Q 1 , β 1 ) and (Q 2 , β 2 ) is condensed, we will say (Q, β) is condensed. This definition is more natural than requiring that all realizations are condensed, as can be seen in Section 6.1 and further explained in Remark 6.6. There, we will also elaborate more on the deep connection between the condensation of quantum currents and the emergent symmetry of gapped quantum phases.

Morphisms between quantum currents
We will show that a morphism in the Drinfeld center defines a reasonable transformation, or morphism, between quantum currents.
A morphism in the Drinfeld center is an intertwiner that commutes with the half-braiding. Suppose we have a realization of quantum current with charge Q and half-braiding β Q,− . Pick a morphism f : (Q, β Q,− ) → (X , β X ,− ) in the Drinfeld center. Inserting f in the graph we obtain a new realization which is associated with charge X and half-braiding β X ,− . We also see that the morphism f in the Drinfeld center can change the intertwiners l, r on the source and target sites. The entire process of dragging f from t to s thus defines a map between (Q, β) s,M,t −,r ′ . Clearly, such map, which happens on the Q or X leg, commutes with the action of local symmetric operators on s, which happens on the H s legs. In other words, f defines an End(H s )-bimodule map. Meanwhile, the map r ′ → r ′ ( f ⊗ id H t ) also commutes with action of End(H t ).

Remark 4.21.
It is desired to have a notion of morphisms between quantum currents, purely from the point of view of symmetric tensors, such that the category of quantum currents is equivalent to the Drinfeld center. We are unclear about such a definition for the moment. The above discussion suggests to define the bimodule maps over local operator algebras as the morphisms between quantum currents, which is the point of view taken in [3,6]. It should work for half-infinitely long quantum currents, but not work well for finitely long quantum currents.

Renormalization of 1+1D lattice system with symmetry
Physically, the renormalization fixed-points represent phases of matter and are thus of great interest. In this section we try to give a rigorous treatment of the renormalization process of 1+1D lattice model. Based on this, we give a general analysis of 1+1D gapped lattice model at fixedpoint.

Renormalization fixed-point and Frobenius algebra
Recall Definition 2.14. For a 1D lattice, without losing generality, we may think the set L of lattice sites as a linearly ordered set.  Hilbert space renormalization is a pair ( f , {U i }), a lattice renormalization f with a collection of intertwiners indexed by L ′ ,
3. Let U = ⊗ i∈L ′ U i . When the following condition holds, where ∆E is some energy zero point shift, we call the pair ( f , {U i }) a (system) renormalization, and meanwhile call (L ′ , H K ′ , H ′ ) the renormalized system of (L, H K , H) after ( f , {U i }). 3) When f is not surjective, for f −1 (i) = , U i : H = → H i just fixes a state in H i . In other words, the non-proper Hilbert space renormalization adds sites in fixed states to the original system.

Remark 5.3.
There is a natural composition of Hilbert space renormalizations Graphically for example, We can speak of whether a system (L, H K , H) is invariant with respect to a renormalization ( f , {U i }). However, in practice we expect to learn from renormalization some information of quantum phases, which are quantum systems at thermodynamic limit (infinite system size) whose interactions are local. To this end, we focus on the special case L = , i.e. infinite chain.
The quantum phases are also almost uniform with (at-least emergent) translation invariance. For translation invariant systems, we hope to study the renormalizations that only depends how many sites are combined into one, but not on positions.
Graphically, for example we sketch a Hilbert space renormalization generated by {m 0 , m 1 , m 2 } on the following local patch, where we use • to depict the empty subset of the lattice whose associated Hilbert space is .
Now we are ready to define renormalization fixed-point. Proof. Recall the composition of Hilbert space renormalization in Eq. (5.4), we have the following equations on renormalizations: The above renormalizations are generated by {m 0 , m 1 , m 2 } as shown in Diagram (5.6). Therefore, we obtain which means that (A, m = m 2 , η = m 0 ) is an unital associative algebra. Taking Hermitian conjugate one has that (A, m † , η † ) is a unital associative coalgebra. By definition m n are partial isometries, while unitality implies that m is an epimorphism 12 and Im m = A, thus i.e., mm † = id A . To show the Frobenius condition and Then by positive definiteness we conclude (id A ⊗ α)(m † ⊗ id A ) = 0 and thus Corollary 5.9. For n ≥ 3, m n is just the n-ary multiplication, generated by (any sequence of) the binary multiplication m = m 2 : Therefore, for any n ≥ 1, m n is an epimorphism and Im m n = A. In particular, we have m n m † n = id A .
In the following a Frobenius algebra is always assumed to be isometric.
Remark 5.11. For readers familiar with the abstract notions, we have established a connection between the renormalization process in 1+1D and the operad theory. We believe that renormalization fixed-points always have certain (weakly) associative algebraic structures. Proof. From the Frobenius condition (5.12) and the associativity of m † , we have

