Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries

We generalize the electric-magnetic (EM) duality in the quantum double (QD) models to the extended QD models of topological orders with gapped boundaries. We also map the extended QD models to the extended Levin-Wen (LW) models with gapped boundaries. To this end, we Fourier-transform and rewrite the extended QD model on a trivalent lattice with a boundary, where the bulk gauge group is a finite group G. Gapped boundary conditions of the model before the transformation are known to be characterized by the subgroups K ⊆ G supplying the boundary degrees of freedom. We find that after the transformation, the boundary conditions are then characterized by the Frobenius algebras AG,K in RepG. An AG,K is the dual space of the quotient of the group algebra of G over that of K , and RepG is the category of the representations of G. The EM duality on the boundary is revealed by mapping the K ’s to AG,K ’s. We also show that our transformed extended QD model can be mapped to an extended LW model on the same lattice via enlarging the Hilbert space of the latter. Moreover, our transformed extended QD model elucidates the phenomenon of anyon splitting in anyon condensation.


Introduction
Two-dimensional phases of matter with intrinsic topological orders, or topological phases for short, can be well studied by effective topological field theories, whose Hamiltonian extensions are exactly solvable lattice models. Two major families of such models are the quantum double (QD) models [1] and the string-net models or Levin-Wen (LW) models [2]. The QD models have been generalized to be the twisted QD models [3,4], and the LW models have also been generalized similarly [5]. In this paper, we shall not deal with such generalizations. A QD model is a lattice gauge theory with a finite gauge group G as its input data defined on a lattice Γ. The Hamiltonian of the model converts the input data to an output JHEP02(2020)030 data -the quantum double D(G) of good quantum numbers. The quasiparticle excitations of the topological phase described by the model, namely the anyons, carry representations ([c], µ) of D(G), where [c] labels the conjugacy classes of G, and µ labels the irreducible representations of the centralizer of [c] in G. Anyons of the type ([e], µ) in which e ∈ G is the identity element are the pure charges, which live at the vertices of Γ; the anyons of the type ([c], 0) with 0 being the trivial representation of the centralizer of [c] are the pure fluxes, which live in the plaquettes of Γ; the anyons of the mixed type are the dyons. Hence, intuitively, the QD models would exhibit an explicit EM duality. Indeed, to every QD model on a lattice Γ, there corresponds a QD model on the dual lattice Γ * , such that the charges (fluxes) on Γ are the fluxes (charges) on Γ * , and vice versa [6]. This EM duality is immediately understood in the cases of Abelian groups. In such a case, the irreducible representations of G are all 1-dimensional and form a group isomorphic to G itself. Denote the set of all irreducible representations of G by Rep G . While the QD model on Γ takes G as its input data, the dual model on Γ * takes Rep G ∼ = G as the input data. That is, the dual model is truly a QD model, which by definition has a finite group as its input data. If G is non-Abelian, however, Rep G cannot be a group and thus cannot serve as the input data of a QD model. In such cases, the QD models must be generalized to allow Hopf algebras as their input data to exhibit the EM duality [6].
On the other hand, a QD model with a finite group G can also be mapped via Fourier transform to an LW model with Rep G as its input data on the same lattice [7]. A subtlety is that when G is non-Abelian, the mapping must come with a truncation of the Hilbert space of the QD model [7] unless one enlarges the Hilbert space of the LW model [6,8]. Via this mapping, the LW model with input data Rep G bears an EM duality too.
The EM duality in the QD models and the aforementioned mapping to the LW models are restricted to topological phases on closed surfaces only. Nevertheless, topological phases on surfaces with boundaries are of more practical and theoretical importance because 1) materials with boundaries are much more available than the closed ones, 2) boundary modes are easier to measure experimentally, and 3) a dynamical theory of topological phases is incomplete if unable to encompass different boundary conditions. Recently the QD models and LW models have been extended to be defined on lattices with boundaries by adding appropriate boundary Hamiltonians and are called the extended QD models [9,10] and the extended LW models [11,12]. We are thus motivated to examine in this paper whether and how the extended QD models still possess an EM duality, in particular along the boundaries, and can be mapped to the extended LW models.
We define our extended QD models on a honeycomb lattice Γ with a single boundary (see figure 1). This convenient choice of lattice leads to no loss of generality because the topological features of the models are invariant under the Pachner moves [10,12], which preserve the topology of the system despite altering the lattice structure. The input data of an extended QD model on Γ is still a finite group G; however, the boundary Hamiltonian projects the boundary degrees of freedom into a subgroup K ⊆ G. Each subgroup K characterizes a gapped boundary condition. Moreover, the boundary Hamiltonian should retain the exactly solvability of the model to guarantee that boundary theory is gapped. It was thought that a fixed boundary subgroup K alone might not be able to fully specify JHEP02(2020)030 a boundary condition because different choices of a 2-cocycle β ∈ H 2 [K, U (1)] might be used to define different gapped boundary Hamiltonians [9]. This subtlety has a cure, fortunately, as two of us have shown that any two boundary theories specified by (K, β) and (K, β ) with β β can be continuously connected without a phase transition [13] and are thus equivalent. Therefore, we can simply take (K, 1) with 1 the trivial 2-cocycle to defined the boundary Hamiltonian, or simply just forget about the 2-cocyles. We first Fourier-transform the Hilbert space on Γ following the similar process as done in the bulk [7,8] and generalize the method to the boundary. The Fourier-transformed Hilbert space basis begs us to rewrite the Fourier-transformed model on a slightly different latticẽ Γ, which modifies Γ by adding near to each vertex of Γ a dangling edge (see figure 2(d)). While the bulk degrees of freedom after the Fourier transform become Rep G , the boundary degrees of freedom would be projected by the Fourier-transformed boundary Hamiltonian into a Frobenius algebra A G,K = (C[G]/C[K]) * , the dual space of the quotient of the group algebra of G over that of a given K. When G is Abelian, the emergent Frobenius algebra A G,K happens to be an Abelian group too; hence, the boundary EM duality can be understood as one between the Fourier-transformed extended QD model onΓ with boundary condition specified by A G,K and the extended QD model with boundary condition specified by K. When G is non-Abelian, A G,K ceases to being a group but truly an algebra.
We also show that our Fourier-transformed extended QD model onΓ can be mapped to an extended LW model on the same lattice. In doing so, instead of truncating the Hilbert space of the extended QD model, we enlarge that of the extended LW model. This enlargement is necessary because the Hilbert space of the original extended LW model is too small to contain the full spectrum of excited states with charges; this has already been done for the original LW model on a closed surface [8]. Since the extended QD models and the extended LW models are Hamiltonian extensions of the extended Dijkgraaf-Witten and extended Turaev-Viro types of topological field theories, our results also offer a correspondence between the two types of topological field theories.
The EM duality on the boundary and the mapping to the extended LW model can be revealed by mapping a K ⊆ G to (C[G]/C[K]) * . Three cases of such mappings are listed in table 1. We explain them in order.
(a) The extended QD model for any G has a rough boundary condition specified by K = {e}, implying charge condensation at the boundary. By a Fourier transform, the extended QD model is mapped to the extended LW model with a boundary Hamiltonian specified by the Frobenius algebra A G,{e} = C[G] * . As the function space over G, C[G] * is the regular representation in Rep G , which has a canonical Frobenius algebra structure that has a decomposition The smooth boundary condition is specified by K = G, implying flux condensation at the boundary. The Fourier-transformed boundary Hamiltonian has the trivial input Frobenius algebra A G,G = 0, the trivial representation in Rep G .