Commuting projector Hamiltonian and ground states
, which implies m † m on neighboring sites commutes. And from Eq. (5.11), we have i.e., m † mm † m = m † m, which implies that m † m is a projector. Next, we show ( , A, −m † m) is a fixed-point model. Recall Definitions 5.10 and 5.1. Let j ∈ be a site. Consider a special lattice renormalization All other terms obviously commute with m 2 . Therefore, m 2 H m † 2 = H − 1. which adds an extra site (in the state of the unit of the algebra) to the model. We may try to include more interactions, such as m † n m n . Straightforward calculation shows that the interaction m † n m n overlapping with this extra site gets renormalized to the interaction m † n−1 m n−1 . Moreover, to preserver translation invariance, we necessarily needs to add extra sites by non-proper renormalizations in a translation invariant way. Such fixed-point in a stronger sense is possible to be defined, but greatly complicates the analysis. We thus focus on the nearest neighbour interactions which is at fixed-point in a weaker sense (only invariant under proper renormalizations).
. Since all terms of H are commuting projections, the ground states of H are given by the common eigenstates of (m † m) i for all i with eigenvalue +1, i.e., the ground state subspace is

Fixed-point boundary conditions
Now we extend the above discussion to include boundaries of 1D lattice. Without losing generality, let's consider the special case that the lattice is L = . • H = Q + i≥1 P i where P i supported in {i, . . . , i + n − 1}, P i = P ⊗ id for some P ∈ End(A ⊗n ), and 0 is in the finite support of Q.
We may simply denote this system by ( , A, P, M , Q).
Graphically, for example we sketch a Hilbert space renormalization generated by {ρ 1 , m 2 } on the following local patch, . P is a Hermitian projection P † = P and P 2 = P.
(e) Due to (d), one can take the isometric image decomposition of P, i.e., r : M ⊗ N → Im P such that r † r = P and r r † = id Im P . Graphically, and together with r = r r † r = r P we know that We need to show that there exists a uniquef : Im P → X such thatf r = f . Note that The existence is guaranteed and we only need to prove the uniqueness. This is easy, since for anȳ f satisfyingf r = f , one must havef =f r r † = f r † . Therefore, (Im P, r) is the coequalizer: Similarly to Theorem 5.20(a), the ground state subspace is given by Im(ρ † ρ) 0 i (m † m) i (λ † λ) J −1 . By repeatedly applying the Frobenius conditions one can show that We may fix the bulk and consider the boundary change: Now we consider fixing an Frobenius algebra A and collect all possible fixed-point boundaries (i.e., all right A-modules, and A-module maps) to form a category, denoted by C A (Definition C.9).

Remark 5.25.
In the above analysis, the action ρ = ρ 1 and the boundary change U 0 are assumed to be partially isometric. By Proposition F.14, any A-module is isomorphic to a sub-module of a free module M ⊗ A for some M ∈ C. The action of the free module M ⊗ A is just id M ⊗ m, which is partially isometric, (id M ⊗ m)(id M ⊗ m) † = id M ⊗A . Therefore, any A-module is isomorphic to one with an partially isometric action. Also since C A is semisimple and unitary, partially isometric A-module maps and general A-module maps differ by at most a scaling. Thus when considering all possible boundary conditions, we can safely drop the requirement for the actions and module maps to be partial isometries. Example 5.27. If we take A = 1 the tensor unit in C (A = the trivial representation in Rep G), we have C 1 = C. When viewing C itself as C-module, we have a self enriched category C C where the internal hom is [V, W ] = W ⊗ V * . As we have explained before, tensor product is the "addition" of symmetry charge, and the dual representation is the anti charge, thus the (internal) hom is the "subtraction" of symmetry charge. For vector spaces, we have Hom Vec (V, W ) = W ⊗ V * ; the Hom C in C can be viewed as an internal hom in Vec. We may write such intuition as With such intuition, the internal hom adjunction Eq.(5.40) is "translated" to This "translation" is a good way to understand the idea behind the internal hom adjuction. However, there are different Hom's (in different categories) involved. Note that for the self enriched category C C, (5.45) By Yoneda Lemma we know that when using only internal hom, we have, rigorously (5.46) If we consider C = Rep G as a usual category, Hom C (V, W ) = W ⊗ V * does not subtract symmetry charge. To find the current and determine the local conservation of charge, we have to compute the difference of symmetry charge, and thus enriched category and internal hom is a must.
Example 5.28. The Drinfeld center Z 1 (C) has a natural action on C, by forgetting the half-braiding and then taking the tensor product. Thus we can also talk about the enriched category Z 1 (C) C and the interal hom [−, −] Z 1 (C) in Z 1 (C). In this case, the internal hom not only computes the charge difference, but also how the charge flows (i.e. the half-braiding). In other words, internal hom computes the quantum current. With such interpretation, we call the adjunction Note that here we made use of the action of Vec on any semisimple category, that given an n-