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boundary condition extended QD model extended LW model (c) For a nontrivial subgroup K ⊂ G, the corresponding Frobenius algebra is (C[G]/ C[K]) * , which is defined by the function space {f |f (kgk ) = f (g)∀g ∈ G, ∀k, k ∈ K} and is the largest sub-representation space of C[G] * , such that ρ(k) = id, ∀k ∈ K.
Another understanding of the boundary EM duality would require generalizing the entire extended QD model to one with input data being a Hopf algebra, similar to what is done only to the bulk as in ref. [6]. We shall not discuss such generalizations in this paper but offer a justification of the boundary EM duality when G is non-Abelian.
According to the mechanism of anyon condensation [14], a gapped boundary condition of a topological phase can be accounted for by a boundary condensate formed by certain types of anyons from the bulk [15,16]. To condense at the boundary, certain types of bulk anyons would have to first split into a few pieces, several or all of which are allowed to condense at the boundary, depending on the structure of the condensate. Current understandings of this phenomenon are categorical and abstract. It would be interesting to understand the phenomenon of splitting and partial condensation based on concrete lattice models of topological phases. As we will show, our Fourier transform of the extended QD model explains this phenomenon in terms of solely the input data of the model.
To facilitate our studies in the paper, we introduce also a graphical tool of group representation theory. We provide concrete examples, one for Abelian groups G and one for the non-Abelian group S 3 , to illustrate our results.
Our paper is organized as follows. section 2 reviews the extended QD model. section 3 Fourier-transforms and rewrites the extended QD model. section 4 verifies the emergent Frobenius algebra structure on the boundary and elucidates the phenomenon of anyon splitting in boundary anyon condensation. Section 5 illuminates the boundary EM duality. Section 6 maps the Fourier-transformed extended QD model to the extended LW model on the same lattice. Section 7 provides two concrete examples of our results. Finally, the appendices collect a review of the extended LW model and certain details to avoid clutter in the main text.

Extended quantum double model
In this section, we briefly review the extended QD model. An extended QD model [9,10] is an extension of the QD model to the case with boundaries by adding a boundary Hamiltonian to the QD Hamiltonian. The model is a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theory with a finite gauge group. The model can be defined on an arbitrary lattice with one or multiple boundaries. Topological invariance under the Pachner JHEP02(2020)030 Boundary Figure 1. A portion of an oriented trivalent lattice, on which the extended QD model is defined. Each edge of the lattice is graced with a group element of a finite gauge group G. Grey region is the bulk, to the left of the boundary (thick line). moves allows the model to be defined on a fixed lattice for computational convenience. In this paper, we consider an oriented trivalent lattice Γ, part of which is shown in figure 1.
The input data of the model is a finite gauge group G, whose elements are assigned to the edges of Γ. The total Hilbert space is spanned by all possible configurations of the group elements of G on the edges of Γ and is the tensor product of the local Hilbert spaces respectively on the edges. Namely, where e is an edge in Γ. Note that reversing the orientation of an edge graced with a group element g changes the group element toḡ := g −1 ; however, we work with a fixed lattice with a fixed orientation. The Hamiltonian of the model adds a bulk Hamiltonian and a boundary Hamiltonian: The bulk Hamiltonian consists of two sums of local operators: where the two sums are respectively over all vertices and all plaquettes in the bulk of Γ. A local vertex operator A QD v acts locally on the three edges incident at the vertex v as follows.
which is understood as a discrete gauge transformation averaged over G. = δ g 1 ·g 2 ·g 3 ...g 6 ,e p g 1 which is also a projection. In gauge theory terminologies, B QD p imposes a local flatness condition in p. All plaquette operators and vertex operators commute. A gapped boundary condition is specified by a subgroup K ⊆ G. The boundary Hamiltonian comprises boundary local operators: where the two sums are respectively over all vertices and virtual plaquettes (to be defined shortly) on the boundary ∂Γ of Γ. We define which is again a gauge transformation averaged instead in a subgroup K ⊆ G. An A QD v is clearly also a projector that projects out any non-invariant states under its action. All boundary vertex operators commute with each other and with all other operators in the total Hamiltonian. An operator B QD v simply does the following projection.
where the trace is taken over the total Hilbert space (2.1). Quasiparticle excitations of the model are charges on the vertices, fluxes through plaquettes, and dyons as bound-states of charges and fluxes. A charge at a vertex v arises when the local Gauss constraint is violated; a flux through a plaquette p occurs when the local flatness condition is violated; when both constraints are violated in p, a dyon shows up in p. Other properties of the model and topological phases that are classified by this model can be found in ref. [10] and references therein.