Fixed-point defects
We can further consider the fixed-point defects between two fixed-point models given by two Frobenius algebras A and A ′ . In particular, excitations are viewed as defects in the same model (i.e., between A and A). By similar arguments as our previous discussions on fixed-point boundaries, we have

4). We may also denote the bimodule by A B A ′ for clarity.
Proof. Similar to the proof of Theorem 5.8, by composition of Hilbert space renormalization, we can all be proved similarly as before. Proof. Similarly to Corollary 5.22, the ground state subspace is given by the image of the following intertwiner In fact, Corollary 5.22 is a special case by taking A ′′ = A ′ = 1.
(5.50) Thus, although we build our model using symmetric operators in C, after renormalization of A, we find that the category of excitations becomes A C A . As symmetry is reflected by how excitations are "added up" or "fused", we should say that there is a new emergent symmetry A C A at low energy.

Theorem 5.34. A defect change between fixed-points is an A-A
Observe that in this model ( , A, −m † m), the Hamiltonian m † m is a symmetric operator in C, while the emergent symmetry at fixed-point is no long C. Such phenomenon is in line with the usual spontaneous symmetry breaking (SSB) where the Hamiltonian has symmetry but the ground state breaks symmetry. Therefore, we consider our models as generalized spontaneous symmetry breaking phases; we call A C A connected to C by (generalized) SSB, i.e., by a model whose Hamiltonian has symmetry C but the ground state subspace is A and excitations have symmetry A C A . The ground state breaks symmetry unless A is the trivial algebra.

Example 5.36. The usual complete SSB is achieved by taking
Physically, we expect that A C A should have the same quantum currents as C. Indeed,

Theorem 5.37 (EGNO15 [24]). As a unitary fusion category
is the category of C-module functors, and rev means reversing the tensor product. A C A is (categorically) Morita equivalent to C; they have the same Drinfeld center Z 1 ( A C A ) ∼ = Z 1 (C). Moreover, any unitary fusion category that is Morita equivalent to C, is equivalent to A ′ C A ′ for some algebra A ′ ∈ C.
Corollary 5.38. Connection by spontaneous symmetry breaking is just Morita equivalence.

The fixed-point model
The Levin-Wen or string-net models [25] in 2+1D can be understood as taking a gapped boundary condition (boundary excitations are described by a UFC) as input, and producing a lattice model for the bulk topological order [26,27], which exhibits boundary-bulk correspondence. In the last section we have figured out the gapped fixed-point of 1+1D lattice model with a given symmetry, together with all the possible boundary conditions, as well as excitations. Below we will further analyse the fixed-point Hamiltonians given by Frobenius algebras in C. In these models, we can verify the holographic principle: boundary determines bulk, or, bulk is the center of boundary [15][16][17], in the enriched setting [23,28]: Here • A is a Frobenius algebra in C; • C A is the category of boundary conditions, which is a C-module, and by the canonical construction (see Remark 5.26), the boundary conditions form the enriched category C C A ; • Fun C (C A , C A ) ∼ = A C rev A describes excitations in the 1+1D lattice model; • Z 1 (C), which we identify with quantum currents, are operators that transport the excitations; is the E 0 -center of C C A defined in the 2-category of enriched categories [23].
is the data describing the bulk. Again by the canonical construction, what we need to verify is that the excitations (i.e., Fun C (C A , C A )) naturally form a monoidal [23] module over the quantum currents Z 1 (C). We further list the holographic categorical symmetry correspondence between our 1+1D model with symmetry and 2+1D topological order with gapped boundaries described by the string-net model, both of which can be characterized by Eq. (6.1) that taking the E 0 -center of an enriched category, in Table 2.
C A 1+1D gapped quantum system with symmetry described by UFC C A 2+1D topological order described by string-net model with input UFC C C A 0+1D fixed-point boundary conditions 1+1D boundary string-net model Condensed quantum currents Condensed bulk excitations Fixed-point excitations 1+1D boundary excitations Here [A, A] is the internal hom of Frobenius algebra A (viewed as the trivial A-Abimodule), which is a Lagrangian algebra in Z 1 (C) describing condensed quantum currents. We will introduce it in subsection 6.1 below.
Remark 6.1. A finite semisimple left C-module M is equivalent to C A for some algebra A ∈ C.
Remark 6.2. Similar constructions of 1+1D lattice model can be found in Refs. [6,21,29]. We motivate the construction from the idea of renormalization and give analysis on excitations and quantum currents.

Condensation of quantum currents
In the following we show that, in the fixed-point model ( , A, −m † m) determined by the Frobenius algebra (A, m, η), the condensed quantum currents form a Lagrangian algebra in the Drinfeld center Z 1 (C). Mathematically, the category of modules over the Lagrangian algebra in Z 1 (C) is automatically a monoidal module over Z 1 (C). As we will see soon, this category is also exactly the category of excitations.
Recall Definition 4.19. Supposing that the quantum current (Q, β) is condensed in the model ( , A, −m † m), then there exists a realization O ∈ (Q, β) whose target intertwiner commutes with the two terms of the form m † m supported around the target site. That is to say, there is r ∈ Hom(Q ⊗ A, A) such that Note that when such r exists, we can choose the source intertwiner l of a realization to be such that this realization commutes with the Hamiltonian terms around its source site. Therefore, we have the following theorem (1) There exists non-zero r ∈ Hom C (Q ⊗ A, A) satisfying Eq. (6.2) and (6.3).
(2) There exists non-zero r ∈ Hom C (Q ⊗ A, A) satisfying (3) There exists non-zero r ∈ Hom C (Q ⊗ A, A) which is an A-A-bimodule map, with respect to the free A-A-bimodule structure on Q ⊗ A.

Proof. The left and right actions of
Then, r being a bimodule map simply means that Then, the equivalence between (1)(2)(3) is an easy exercise using the unitality and Frobenius condition of A and the naturality of β.
Remark 6.4. This result is physically reasonable, that a realization of quantum current is condensed if and only if its source and target intertwiners (in C) are also morphisms or symmetric operators (i.e., A-A-bimodule maps) in the new emergent symmetry A C A .
Note that the free bimodule functor with bimodule structures given above, is moreover a central functor [30] which can be identified be the right adjoint of − ⊗ A, by results in [30], [A, A] has a canonical structure of a Lagrangian algebra in Z 1 (C), and A C A is identified with the category of right [A, A]-modules in Z 1 (C). We now rephrase the results of [30] in the language of internal hom.
Note that given a bimodule B ∈ A C A and an object (Q, β) ∈ Z 1 (C), Q ⊗ B has a natural structure of A-A-bimodule, defined similarly as that of Q ⊗ A. In other words, It is not hard to check that this action is moreover a monoidal functor, and thus A C A is a monoidal module over Z 1 (C). This justifies our notation of internal hom for the right adjoint of − ⊗ A. We now have the internal hom adjunction for any quantum current (Q, β) and A-A-bimodules B, B ′ : Be reminded that when computing this internal hom, B, B ′ are viewed as A-A-bimodules, instead of objects in C. 16 Given three bimodules B, B ′ , B ′′ , there is a canonical associative composition morphism defined as the image of the following morphism under the internal hom adjunction (6.10) . Finally, the monoidal equivalence between A C A and Z 1 (C) [A,A] is given by the functors [30,31]  A is obviously monoidal, and so is its inverse

Coming back to the trivial A-A-bimodule A, the adjunction (− ⊗ A) ⊣ [A, −] reads
by which know that the A-A-bimodule maps between Q ⊗ A and A are in natural bijection with the morphisms between (Q, β) and [A, A] in Z 1 (C). Physically, Hom A C A (Q ⊗ A, A) is the ways how (Q, β) can be condensed, which is the same as the ways (Q, β) can be mapped into [A, A], i.e., Hom Z 1 (C) (Q, [A, A]). In other words, the quantum current [A, A] provides the universal answer to how an arbitrary quantum current (Q, β) can be condensed. Therefore Theorem 6.5. The Lagrangian algebra [A, A] ∈ Z 1 (C) is the universal quantum current that is condensed in ( , A, −m  † m). The excitations are related to the condensed quantum currents via The first equivalence is given by Remark E.4. Remark 6.8. Conversely, suppose that A L is a Lagrangian algebra in Z 1 (C). Z 1 (C) A L is then a unitary fusion category, that is Morita equivalent to C [30,33], and there must exist an algebra Based on the above analysis, we see that the excitations in the fixed-point model indeed form a monoidal module category over the quantum currents Z 1 (C). By the canonical construction, we have verified that the excitations are described by the enriched fusion category Z 1 (C) Fun C (C A , C A ).