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3 Fourier transforming and rewriting the extended QD model In this section, we Fourier-transform the basis of the Hilbert space of the extended QD model from the group space to the representation space, and as urged by this transformation, rewrite the extended QD model on a slightly different lattice. The Fourier transform of the bulk theory only has been done in refs. [7,8]; hence, we shall focus on Fouriertransforming the boundary in this section.

A graphical tool for group representation theory
To facilitate the derivations in this paper, in particular the Fourier transform, we introduce the following graphical tool for group representation theory [17]. Let L G be the set of all (representatives of equivalence classes of) irreducible representations of a finite group G.
For simplicity, we shall define d µ = dim V µ for µ ∈ L G . A representation matrix D µ mµnµ (g) acting on V is depicted as g nµ mµ µ . (3.1) Here, the line is oriented from the right index n µ to the left index m µ of the representation matrix. The inline g insertion indicates the group action by g. In this graphical presentation, multiplying two representation matrices is synonymously concatenating two such lines: where the arrow is reversed after the complex conjugation. This way, the complexconjugated group action should be read upward as D µ mµnµ (g) * = D µ nµmµ (ḡ), withḡ := g −1 , because µ is unitary. As such, the great orthogonality theorem is presented by (3.6) Every irreducible representation µ ∈ L G has a dual µ * ∈ L G , such that µ * is equivalent to (but not necessarily identical to) the complex conjugate representation of µ. There is an invertible duality map where the Ω µ mµn µ * is a complex matrix satisfies normalization Ω † Ω = I and maps µ to µ * by similarity transformation (Ω µ ) −1 D µ (g)Ω µ = (D µ * (g)) * . (3.8) Graphically, the duality map and its inverse are And the similarity transformation in eq. (3.8) is presented by (3.10) For the duality map to be unique, the matrix Ω µ is either symmetric or antisymmetric depends on whether µ is pseudo-real or not. This is an intrinsic property specified by a number β µ called the Frobenius-Schur (FS) indicator. We have (Ω µ * ) T = β µ Ω µ with β µ = ±1 if µ is real or pseudo-real. Frequently in later derivations, we will need 3j-symbols to deal with the coupling of three representations of G. A 3j-symbol is a tensor w abc mam b mc that is defined as an intertwiner: (3.13) In this paper, we will also use Clebsch-Gordan coefficients, which are the values of intertwiners and can be defined using the 3j-symbols (3.12) and duality map (3.9) as (3.14) For later convenience, we list a few properties of the 3j-symbols (Clebsch-Gordan coefficients) as follows. For a generic finite group G, we can always construct such 3j-symbols satisfying the following properties [17].

Fourier transform on the Hilbert space
Let us first Fourier-transform the Hilbert space of an extended QD model on Γ with a finite gauge group G. The total Hilbert space of the model is defined in eq. (2.1). Since the degrees of freedom of the extended QD model live on the edges of the lattice, the total Hilbert space H is the tensor product of all local Hilbert spaces H e on the edges. The local basis state on an edge is |g , which can be Fourier transformed to the basis states in terms of representations |µ, m µ , n ν , called a rep-basis, by performing the following Fourier transform (FT): where (µ, V µ ) pairs an irreducible unitary representation of G with representation space V µ , v µ = d µ , |G| is the order of G, and D µ mµnµ is the representation matrix in V µ . The local rep-basis and the group-basis have the same dimension because µ d 2 µ = |G|. Note again that the total Hilbert space of the model is defined regardless of the Hamiltonian, which when imposed would separate the Hilbert space into ground and excited states. Hence, the Fourier transform of the group-basis of total Hilbert space on Γ can be done by transforming the local basis of the H e on each individual edge independently. Figure 2 Rewrite all vertices µ 2 , µ 3 , and µ 5 in figure 2(b); however, there is no a priori a reason the three lines graced with µ 2 , µ 3 , and µ 5 should simply just meet at v 2 . If we did so and removed the matrix indices n 2 , m 3 , and n 5 , one would misinterpret the vertex v 2 as an intertwiner of the three representations because the dimension of the basis would be lower than that of the groupbasis. Hence, we leave the vertex v 2 and all other vertices open in figure 2(b), and the basis becomes detached from the original lattice. Now the question is: Can we rewrite the basis in figure 2(b) to one that has the same dimension and is still attached to an actual trivalent lattice? The answer is "Yes". Let us again stare at the vertex v 2 in figure 2(b). The strategy is to fuse (i.e., couple) the three representations in an order at v 2 following the fusion rules defined by µ ⊗ ν = η N η µν η, where N η µν are the fusion coefficients (see also appendix A). Since coupling representations is associative and linear, we can choose the order. Let us first fuse µ 2 and µ 5 by contracting their indices n 2 and n 5 , resulting in a linear combination of irreducible representations {η}, each member of which is graphically a line with a free end labeled by m η . Then, we can fuse an η with µ 3 by contracting m η and m 3 , resulting in a linear combination of representations {s}, each member of which is a line with a free end labeled by m s . For each η and each s, we obtain a new basis state, as in figure 2(c). A pivotal point is that the degrees of freedom η and s are both local with respect to the vertex v 2 because they arise from fusing the three representations µ 2 , µ 5 , and µ 3 at v 2 . Hence, the line graced with s can be considered an induced degree of freedom at v 2 , and we place the line as a dangling edge very close to v 2 . This procedure of rewriting the basis causes no loss of dimension and is thus a linear transformation because the degrees of freedom associated with v 2 in figure 2 , which agrees with that contributed by g 2 , g 3 , and g 5 in figure 2(a) and that contributed by µ 2 , µ 3 , and µ 5 in figure 2(b). Here we used the identities (A.3) and a d 2 a = |G|. We can go through the procedure above on all the other open vertices in figure 2(c) and obtain the basis states in figure 2(d), which we shall call the rep-basis states. Now the basis states are again defined on an actual trivalent latticeΓ, which differs from the JHEP02(2020)030 original lattice Γ by having a tail anchored near each of the original vertices. The tails are necessary to maintain the correct number of local degrees of freedom. On the new latticẽ Γ, each vertex can indeed be interpreted as where three representations fuse. The latticẽ Γ is in fact the right lattice for defining a LW model, such that its Hilbert space contains both charge and dyonic excitations [8]. We will come back to this point later when we map our Fourier-transformed model to the extended LW model in section 6.
Since the vertex and plaquette operators comprising the Hamiltonian of the extend QD model are local operators, the action of such an operator affects only a few local degrees of freedom but not any other. Consider a boundary vertex operator (2.7) acting on the vertex v 2 in figure 2(a), it affects only the three group elements meeting at v 2 . When transformed into the basis in figure 2(d), the vertex operator acting on v 2 would possibly affect at most the degrees of freedom µ 2 , µ 3 , µ 5 , η, s, m s , which are local at v 2 . In other words, the other degrees of freedom other than these six are diagonal indices when the vertex operator is represented in the Hilbert space. Therefore, when studying a vertex operator action at a vertex, we can simply single out a local basis-state consisting of only the degrees of freedom local at the vertex.
Let us study the Fourier transform and basis-rewriting depicted in figure 2 by focusing on a local state at a boundary vertex in detail to retrieve the linear transformations of basis with precise coefficients. Consider a local basis-state |g, h, l drawn on the right hand side of eq. (3.22). In our graphical presentation, Fourier-transforming this basis-state to one in the representation space is as trivial as in eq. (3.22), and the inverse transformation in eq. (3.23). (3.23) In the basis state on the left hand side of eq. (3.22), for any fixed µ, ν, and λ, the degrees of freedom are only at the ends of the three lines. We then rewrite the local basisstate on the left hand side of eq. (3.22) by first fusing the two representations µ and ν via contracting their indices m µ and m ν , resulting in a set of representations {γ}, Then, we can fuse a γ with λ by contracting m γ and m λ , resulting in a set of representations {s}. This procedure yields two 3j-symbols with a pair of indices contracted, resulting in the coefficients of the expansion on the right hand side of eq. (3.24) (also recall figure 2(d)). In the third equality in eq. (3.24), we bend the lines labeled by λ and µ in the coefficients using eq. (3.4). Such bending of lines would make future calculations easier by using eq. (3.17).