The universal model
Let C = Rep G. First, we consider Frobenius algebras A := Fun(G/H) in Rep G (Proposistion F.8 and Example F.9), which are -valued functions on cosets G/H. If we just take local Hilberts space to be A, the model ( , A, −m † m) may be too small to for us to see all possible excitations. Thus, we prefer to take ( , Fun(G), x (6.17) is the embedding of A into Fun(G). Graphically, the local term of the Hamiltonian is . For the Frobenius algebras (H, ω 2 ) with nontrivial ω 2 as in Example F.10, they can be embedded into two adjacent sites Fun(G) ⊗ Fun(G). In other words, one can similarly define a Hamiltonian on ( , Fun(G)) that renormalizes to ( , A, −m † m) using the embedding in Example F.10. These pairs (H, ω 2 ) classify all 1+1D bosonic gapped phases with G-symmetry [34][35][36]. Therefore, the lattice ( , Fun(G)) can host all possible G-symmetric gapped phases.
Moreover, Fun(G) as the regular group representation, contains all possible irreps of G Fun(G) ∼ = ⊕ i∈Irr(Rep(G)) i ⊕d i , (6.19) where d i is the dimension of irrep i. The simple fixed-point boundary conditions and fixed-point defects, can be embedded into a suitable free module i ⊗ A or free bimodule A ⊗ i ⊗ A ′ (see Remark F.15) and then embedded into several (at most five) Fun(G) sites. Therefore, we know in the models constructed on the lattice ( , Fun(G)), indeed all possible fixed-point boundary conditions and fixed-point defects/excitations can be seen.  Table 3. 17 The whole universal model for G = 2 can be summarized by the transverse field Ising model with Hamiltonian
We depict the 2 action on A by the following intuitive diagram: |1〉 |ζ〉 ζ ζ , (6.26) where the general rule and a more nontrivial example is explained in Convention 2 We now compute the simple right A-modules and A-A-bimodules in Rep 2 . By Remark F.15, we only need to decompose the free (bi)modules. Denote the basis of irreps of 2 by We check the free right A-modules: The nontrivial A-module action of α + ⊗ A is We conclude that (Rep 2 ) A ∼ = Vec.
Then we check the free A-A-bimodules: Therefore, we conclude that A (Rep 2 ) A ∼ = Vec 2 as fusion categories.
In the end we compute the condensed quantum currents. In order to compute the internal hom The left action on where all other left actions are zero. We see (C ζ , α + ) ⊗ A ∼ =ζ as A-A-bimodules as they have the same bimodule action.
2. For e ⊗ A = (C 1 , α − ) ⊗ A, the half-braiding is The left action on where all other left actions are zero. We see Therefore using the adjunction (6.39) we conclude that which corresponds to condensing 2 charges. Example 6.10. Let A = α + the trivial algebra. The multiplication m of A is trivial, and the embed-

The Hamiltonian is
It is clear that this Hamiltonian describes the same phase as the polarized Ising model (they share the same ground state and the same classification of excitations):   which is a sub Frobenius algebra of Fun(S 3 ), and we omit the Dirac notation in the following. For simplicity of later computation, we denote these three basis vectors in A as

45)
The multiplication m : A ⊗ A → A in the above basis is The group action of S 3 on A is (we list only a, b as S 3 is generated by a, b) We have A ∼ = λ 0 ⊕ Λ as S 3 representations 18 and A is cyclic.

Convention 2.
We introduce the following Cayley-like diagram to represent cyclic representations: • each node is a vector in the representation; • each node has outgoing arrows labeled by generators of the group, which is a, b of S 3 here; • the number of outgoing arrows at each node is equal to the number of generators; • for an arrow labeled by g, whose source node is vector n, the target node is gn. We check the free right A-modules: 18 A = 〈x, y, z〉 ∼ = 〈x + y + z〉 ⊕ 〈x + ω 2 y + ωz, x + ω y + ω 2 z〉 = λ 0 ⊕ Λ.
1. λ 0 ⊗ A with basis (we omit tensor product between vectors in the following computations) e x, e y, ez, (6.51) is simple. This is easy to see from the multiplication of A, which is like delta-functions that picks out 2. More generally, the special form of the multiplication of A implies that any nonzero submodule of a free module X ⊗ A must contain w x, b(w ) y, b 2 (w )z for some non-zero w ∈ X .
4. Λ⊗A = 〈0x, 0 y, 0z, 1x, 1 y, 1z〉 is isomorphic to (λ 0 ⊗A)⊕(λ 1 ⊗A). The symmetric A-module maps are as follows: The two maps are symmetric, which can be easily seen from the following diagrams: (6.55) 19 Or in other words, we cannot find a subspace for 〈e x, e y, ez〉 such that this subspace is invaraint under both the right A-module action (i.e., the multiplication of A) and the group action (almost the same group action as in Diagram (6.49)).

Now we check the free A-A-bimodules:
1. A⊗ λ 0 ⊗ A has 9 basis vectors. As discussed before, xe x must be in a non-zero sub bimodule. We may check the cyclic S 3 representation generated by xe x: This sub A-A-bimodule is simple and isomorphic to A. It will be denoted byλ 0 (we will soon see why this notation is reasonable). Then we check the cyclic S 3 representation generated by xe y: This sub bimodule is also simple and will be denoted byΛ. Thus A ⊗ λ 0 ⊗ A ∼ = 〈xe x, ye y, zez〉 ⊕ 〈xe y, xez, yez, ze y, ze x, ye x〉 =λ 0 ⊕Λ.

2.
A ⊗ λ 1 ⊗ A has 9 basis vectors. The cyclic S 3 representation generated by x o x is It is simple and will be denoted byλ 1 . The cyclic S 3 representation generated by x o y is: It is isomorphic toΛ (just identify vectors at corresponding nodes in the two diagrams). In the end we compute the condensed quantum currents. As an example we compute the bimodule The right A action is easy. To compute the left A action (the first diagram in Diagram (6.6)), note that the half-braiding is just the group action on A (recall Eq. (A.15)): Thus the left action is while all others are zero. We can then identify, for example, b ⊗ x with ye x, just by checking the two-sided action of A. It is then easy to see (C b , 1) ⊗ A ∼ =Λ as bimodules. We can similarly compute Therefore using the adjunction which corresponds to condensing pure S 3 fluxes. It is not just a coincidence that exchanging (C 1 , Λ) and (C b , 1) in the Lagrangian algebra leads to the same category of excitations; in fact, exchanging (C 1 , Λ) and (C b , 1) is moreover a braided auto-equivalence of Z 1 (Rep S 3 ).

and in this case
which corresponds to condensing all S 3 charges.

and in this case
Note that the two Lagrangian algebra for Fun(S 3 ) and Fun(S 3 / 3 , ) again differ by exchanging (C 1 , Λ) and (C b , 1). Remark 6.15. If we begin with C = Vec S 3 and follow the same algorithm to compute the fixedpoint models, we will obtain the four models in the same phases as those obtained from Rep S 3 . Usually we think Rep S 3 as the category of symmetry charges while Vec S 3 as the category of symmetry defects. It turns out symmetry charge becomes a relative notion; we can equally consider Vec S 3 as the symmetry charges and correspondingly Rep S 3 as the symmetry defects. The two perspectives lead to the same classification of fixed-point models (or gapped phases). Choosing the category of symmetry charges is like choosing an inertial frame of reference.