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To understand the difference between the graphs of the states and those of the coefficients, note that the coefficients are in fact wavefunctions defined by Φ(Γ) := Γ|Φ with Γ a graph. The Γ in a state |Γ differs from the Γ in Γ| by parity and opposite edge orientations. (3.25) Each black dot at a vertex in eqs. (3.24) and (3.25) and in the derivations hereafter represents a 3j-symbol that ensures an intertwiner space at the vertex. Explanations of the notation |Ψ sms must follow. The linear transformation in eq. (3.24) rewrites only a subspace of the local Hilbert space spanned by the basis on the left hand side of the equation. The subspace is the one comprised by the degrees of freedom m µ , m ν , and m λ , which are transformed into γ, s, and m s . Hence, the two bases before and after the rewriting both have the same labels µ, ν, λ, n µ , n ν , and n λ . Moreover, staring at the right JHEP02(2020)030 hand side of eq. (3.24), one can see that the two degrees of freedom γ and s cannot be both independent. If we choose s independent, then γ is determined by µ, ν, λ, and s. Therefore, we can denote the new basis after the rewriting by |Ψ sms for simplicity, while keeping all the other labels in the graph inexplicit. This simplification causes no confusion because in actual calculations, e.g., in computing an inner product of two such local basis states, i.e., Ψ s m s |Ψ sms , the prime in Ψ implies that the hidden labels in Ψ should all be the primed version of those in Ψ. In the next subsection, we will see another advantage of this simplified notation.
Equations (3.25) and (3.23) lead to a direct linear transformation between the local basis-states in the rep-basis and those in the group-basis at a boundary vertex v: (3.26) Equations (3.26) and (3.22) in turn can lead to the correct coefficients in eq. (3.24), as shown in appendix C, where we also verify that the local basis states |Ψ sms indeed form a well-defined local basis by showing that it is orthonormal and complete.

Fourier transform of the vertex operators
We are now ready to study how a boundary vertex operator A QD v acts on a local basis-state |Ψ sms in the rep-basis. In other words, we need to find the Fourier-transformed version Two subtleties are in order. First, a boundary vertex operator A QD v acts on a group-basis local state |g, h, l ∈ H v by modifying the group elements g, h, and l on the three edges incident at v via gauge transformation. When the group-basis local state is expanded in terms of the local states |Ψ sms , the action of A QD v are expected to spread over all the degrees of freedom µ, ν, λ, γ, s, and m s local to the vertex v. Nevertheless, as we will see, the indices µ, ν, λ, γ are all readily diagonal with respect to the A QD v represented in this local Hilbert space. This further renders |Ψ sms a good notation of the local basis states for A QD v , as the other advantage of the notation alluded to earlier.
Second, recall that a bulk vertex operator A QD v (2.4) acts as a gauge transformation averaged over the entire group G; it is a projector that projects out non-intertwiner states of µ, ν, and λ but keeps the intertwiner states as its +1 eigenstates. A boundary vertex operator A QD v (2.7) however performs a gauge transformation averaged over a subgroup may not project out all non-intertwiner states of µ, ν, and λ.
To see this point, let us consider two extreme cases. When K = G, A QD v acts exactly the same as a bulk vertex operator and will project out all non-intertwiner states of µ, ν, and λ, and leaves no degrees of freedom for s. That is, the s on the dangling edge (tail) attached JHEP02(2020)030 to v must be the trivial representation 0 of G. Note that the γ, as a degree of freedom on an internal edge, is not independent; hence, if s is trivial, γ = ν. When K = {e}, the action of A QD v is trivial and thus does not project out any states in the fusion space of µ, ν, and λ. That is, the s can be any irreducible representation of G. Now for a generic nontrivial subgroup K ⊂ G, the action of A QD v may render s taking value in certain subset of the set L G of all irreducible representations of G. Moreover, certain states in V s may also be projected out. Therefore, the states that survive the action of A QD v may be labeled by pairs (s, α s ) and collected into a set L A = {(s, α s )}, which is to be defined shortly. The intuitive argument above can be explicitly verified as follows, using eqs. (2.7) and (3.26), and the completeness of the local rep-basis. Note that for certain s ∈ L G , P s K may project out the entire H s v ; therefore, the set L A collects all the +1 eigenstates of A QD v or its representation P s K for all possible s that is not JHEP02(2020)030 annihilated by P s K . That is, The states |Ψ sms with (s,m s ) / ∈ L A are the zero eigenstates of A QD v ; they have higher energy than the +1 eigenstates according to the Hamiltonian (2.6) and thus are excited states. The excitations appear at the end of the dangling edge graced with representations s and are the point-like charge excitations at the boundary.