Conclusion
In this paper, we established the general formulation for quantum currents. Given the category C whose objects are symmetry charges and morphisms are symmetric operators, we showed that quantum currents can be identified with the Drinfeld center Z 1 (C) of C.
We also gave a rigorous analysis on the renormalization process and fixed points in 1+1D. We showed that the fixed-points correspond to Frobenius algebras in C, and in turn Lagrangian algebras in Z 1 (C). From the quantum current point of view, it is the condensation of quantum currents that determine the fixed-points. Since fixed-points represent phases of matter, the condensation of quantum currents also determines gapped phases.
The Frobenius algebra fixed-point model is constructed by symmetric operators in C, so, a priori, it has the symmetry C. But in the end, the symmetry C is (partially) spontaneously broken; a new emergent symmetry (the category of excitations) is observed, which turns out to be the category of bimodules over the Frobenius algebra. Quantum currents provide an invariant for all gapped phases arising in this way. Mathematically, the fusion categories of excitations in these phases are Morita equivalent. Physically, these phases share the same holographic categorical symmetry; the holographic categorical symmetry remains the same upon spontaneous symmetry breaking.
Let's collect all relevant notions regarding the phases sharing the same holographic categorical symmetry Z 1 (C) for a global view. We begin with one of them exhibiting the category of excitations as C. First, there is 2-category Alg(C) whose objects are Frobenius algebras in C, 1-morphisms are bimodules in C and 2-morphims are bimodule maps. Second, there is a 2-category C 2Vec whose objects are left C-module categories in 2Vec, 1-morphisms are C-module functors and 2-morphisms are C-module natural transformations. These two 2-categories are equivalent under the following identification Thus we introduce the notation ΣC ∼ = Alg(C) ∼ = C 2Vec, known as the delooping or condensation completion [2,10,15,37] of C. Moreover, ΣC does not depend on the beginning choice: for any D that is Morita equivalent to C, ΣD ∼ = ΣC (this is in fact an alternative definition of Morita equivalence). Physically, this result indicates that Morita-equivalent D and C should be viewed on equal footing; we can equally call objects in D as symmetry charges and consider C as a (generalized) SSB phase of D. The collection of all generalized SSB phases ΣC ∼ = ΣD does not depend on which we call as symmetry charges. Note that the physical interpretations of the two realizations of ΣC are not exactly the same. The invariant Z 1 (C) of Morita equivalence is given by [11] Z 1 (C) ∼ = ΩZ 0 (ΣC) := Hom Fun(ΣC,ΣC) (id ΣC , id ΣC ). (7.2) We conclude these notions (together with the physical interpretations in parenthesis), in Table 5.  Note that in 1+1D, we can only talk about the total charge transported between subsystems. In higher dimensions, however, the charge distribution, as well as current density, is also of interest. Moreover, for higher symmetries, there can be extended charged object of intrinsic higher dimensions [2].
A simple object (C x , τ) also carries a representation of G. To describe such representation, we need to make some auxiliary choices. First, it is clear that G/N x ∼ = C x . Let {z i } z i ∈i,i∈G/N x be a chosen set of representatives of left cosets (Definition D.1). Then for any group element h ∈ G, there is a unique pair i ∈ G/N x , h ′ ∈ N x such that h = z i h ′ . Now we form a vector space C(C x ) ⊗ V and define the group action of G on it by where z k , h are determined by the unique coset decomposition of gz i , i.e., gz i = z k h, h ∈ N x . It is not hard to check that different of choices of z i lead to isomorphic G-representations on C(C x )⊗V .
where we can choose the representatives of these three left cosets to be a, a b 2 , a b respectively. Thereby, any group element in S 3 can be expressed by a representative in {a, a b 2 , a b} multiply with an element in subset {1, a}.
Next, we will recover the above results by directly solving the half-braiding conditions. Given object (Q, β Q,− ) ∈ Z 1 (Rep G), it suffices to check the half-braiding β Q,R between Q and R = C(G), the regular representation (Definition D.3), as R is "universal" in Rep G: R contains all possible irreducible representations. Our convention for group actions on R is by left multiplication: g ⊲h = gh. It is easy to check that the intertwiners between R and R itself is Hom(R, R) ∼ = C(G), by right multiplication. We will write for w ∈ C(G), r w ∈ Hom(R, R), Moreover, we have intertwiners for each y ∈ G. They satisfy by which we know that R ⊗ R is the direct sum of |G| copies of R. With these preparations, we now examine the form of Q and β Q,R . Firstly, since β Q,R is symmetric and natural in the R component, we have for any a ∈ Q, Second, we want to prove that Q is graded by G. Consider the following linear map for each h ∈ G, Thus we conclude that P h are mutually orthogonal projections. Also using the fact that h∈G δ h is an intertwiner, we have h P h = id Q . Therefore, Q = ⊕ h∈G P h Q, i.e., Q is graded by G. Now we focus on the subspace P x Q. Let a ∈ P x Q, i.e., (δ x ⊗id Q )β Q,R (a⊗e) = a or β Q,R (a⊗e) = x ⊗ a. We have which means that ga ∈ P g x g −1 Q. It is then clear that P x Q carries a representation of N x . We make the choice z i for representatives of G/N x as before.
The representation carried by P z i xz −1 i Q is also is isomorphic to that carried by P x Q. To see this, suppose b ∈ P z i xz −1 i Q, then z −1 i b ∈ P x . For h ∈ N x we have A right C-module is defined similarly. such that ∀X , Y, Z, W ∈ C M, the following diagrams commute: (B.5) C is called the background category of C M. • Morphisms: Hom Vec G (V, W ) := { f ∈ Hom Vec (⊕ g∈G V g , ⊕ h∈G W h )| f (V g ) ⊂ W g }.
• Tensor product: (V ⊗ W ) g = ⊕ a b=g V a ⊗ W b .
• Tensor unit: One dimensional vector space graded by e ∈ G the identity element.
A simple object is a one-dimensional vector space graded by g ∈ G. One-dimensional vector spaces are isomorphic if and only if they are graded by the same g.
Definition B.6 (Monomorphism). In category C, a morphism f : X → Y is called a monomorphism if it is left-cancellative, i.e., ∀Z ∈ C and ∀g 1 , g 2 : Z → X , Definition B.7 (Epimorphism). In category C, a morphism f : X → Y is called an epimorphism if it is right-cancellative, i.e., ∀Z ∈ C and ∀g 1 , g 2 : Y → Z, Definition B.8 (Image). Let f : X → Y be a morphism in C. The image of f is an object Im f ∈ C together with a monomorphism j : Im f → Y satisfying • There exists a morphism i : X → Im f such that f = ji, called a factorization of f .
• For any triple (I ′ , i ′ : X → I ′ , j ′ : I ′ → Y ) where j ′ is a monomorphism and f = j ′ i ′ , there exists a unique morphism v : Im f → I ′ such that j = j ′ v.
The universal property of kernel can be depicted by the following commutative diagram, If Im f exists, it is unique up to a unique isomorphism, and the following defined (co)equalizer and (co)kernel all have this property.
Definition B.9 (Coequalizer). Let f , g : X → Y be a pair of morphisms in C. The coequalizer of f , g is an object C together with a morphism c : Y → C such that • c f = c g.
• For any pair (C ′ , c ′ : Y → C ′ ) such that c ′ f = c ′ g, there exists a unique morphism γ : C → C ′ such that c ′ = γc.
In terms of diagram, (B.10) Denote the coequalizer of f and g as coeq( f , g). The equalizer eq( f , g) is similarly defined.
Example B.10. In the category of sets denoted as Set, a coequalizer of two maps f , g : X → Y is the quotient set Y / ∼, where the equivalence relation ∼ is generated by f (x) ∼ g(x), ∀x ∈ X .
Below we always assume C to be an additive category.
Definition B.11 (Kernel). Let C be an additive category and f : X → Y is a morphism in C. The kernel of f is an object ker f together with a morphism k : ker f → X such that • f k = 0.
• For any pair (K ′ , k ′ : K ′ → X ) such that f k ′ = 0, there exists a unique morphism l : K ′ → ker f such that kl = k ′ .
In terms of diagram,