Fourier transform of the plaquette operators
We can now proceed to check how a boundary plaquette operator B is represented in the Hilbert space. Hence, following the same logic as that of constructing the local basis states for the boundary vertex operators, we can denote the local basis states to be acted on by a boundary plaquette operator by |Ψ ηλ rmr;sms , defined as follows.
Here,ṽ j = d j withd j = β j d j , and the coefficients G abc def are the symmetric 6j-symbols of the irreducible representations of the group G; their properties can be found in appendix A or in ref. [17]. The full derivation of the equation above is found in appendix C. The local basis states | Ψ ηλ rmr;sms may be +1 or zero eigenstate of the boundary vertex operators acting on the relevant vertices. It is useful to study the matrix elements of B  We have successfully Fourier-transformed and rewritten the extended QD model on a trivalent latticeΓ. In the following sections, we shall study the physical consequences.

Emergence of Frobenius algebras and anyon condensation
Interestingly, the Fourier transform and rewriting of the extended QD model leads to an emergent Frobenius algebra structure on the boundary ofΓ. Namely, the set L A defined in eq. (3.29) together with the symbols f defined in eq. (3.33) form a Frobenius algebra, as an object in the UFC Rep G -the category of linear representations of G, which is the tensor category generated by L G -the set of irreducible representations of G.
Before we show the emergent Frobenius algebra, let us review what Frobenius algebras are. A Frobenius algebra A is a pair (L A , f ), where L A is the set of elements of A, and f is the multiplication. An element of L A is a pair (s, α s ) (or just sα s for short), where s is a simple object of a UFC F. We denote the number of different pairs (s, α s ) with JHEP02(2020)030 the same s by |s| and call it the multiplicity of s in A. The multiplication f is a map: L A × L A × L A → C that satisfies the following associativity and non-degeneracy. cαc f aαabα b c * αc f cαcrαrs * αs G abc * rs * tṽ cṽt = αt f aαatαts * αs f bα b rαrt * αt , (4.1a) where 0 is the unit element of A and has multiplicity 1, i.e. 0 = (0, 1). Here,ṽ c = d c , which is defined in appendix A. That the Frobenius algebra A defined above is an object of the corresponding UFC F is understood by writing A as A = s| (s,αs)∈L A s ⊕|s| , which is in general a non-simple object in F. For the sake of computation, one may also write A = (s,αs)∈L A s αs , explicitly taking different appearances of s as distinct elements of A.
We can show that the symbols f aαabα b cαc defined in eq. (3.33) indeed satisfy the defining conditions (4.1) of a Frobenius algebra. We first prove the associativity (4.1a) as follows.
In the first equality above, we simply rewrite the left hand side using the definition (3.33). The result is then graphically presented via the second equality. The third and fourth equalities are due to eq. (3.15), while the fifth equality due to eq. (3.16). The last equality is again rewriting using definition (3.33). To prove the non-degeneracy condi-JHEP02(2020)030 tion (4.1b), note that since ñ c * ũ bũc (w 0bc * 0α bñc * ) * (Ω c ) −1 n c * αc ∝ δ b,c δ α b ,αc due to Schur's lemma, det( ñ c * ũ bũc (w 0bc * 0α bñc * ) * (Ω c ) −1 n c * αc ) = 0 . Therefore, for any finite group G and a subgroup K ⊆ G, the set L A defined in eq. (3.29) equipped with the multiplication f defined by eq. (3.33) indeed form a Frobenius algebra A G,K = (L A , f ) G,K .
Recall that according to eq. (3.29), each pair (s, α s ) labels a local +1 eigenstate |Ψ sαs at vertex v of A QD v , which is a projector. All such eigenstates sharing the same hidden labels span a subspace in the d s -dimensional representation space V s . The dimension |s| of this subspace is the number of pairs (s, α s ) with the same s. A salient point of our recognition of an emergent Frobenius algebra out of the Fourier transform of the extended QD model with gauge group G is that it identifies this dimension |s| as the multiplicity |s| of s appearing in A (see above eq. (4.1a)). This identification is intricately related to the mechanism of anyon condensation in topological phases. Here we briefly describe this relation and shall report the detailed studies elsewhere.
Anyon condensation has been extensively studied recently (see refs. [14,[18][19][20] and references therein). In a topological phase C, certain types of anyons may condense and cause a phase transition that breaks the topological phase to a simpler child topological phase U . In an extreme case, U may be merely a vacuum, and the original topological phase is said to be completely broken. An alternative and equivalent perspective is that there is a gapped domain wall separating C and U . In particular in the case where U is a vacuum, we say certain types of anyons of C can move to and condense at the gapped boundary between C and the vacuum. An interesting phenomenon is that certain types of anyons do not condense at the boundary straightforwardly; rather, such an anyon may split into a number of pieces at the boundary, and not all of these pieces can and necessarily condense. If an anyon splits into two pieces, it is said to have multiplicity one (two) in the condensate if only one (both) of the two pieces condense. Anyon splitting only occurs to anyons with quantum dimension greater than or equal to 2. So far, the understanding of anyon splitting is categorical; hence, it would be important to understand such splitting in a concrete lattice model of topological phases with gapped boundaries in terms of the input data of the model.
For the extended QD model with input gauge group G, the gapped boundary conditions are specified by the subgroups of K ⊆ G. It is known that K = {e} corresponds to condensing all the G-charges at the boundary, K = G corresponds to condensing all the G-fluxes, and any K in between corresponds to condensing certain types of dyons. Nevertheless, by the subgroup K alone one cannot immediately tell whether a type of condensed anyons should have a non-unit multiplicity in the condensate. In the Fourier-transformed picture, as we now show, anyon splitting and multiplicity become lucid. According to our discussion earlier in this subsection, a subgroup K on the boundary of Γ gives rise to a Frobenius algebra A G,K = (L A , f ) G,K at the boundary ofΓ. Consider the case where K = {e}, i.e., charge condensation at the boundary. Then by eqs. (3.28) and (3.29), all irreducible representations must appear in L A . Namely, for any s ∈ L G , the set of all irreducible representations of G, we have {(s, α s )|α = 1, 2, . . . , d s } ⊂ L A . Since s labels a type JHEP02(2020)030 of pure charge excitations, and since each pair (s, α s ) is an independent element of A G,K , the pure charge s splits into d s pieces, each of which condenses at the boundary. Thus, the multiplicity of the charge s in the boundary condensate is d s = |s|, the multiplicity of s in the Frobenius algebra A G,K . In the case where K is a nontrivial subgroup, we may have for some s, only a set {(s, α s )|α = 1, 2, . . . , |s| < d s } ⊂ L A . That is although the pure charge s splits into d s pieces, only |s| pieces of them contribute to the boundary condensate. In section 7.2, we shall see both possibilities in a concrete example.
The Frobenius algebra A G,K could be understood more intuitively in the language of group algebras C where V αs s denotes the α s -th copy of V s that appears in A G,K with the multiplicity index α s . The multiplication is also the 3j-symbol (up to some normalization factor) as defined in eq. (3.33). By the definition of L A in eq. (3.28), α s is given by a basis vector such that ρ s (k)|s, α s = |s, α s for all k ∈ K. The space A G,K spanned by |s, α s is thus identified with (C[G]/C[K]) * defined above.

EM duality in the bulk
The QD model exhibits an EM duality [6][7][8]. Let us recap the known results here using our notation. Consider the QD models with finite Abelian groups G. In such a case, all the irreducible representations of G are 1-dimensional and form a group, with the group multiplication defined as the tensor product of representations. For example, the irreducible representations of G = Z n form the group Rep G = Z n . Let g ∈ {0, 1, . . . , n − 1} JHEP02(2020)030  Figure 3. A closed trivalent graph Γ (a) and its dual graph Γ * (b). A vertex v (a plaquette p) of Γ becomes a plaquette p * (a vertex v * ) of Γ * . and g · h = g + h mod n. Then the irreducible representations are j ∈ {0, 1, . . . , n − 1} and j · k = j + k mod n in the sense that ρ j (g)ρ k (g) = ρ j·k (g) for all g ∈ G.
As reviewed in section 2, the QD model on a closed trivalent graph Γ consists of vertex and plaquette operators. For example, in figure 3(a), we have an operator at vertex v = 1: where g 1 , g 2 , g 3 ∈ G are the group elements on the three edges meeting at v. At plaquette p = 2, we have B p |g 1 , g 2 , g 3 , g 4 , g 5 , g 6 = δ g 1 ·g 2 ·g 3 ·g 4 ·g 5 ·g 6 ,0 |g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , where the g s are the group elements on the edges outlining the plaquette p.
We also Fourier transform the basis of Hilbert space H Γ on Γ from the group space to the representation space. Hence, the vertex and plaquette operators act on the new basis states as A v |j 1 , j 2 , j 3 v=1 = δ j 1 ·j 2 ·j 3 ,0 |j 1 , j 2 , j 3 v=1 , where j 1 , j 2 , j 3 ∈ RepG are on the three edges meeting at the vertex v = 1 in Γ, and B p |j 1 , j 2 , j 3 , j 4 , j 5 , j 6 p=2 = 1 |G| k∈Rep G |k · j 1 , k · j 2 , k · j 3 , k · j 4 , k · j 5 , k · j 6 p=2 , (5.4) where the j 's are graced on the six edges outlining the plaquette p = 2 on Γ.
On the other hand, we can draw the dual lattice Γ * ( figure 3(b)) of Γ and place an element of Rep G on each edge of Γ * . Since Rep G is naturally defined on Γ * . The basis states |j 1 , j 2 , j 3 v=1 and |j 1 , j 2 , j 3 , j 4 , j 5 , j 6 p=2 on Γ are the same as the basis states |j 1 , j 2 , j 3 p * =1 and |j 1 , j 2 , j 3 , j 4 , j 5 , j 6 v * =2 on Γ * . Hence, we have

JHEP02(2020)030
where B * p * and A * v * are the plaquette and vertex operators of the dual QD model. Comparing eqs. (5.3) and (5.4) to (5.5) and (5.6), and since the above discussion applies to any vertex and plaquette on Γ, we conclude that the QD model with a group G on Γ is mapped to the dual QD model with a group Rep G on Γ * , with H Γ ∼ = H Γ * , and Since A v measures the gauge charges (which have an electric nature) and B p measures the gauge fluxes (which have a magnetic nature), the above duality is an electric-magnetic duality: an electric charge/magnetic flux on Γ is mapped to a magnetic flux/electric charge on Γ * .
For non-Abelian cases, Rep G is no longer a group. In a generalized context using quantum groups (Hopf algebra), we can still define the charges and fluxes of Rep G , and thus are able to study the EM duality [6] in the QD model on a closed surface. In ref. [8], the authors have also checked the EM duality in the enlarged Hilbert space of the LW model on a closed surface.

EM duality on the boundary
Our results in sections 3 and 4 enable us to extend the EM duality in the bulk to one that is on the boundary. We again consider the Abelian cases first. As reviewed in section 2, on the boundary, given a subgroup K, there are two types of operators,Ā v acting on the boundary vertices, andB p on the boundary edges, as illustrated in figure 4(a).
To reveal the EM duality on the boundary, we need to extend the lattice Γ in figure 4(a) to the latticeΓ in 4(b), following the procedure illustrated in figure 2. As explained in section 3.2, this extension preserves the Hilbert space of the extended QD model, i.e., H Γ = HΓ, and is merely a basis transformation. That is, the basis of the Hilbert space is transformed from the group space to the representation space.
Then according to eqs. (2.7) and (2.8) and the basis transformation, a boundary vertex operatorĀ v and plaquette operatorB p acts respectively on the highlighted dangling edge and open plaquette onΓ in figure 4(b) as where j ∈ Rep G is graced on the red dangling edge in figure 4(b), and where j 1 , j 2 , j 3 , j 4 ∈ Rep G are respectively placed on the four blue edges outlining the open plaquette in figure 4(b), and L A is a subgroup of Rep G defined by We then draw in figure 4(c) the dual latticeΓ * of the extended latticeΓ. This dual latticeΓ * is a triangular lattice with a boundary and naturally defines a dual extended QD model with Rep G being the gauge group. The dual boundary operators B * p * and A * v * act on the dual boundary edges and vertices as (5.14) Comparing eqs. (5.10) and (5.11) to (5.13) and (5.14), we conclude that the extended QD model with a group G on Γ is mapped to the dual extended QD model with the group Rep G onΓ * , with H Γ ∼ = HΓ * , and . Note that as defined in eq. (3.28), here we choose the indicesm s i andm q i that diagonalize the vertex operators A v and A v . The original LW model [2] is known to have a Hilbert space smaller than that of the QD model [7] because the Hilbert space of the original LW model does not contain excited states with charges or dyons [8]. In ref. [8], the authors enlarged the original LW model by adding a tail to each vertex in the trivalent lattice of the model. The enlarged LW model has a larger Hilbert space encompassing a full dyon spectrum due to the tails, which play the same role as the dangling edges in figure 5. Therefore, to map the Fourier-transformed extended QD model to the extended LW model with an enlarged Hilbert space, there is no need of reducing the Hilbert space of the extended QD model, in contrary to what was done in ref. [7].
What acts on the basis states in figure 5 is the Fourier transformed version of the Hamiltonian of the extended QD model. Namely, where use is made of P s G = 1 |G| g D s msm s (g) = δ s,0 , with 0 being the trivial representation of G. Clearly, when s = 0, m s is not a degree of freedom and can be omitted. As to a bulk B QD p operator, its action on a bulk plaquette would be straightforwardly obtained and would be similar to that of the boundary B QD p but with seven more 6j-symbols because a bulk plaquette has seven more vertices on its perimeter. Besides, there is no restriction to L A in a bulk B QD p operator. By comparing to the boundary vertex and plaquette operators in the extended LW model reviewed in appendix B or in refs. [11,12] and the bulk operators in the enlarged LW model in ref. [8], we can actually identify the model H QD,Γ L G ,A G,K defined onΓ as the extended LW model H LW,Γ Rep G ,A with A = A G,K defined on the same latticeΓ. That is, The two models have the same Hilbert space and Hamiltonians term by term.

Examples
In this section we offer two explicit examples, one for G being Abelian and one for G = S 3 , to aid the understanding of the results.
Since the irreducible representations are 1-dimensional, the representation matrices are merely complex numbers; hence, all the matrix indices can be removed. The Fourier transform of the local group-basis states on an edge takes the simple form: The rep-basis local states (3.25) of the vertex operators are now denoted simply by |Ψ s . The fusion of three irreducible representations µ, ν, and λ is determined by the delta function δ µνλ just introduced. Note that in eq. (7.2), the group elements and irreducible representations are on equal footing, the fusion of irreducible representation is the same as the multiplication of group elements. Indeed, for an Abelian group G, Rep G has a group structure and is isomorphic to G itself. This fact results in the self duality under Fourier transforming the extended QD model with an Abelian gauge group G. This self duality is actually the EM-duality in eq. (5.15).

(7.5)
This result can be understood in the following way. Since G ∼ = Rep G in this case, the constraint δ t * s * s = 1 indicates the equality s = s·t * , where the dot is the group multiplication.
Then, a B QD p acts on its local basis states as   For notation simplicity, we rename the three irreducible representations of S 3 by 0, 1, and 2, respectively corresponding to the 1, sign, and π in the table above. One set of JHEP02(2020)030 Table 3. Four emergent Frobenius algebras A S3,K = (L A , f ) S3,K of the Fourier-transformed extended QD model with G = S 3 . In the third column, only the non-vanishing symbols f are shown.
irreducible representations matrices is listed as follows.
One thus can forget about A 1 . Here we can try to understand explicitly the relation between the emergent Frobenius algebras at the boundary and anyon condensation, in particular anyon splitting and multiplicity.

Condensation type Boundary condensate Boundary
In the current example, there are four different gapped boundary conditions respectively characterized by the four subgroups 1 of S 3 , imposed by the boundary plaquette operators B QD p of the model. These four gapped boundary conditions respectively correspond to four different boundary condensates composed of different types of anyons condensed at the boundary. These correspondences are shown in table 5. One can see in the table that the type C anyons, which are pure charges ([e], 2), appear with multiplicity 1 in the condensate A 3 but multiplicity 2 in the condensate A 4 .
Condensate A 3 corresponds to the emergent Frobenius algebra A 3 = (0, 1)+(2, 1). The element s = 2 in the pair (2,1) in A 3 actually is the 2-dimensional irreducible representation of S 3 and thus can be directly identified with the pure charge C in the condensate A 3 . Note that for each K ⊆ S 3 in table 3, the pairs (s, α s ) in the corresponding row each labels a +1 local eigenstate |Ψ sαs of the A JHEP02(2020)030 Table 6. The GSDs of the extended QD model with G = S 3 on a cylinder, whose two boundaries are specified by two subgroups of S 3 . where we didn't differentiate a matrix and its matrix elements labeled by m s m s to emphasize that the operator is not diagonalized yet. We can diagonalize the projector in the basis states em s labeled bym s and find the +1 eigenstate e 1 =

Frobenius algebras
T . Hence, only α s=2 :=m s = 1 is allowed. That is, the pair (2, 1) ∈ L A labels the only +1 local eigenstate |Ψ 21 of the A QD v . Hence, only half of the 2-dimensional space V s or half of the charge-2 condenses at the boundary. As such, we can say that the charge-2 splits: 2 = (2, 1) ⊕ (2, 2) but only the half (2, 1) condenses. That is why only (2, 1) appears in the A 3 in table 3, and type C anyons appear with unit multiplicity in A 3 in table 5. If we denote (2, 1) by 2 1 and (2, 2) by 2 2 , we found that the Frobenius algebra object A 3 in table 4 should be rigorously written as A 3 = 0 ⊕ 2 1 ; however, one often just write A 3 = 0 ⊕ 2 customarily. Now in the case with A 4 corresponding to A 4 , the subgroup is K = {e}. Then, for s = 2, an operator A QD v is represented in the 2-dimensional local space V s by the matrix P s=2 K={e} = D s msm s (e) = δ msm s = δm sm s , which is automatically diagonal and in fact a 2 × 2 identity matrix. Thus both local eigenstates of A QD v in V s are +1 eigenstates, which can be labeled by (2, 1) and (2,2). Note that these two eigenstates are not those of the operator in eq. (7.9). As such, the charge 2 splits as 2 = (2, 1) ⊕ (2, 2), but both pieces condense at the boundary. Therefore, in the boundary condensate, the type C anyons count twice, rendering A 4 = A ⊕ B ⊕ 2C. If we write (2, 1) and (2, 2) as 2 1 and 2 2 , we can identify the emergent Frobenius algebra A 4 with the Frobenius algebra object A 4 = 0 ⊕ 1 ⊕ 2 1 ⊕ 2 2 in table 4.
As a further corroboration of the mapping between the extended QD model and the extended LW model, we can compute the GSDs of both models on a cylinder, whose two boundaries may not necessarily possess the same boundary conditions. Using eqs. (2.9) and (B.10), we obtain respectively    where N k ij are the fusion coefficients satisfying Here, j * is the dual of j defined by the second row in the equation above. Each object j also has a nonzero characteristic numberd j , called quantum dimension of j, such that d j =d j * andd In particular,d 0 = 1. Let β j = sgn(d j ), then β i β j β k = 1 is satisfied if N k ij > 0. When F = Rep G for a finite group G, we haved j = β j d j , and β j is identified with the FS indicator of the representation j of G. Now we can define tetrahedral-symmetric unitary 6j-symbols as the coefficients of the map G : L 6 → C that satisfy the following conditions

B Extended LW model
In this section, we review the basic ingredients of extended LW model. The extended LW model is defined on an oriented trivalent latticeΓ , part of which is depicted in figure 6. The Hilbert space is spanned by all configurations of assigning of the simple objects of a UFC F on the oriented edges ofΓ . Conventionally, we call these simple objects string types and label them by integers in L = {0, 1, . . . , N }. Each string type j has a dual j * , which is also an element of L and satisfies j * * = j. If we reverse the orientation of the edge and conjugate the string type j → j * associated with the edge at the same time, the state remains the same. There is always a trivial (unit) element 0 ∈ L satisfying 0 * = 0.
The set {N k ij ,d j , G ijk lmn } is the input data of the extended LW model, which can be derived from representation theory of a finite group or quantum group. In this work, we take the string types µ to be the irreducible representations of a finite group G and label them with Greek letters. The trivial representation is µ = 0. The set L is identified with L G -the set of irreducible representations of G. The fusion coefficients N λ µν are the numbers of trivial representations appearing in the decomposition of µ ⊗ ν ⊗ λ. In such cases, as mentioned earlier,d µ = β µ d µ with β µ the FS indicator. The dual representation µ * of µ satisfiesd µ * =d µ . The 6j-symbols G µνλ κηγ are the same as the symmetrized Racah 6j-symbols of G. Similar to the extended QD models, there are also two types of local JHEP02(2020)030 operators in the bulk Hamiltonian of the extended LW model, namely and B LW p are respectively the vertex and plaquette operators. The action of A LW v on a local trivalent state as where δ µ 1 µ 2 µ 3 is delta function that is equal to 1 if N µ 3 µ 1 µ 2 > 0 and otherwise 0. The action of B LW p is more involved and reads The bulk Hamiltonian is exactly solvable because all the operators therein are commuting projectors.
On the boundary ∂Γ ofΓ however, we need an extra input data for the extended LW to be well-defined. This extra input data is the maximal Frobenius algebra object in Rep G . This maximal Frobenius algebra object can be identified with the canonical Frobenius algebra defined in eq. (4.3). But to comply with the definition of boundary vertex operators in refs. [11,12], where the extended LW model was constructed. We rewrite this canonical Frobenius algebra as As in the main text, here d s counts the multiplicity of s. Note that refs. [11,12] actually did not consider the cases with multiplicities in a Frobenius algebra and thus did not require an extra input data on the boundary. In this review, we offer a generalization to such cases. We emphasize that in an extended LW model, the boundary ∂Γ consists of the dangling JHEP02(2020)030 edges only. As far as boundary degrees of freedom are concerned, we regard A canonical as a set and allocate to each tail on the boundary an element sα s . The boundary Hamiltonian of the model is then defined on the boundary lattice ∂Γ ofΓ by where the A LW n operator acts on the tail n on the boundary and projects it to a Frobenius algebra object A ⊆ A cananical of Rep G : s α s ,η ,λ ,r α r f t * αts * α s sαs f rαrtαtr * α r ũ rũsũr ũ s × G ρ * ηs * t * s * η G κλη * t * η * λ G γλ * r * t * r * λ * ṽλṽηṽλ ṽ η where the second line uses eq. (3.25) and last equality uses eq. (3.17). To prove the completeness, we check that µ,ν,γ λ,s,m s nµ,nν ,n λ g,h,l x,y,z |g, h, l g, h, l|Ψ sms Ψ sms |x, y, z x, y, z| = µ,ν,γ λ,s,ms nµ,nν ,n λ g,h,l x,y,z g,h,l x,y,z,g ,g |g, h, l x, y, z|δ g ,e δ g ,g δȳ h,g δ gx,g δz l,g = g,h,l x,y,z |g, h, l x, y, z|δ g,x δ h,y δ l,z = g,h,l |g, h, l g, h, l|, where use is made of the great orthogonality theorem (3.6)