C Algebra and module category
Definition C.1 (Algebra). Let C be a monoidal category. An (associative unital) algebra in C is a triple (A, m, η), which is an object A ∈ C together with a multiplication m : A ⊗ A → A and a unit morphism η : 1 → A satisfying associativity and identity: where id A : A → A is the unique identity map on A.
Definition C.2 (Algebra homomorphism). Given two algebras (A 1 , m 1 , η 1 ) and (A 2 , m 2 , η 2 ) in C, an algebra homomorphism between them is a morphism f : A 1 → A 2 such that Proof. We show that (M , ρ) is the coequalizer: Definition C.9 (Category of modules over an algebra). Given an algebra A in C, the category of right A-modules C A consists of: • Objects: Right A-modules.
The category of left A-modules denoted as A C is defined similarly. And given another algebra B in C, the category of A-B-bimodules denoted as A C B consists of A-B-bimodules as objects and A-B-bimodule maps as morphisms.
Remark C.10. Given a monoidal category C and a algebra A in C, the category of right A-modules C A is a left C-module category. Explicitly, the module action functor is defined by 1 → e, (C.11) where e ∈ G is the identity element.
Remark C.12. The category of all left [G]-modules [G] Vec is exactly Rep G.
Remark C.13. For any algebra A ∈ C, for each X ∈ C we can construct a free A-module through the C-module functor (C itself is a C-module category) There is also a forgetful C-module functor Forg : It is a group when H is a normal subgroup gH g −1 = H, ∀g ∈ G, and called a quotient group. Right cosets are similarly defined. F Frobenius algebra, separable algebra and Lagrangian algebra Definition F.1 (Coalgebra). A (unital associative) coalgebra in a monoidal category C is a triple (C, ∆, ε), which is an object C ∈ C together with a comultiplication ∆ : C → C ⊗ C and a counit morphism ε : C → 1 satisfying coassociativity and coidentity: