Bulk-to-boundary anyon fusion from microscopic models

Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2+1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.


Introduction
Topological phases of matter are intriguing zero-temperature quantum phases that are accompanied by robust ground state degeneracy and patterns of long-range quantum entanglement. They constitute cornerstones of modern condensed matter physics [1,2]. At the same time, they play a central role in notions of topological quantum error correction that is widely seen as one of the most promising paradigms for scalable quantum computing, a paradigm in which anyonic defects in the code are suitably manipulated to perform quantum information processing [3,4]. In more abstract terms, a common high-level description of topologically ordered models is via the codimension-2 defects or "excitations". The most famous example for this is the anyons in a 2+1-dimensional model, which are characterized by their fusion and braiding statistics [3]. Such a description via higher-level invariant data can be extended to boundaries or domain walls, by also considering excitations within the boundary. A more concrete lower-level description of topological order is via microscopic fixed-point models, which are exactly solvable due to a notion of discrete topological invariance [5]. Those models allow for the computation of any higher-level invariants usind only finite-dimensional linear algebra. In 2 + 1 space-time dimensions any non-chiral topological order can be represented by such a microscopic fixed-point model, with finite-dimensional Hilbert spaces.
There are two major pictures in which those fixed-point models can be formulated, namely the space (or string-net) picture, and the space-time picture. In the space picture, we start by assigning a Hilbert space to a cellulation of a two-dimensional manifold. The most well-known examples are string-net models defined on a trivalent cellulation [2] or its dual formulation in terms of a triangulation of the same manifold [6]. To achieve topological invariance, these models are also equipped with isometries mapping between the vector spaces of different cellulations. In fact, arbitrary changes of the cellulation are generated by local moves such as Pachner moves, so the model is defined by the associated local isometries. A local ground-state projector, or Hamiltonian, can be defined via a local sequence of moves/isometries which have no net effect on the cellulation.
In the space-time picture, a model is given by a discrete state-sum path integral in Euclidean space-time, the most famous one being the Turaev-Viro state-sum [7]. Such a state-sum is defined on any 3-dimensional cellulation of that space-time, and assigns a finite label set to any edge in that cellulation. The highestdimensional cells carry weights depending on those labels. The state-sum is evaluated by taking the product of all weights and summing over all label configurations, and it must be invariant under changes in the cellulation. The cellulations in the space picture can be interpreted as codimension-1 sections of the cellulations in the space-time picture, giving rise to the equivalence of the two pictures. In this work, we study topological phases on manifolds with boundary using both pictures, since they each have their own up-and downsides. The space/string-net picture is more illustrative to most physicists since it is directly related to the usual quantum mechanical language of Hilbert spaces, states, and Hamiltonians. Hence, we mainly present our constructions in this space picture. However, the mapping between different cellulations in this picture can be quite tedious to work out, so we fall back on the space-time picture to evaluate complicated sequences of moves.
Topological fixed-point models on manifolds with boundaries have been studied in various places in the literature to understand properties of the defects on the boundary and how they interact with each other. Refs. [8][9][10] focus on corners, defects between different types of boundaries or domain walls. In particular, they calculate vertical fusion events of codimension 2 defects along the same domain wall and horizontal fusion of defects on neighboring domain walls. Combining their methods allows them to also calculate the associator of these defects. Ref. [11] gives an algebraic description of what happens to bulk defects when approaching the boundary in terms of a forgetful functor on tensor category describing the bulk anyons. However, they do not give a constructive framework to calculate these properties. Ref. [12] gives a formula to calculate the set of condensable anyons in gauge theory models with trivial 3-cocycle. In general, it is known that the set of condensable anyons have to form a Lagrangian algebra object in the modular tensor category that describes the bulk anyons [13][14][15].
What we have found to be missing is a full description of the fusion of bulk anyons to boundary anyons. More explicitly, we are interested in the dimensions m ij ≥ 0 of the fusion spaces between an (ingoing) bulk anyon i and an (outgoing) boundary anyon j. Importantly, we are interested in constructive formulas to calculate m i,j .
Apart from a mathematical interest, these fusion events find application in topological quantum error correction and computing with boundaries [16]. For example, the logical operators of a topological code on a manifold with boundary are associated to ribbon operators [17] of anyons which connect different boundary segments through the bulk. Moreover, any lattice-surgery-based computation scheme ultimately relies on deforming non-transparent domain walls between code patches into (partly-)transparent ones in a systematic way [18][19][20]. Understanding how the anyon ribbon operators precisely behave close to these domain walls is therefore essential in the design of novel computational protocols in topological codes. In this sense, the work done here is also expected to provide guidance when devising novel schemes of topological quantum computing involving notions of lattice-surgery with codes beyond untwisted quantum doubles.
With the framework established in this work we aim to contribute to a further understanding of topological phases with boundaries by formulating a framework to describe bulk-to-boundary anyon fusion events in topological fixed-point models. We explicitly derive a closed formula for fusion multiplicities of fusion events between bulk and boundary anyons for 2+1-dimensional twisted gauge theory models, also known as Dijkgraaf-Witten state-sums [21]. In this case, the anyons in the bulk as well as on the boundary are classified by irreducible sub-spaces of some special type of algebras, which we call twisted group algebras with action. We show how such an algebra is diagonalized and discover that there is an intimate connection to a group cohomological isomorphism that also appears in the classification of topological boundaries of gauge theories.
This manuscript is structured as follows. In Section 2, we give a general recipe to find the irreducible subspaces of twisted group algebras with action, which characterize both the bulk and boundary anyons in gauge theory models of a finite group. In Section 3, we give a self-contained introduction into string-net fixed-point models for topological phases with boundaries in two spatial dimensions and illustrate the equivalence to spacetime state-sum models. This section is mainly addressed to readers not yet familiar with these models. Readers with a background in both formulations of fixed-point models might want to use that section to get familiar with our notation and conventions in the upcoming sections. In Sections 4 and 5, we classify the anyons in the bulk and boundary. In particular, we focus on gauge theory models of a finite group. The main result of this paper is presented in Section 6, where we define a bimodule that allows to calculate the dimensions of bulk-to-boundary fusion events in any fixed-point model and explicitly derive a closed formula for the gauge theory case. Lastly, we give many examples, that partly already appeared in the literature, but combine them with new calculations to show the wide applicability of our formula to boundaries as well as domain walls. Finally, we conclude the results and give an outlook into possible continuations of this work in Section 7. For a reader interested in the technical details going into the derivation of the bimodule used in Section 6 and tools to diagonalize the group algebras characterizing point defects, we refer to the appendix for further details.
2 Diagonalizing twisted group algebras with action Before we introduce topological fixed-point models, we want to highlight the technical tools used to classify bulk and boundary anyons of gauge theory models in Secs. 4 and 5. In both cases topological invariance of the anyonic subspaces naturally defines a finite-dimensional algebra. In most parts of this paper we focus on gauge theory models, derived from a finite group and a 3-cocycle on it. For these models both bulk and boundary anyons are classified by a special type of algebra. In this section, we will present the technical tools used to find the irreducible sub-spaces of, i.e., block-diagonalize, these algebras.
Consider a finite group G. Let X be a finite (left) G-set, i.e., there exist a map : G × X → X representing the G-multiplication on X, (1) Given a G-set X, we define an algebra A over C X×G via the multiplication (x, g) * (y, h) = δ x,h y Ψ y (g, h)(y, gh), (2) with Ψ : X × G × G → U (1) ⊂ C. For this multiplication to define an algebra it has to be associative. This imposes a non-trivial condition on the phase Ψ, This can be seen as a 2-cocycle condition over U (1) X as a G-module with non-trivial action defined via the G-action on X. Moreover, we require that Ψ x is normalized, i.e., Ψ x (1 G , h) = Ψ x (h, 1 G ) = 1 ∀g ∈ G, x ∈ X.
We can in fact redefine the basis states of A with ξ : G → C × by (x, g) T → ξ(g)(g, h) T to effectively map Ψ x to a normalized 2-cocycle [22,23] so we do not loose generality with this assumption. We call such an algebra twisted group algebra with action. For more details on G-modules and group cohomology, see App. B. The goal of this section is to find the irreducible sub-spaces of A. In particular, we find faithful invariants classifying the sub-spaces, determine their dimension and find the associated central idempotents whose representations project onto the respective sub-spaces. We give a comprehensive summary in terms of a recipe to construct the central idempotents at the end of this section.
First, we note that due to the delta in the multiplication in Eq.
(2), A decomposes over transitive subsets of X, or equivalently, over G-orbits {X i ⊆ X}, so we have an isomorphism For any X i there exists a subgroup K i ⊆ G such that X i is isomorphic to G/K i , the set of left K i -cosets. After this isomorphism, G acts via left translation, g hK i = (gh)K i , g, h ∈ G, onto X i . Since X i is a transitive G-set the stabilizer groups of any element x ∈ X i , are isomorphic. In particular, for x ∈ X, g ∈ G, Hence, we can define an abstract stabilizer group of X, Stab G (X) Stab G (x) for any x ∈ X. In fact, K i is isomorphic to Stab G (X). As a subgroup of G, K i depends on the chosen identification X i ↔ G/K i . We pick a representativex i ∈ X i and define Note that we can obtain the stabilizer group of any other element in X i from K i with Eq. (6). Next, we show how to further decompose each A i into irreducible components. In fact, we find that they are in one-to-one correspondence with irreducible unitary Ψx i -projective representations (IPRs) of K i . To see this, we explicitly construct the indecomposable central idempotents in A i from IPRs of K i . Let {ρx i } be the irreducible Ψx i -projective representations of K i , defined via a representativex i ∈ X i . In particular, they fulfill and are unitary. If not stated otherwise, any representation considered in this manuscript is assumed to be unitary. We can construct an equivalent IPR of any other isomorphic stabilizer group Stab G (y) for y ∈ X i . Concretely, starting from an irrep ρ x of Stab G (x) and k ∈ Stab G (x) we can define the irreducible representations on all of A via which by construction fulfill when the group label is restricted on the respective (isomorphic) stabilizer groups. Note that we act with the inverse group element on the left-hand-side of Eq. (9) because we pull of the G-action on X back onto ρ. This allows us to uniquely construct the IPRs of all stabilizer groups Stab G (x) for x ∈ X i from the IPRs of the stabilizer group of a single representativex i ∈ X i . Given all the equivalent IPRs, we can construct the indecomposable central idempotents in A i . They are labeled by IPRs {ρ i } of K i and given by we can show straight forwardly that the c i,ρi s are idempotent, and central, For now, we have found a set of invariants, {i, ρ i } labeling independent central idempotents. To show that they are complete, i.e., correspond to irreducible invariant sub-spaces of A, we have to show that the c i,ρi s are not only central and idempotent but also indecomposable. To see this, consider the following set of smaller idempotents that are irreducible (by definition ofχ) but not central. In fact, we can get to any value of x ∈ X i via left-or right-multiplication of an element in A which makes c i,ρi irreducible. Furthermore, the isomorphism discussed in App. C provides a 1-1 mapping from irreducible representations of the twisted algebra with action in Eq. (2) and Ψx-twisted group algebras without action of (a collection of) subgroups. In App. E we give an interpretation of this isomorphism in terms of an invertible domain wall mapping between two different kinds of state-sums with boundaries. We summarise this section by giving a recipe on how to find the irreducible representations of an algebra of the form of Eq. (2).
2. For each X i : (a) Pick representativex i ∈ X i and calculate its stabilizer group (c) Use Eq. (9) to derive irreducible representations and character functions,χ x ρi (g) = Tr(ρ x i (g)) for all x ∈ X i .

The indecomposable central idempotents in
Irreps of K i ) and given by Note that the above is not a complete algorithm for finding the algebra irreducible representations but merely reduces it to finding the irreducible representations of a much smaller algebra in step 2b. Those irreducible representations are equivalent to projective group representations which can be found in the literature in many cases.

Models for topological phases with boundary
Topological phases which possess gapped boundaries can be studied using fixed-point models on a discretized space-time. Topological invariance highly restricts the microscopic constituents of these models. In the Turaev-Viro state-sum [7,24], or tensor-network path integrals [5,25] for 2 + 1-dimensional topological order, the topological invariance corresponds to recellulations in a three-dimensional space-time, and the algebraic constraints correspond to the ones defining spherical fusion categories. A different but equivalent picture are Levin-Wen string-net models [2], where recellulations on a two-dimensional space triangulation are represented by linear operators acting on the local degrees of freedom. In this picture, the topological invariance in spacetime takes the form of coherence axioms between different equivalent space recellulations. Since these models are equivalent, one can construct the linear operators implementing topological invariance in Levin-Wen models from a state-sum on particular space-time cellulations.
In this section, we introduce microscopic models for gapped topological phases on manifolds with boundaries. We mainly use the string-net picture, but occasionally refer to the space-time picture where we find it more instructive. An overview of the space-time picture can be found in App. D. First, we introduce the bulk degrees of freedom and how states hosting exact topological invariance are constructed based on a spherical fusion category C. Secondly, we extend the models to boundaries and show how the model is constrained by the bulk data close to the boundary, leading to a description of the boundary in terms of a C-module category.

Bulk
In the bulk, the microscopic model is defined by a spherical fusion category C. We denote the set of (finitely many) simple objects in C by Obj(C) = {1 C , i, j, . . . , k} and fusion multiplicities N k ij ∈ Z ≥0 . These define a fusion-operation A fusion category for which there exist i, j ∈ Obj(C) for which N k ij > 0 for more than one k ∈ Obj(C) is called non-Abelian. Moreover, the fusion above can be equipped with a non-trivial associator capturing the isomorphism between objects obtained from fusing in different orders. The associator is part of the input category C and will be described in terms of so-called F-symbols in microscopic models.
A microscopic (topological) model is defined on a framed 1 trivalent graph (tessellating some two-dimensional manifold) with local Hilbert spaces H i = span(Obj(C)) on each edge. The total Hilbert space is the tensor product space of all the local spaces, H tot = i H i . Obj(C) defines a natural basis on H i and with that on H tot . The fusion multiplicities define a local constraint at each vertex, defining the physical subspace H phys ⊂ H tot . H phys is defined by the span of basis states for which the local labels (i, j, k) at every vertex fulfill N k ij = 0. For N k ij > 1 the vertex itself carries in additional degree of freedom, span({0, 1, . . . , N k ij − 1}). For the rest of this work, we will work within H phys and depict a vertex and its adjacent edge labels (i, j, k) in an allowed basis configuration with Note that, when going around a vertex, one frame has to point in a different direction than the other two. We call this condition local acyclicity and is related to the fact that one has to distinguish the left and the right hand side of Eq. (17) when interpreting it as vertex. Given this prescription, one can tessellate (the bulk of) any two-dimensional manifold with these trivalent vertices, each of which enforce a local constraint on the 1 A framed graph has "flags" on each edge pointing perpendicular to it. The orientations have to be chosen such that the flags do not point in the same direction around any vertex. This induces a local ordering of the faces around any vertex by the number of flags pointing into the faces. Analogously, one can think of such a framing as a branching structure on the dual triangulation, see for example (19). Figure 1: Different sequences of F -moves evaluating to the same transformation on of the graph have to compose to the same map on the associated vector spaces. In particular, the above diagram has to commute, i.e., the composite map corresponding to the top path has to evaluate to the same as composing the maps corresponding to the bottom path. The resulting condition on the F -symbols is called pentagon equation. edge labels. Every allowed configuration defines a basis vector in the state space assigned to the cellulation. To construct a topological fixed-point model, topological invariance is imposed exactly. In particular, one can relate different cellulations (with the same input category C) via Pachner moves, for example an F-move .
The fusion category C defines linear transformations that relate vector spaces of different cellulations to each other. The matrix entries of this linear map in the basis of string labels are complex numbers {F aabe,µη cdf,νρ }, These F-symbols are part of the input category C. Any sequence of topological moves can be represented by a 3-dimensional triangulation derived from the dual cellulation indicated in gray in the above figures. In the multiplicity-free case, where µ, ν, ρ, η are fixed, the F-move above, for example, is represented by a tetrahedron, In general, the faces would carry the additional multiplicity labels, but we will suppress those multiplicity labels for the rest of this paper. One can view the bottom (dotted) edge as the "initial", vertical, edge and the top edge as the "final", horizontal, edge in Eq. (20). The remaining four edges correspond to the outgoing/incoming edges in Eq. (20). Note that the boundary of the complex is the union of the initial and final cellulation. In a similar way, any sequence of Pachner moves can be represented by a three-dimensional cellulation whose boundary is the union of the initial and the final (two-dimensional) triangulation. The associated amplitude is evaluated by taking the product of numbers associated to the subsimplices. For details on the space-time picture for topological moves in our models, see App. D.
The numbers associated to the labeled space-time simplices have to fulfill several consistency conditions to implement exact topological moves. For example, if one combines more than 2 vertices, there are different sequences of Pachner moves that have the same effect on the tesselation, see, for example, Fig. 1. This results in the non-trivial pentagon equation on the F -symbols, (22) for all a, b, c, d, e, f, g, h, i which can be depicted as On the left hand side the diamond formed by the 5 vertices is tessellated with two tetrahedra (top and bottom) whereas on the right hand side the same diamond is tessellated with three tetrahedra around the middle axis. Both should give the rise to the same number when summed over labels in the interior. Strictly speaking, the pentagon equation in Eq. (22) is not the only constraint to the F -symbols. Firstly, for complete topological invariance, we would need to impose the move in Eq. (23) for all possible branching structure configurations. Equivalently, one can add simpler auxiliary axioms specifically targeted to change the branching structure [5]. Secondly, for a physical model we have to impose Hermiticity/unitarity, which means that all triangulations carry an orientation, and orientation reversal equals complex conjugation.
A spherical fusion category C [26] is the mathematical object giving all the data for a consistent definition of topological moves in the bulk. The topological (ground) space on a given manifold is modeled by the space of all labeled modulo topological moves. In practice, one chooses a minimal reference cellulation whose labelings form the basis of the associated vector space. One can use topological moves to relate any (basis) state on a given cellulation to an equivalent one on the reference cellulation.

Boundaries and domain walls
At the boundary, the bulk model is terminated along boundary edges which can host different degrees of freed. However, the there is some (possibly non-trivial) action of the bulk edges connected to the boundary on these new degrees of freedom. In order to have full topological invarince of the model, the action has to fulfill certain consistency conditions. As we will show in this section, a boundary to a bulk with input category C is given by a (left) C-module category C M [8,11,26]. The module category C M is a (semi-simple) category equipped with a (left) C action. On the level of the simple objects of C M, the C action is defined via module fusion multiplicites M β a,α ∈ Z ≥0 , This action has to be compatible with the fusion operation in C, i.e., This gives consistency conditions on {M β aα } in terms of {N c a,b }. Graphically, the C-action can be represented by trivalent boundary vertices where µ is non-trivial for M β aα > 1. We will later resort to the space-time picture where the dual edges of the above cellulation enter. As before, we depict them by dashed edges. Similar to the F -symbols in the bulk, there In the space-time picture this move can be represented by a triangle with additional boundary labels at each of its vertices. Omitting the multiplicity labels it can be represented by The L-symbols have to fulfill an associativity condition similar to the bulk pentagon equation. This can be expressed as a commutative diagram in Fig. 2. The resulting condition on the L-(in relation to the F -symbols) reads The equation on the right shows the corresponding space-time recellulation. On the left, there are two boundary triangles, whereas on the right there is one bulk tetrahedron with two boundary triangles. Note that the edge labeled by e on the right side is not part of a boundary triangle but extends into the bulk. Again, Eq. (29) does not yield full topological invariance, but we have to add additional axioms as well as Hermiticity. For the models we are going to study, these additional axioms will be fulfilled automatically.
Taken together, the mathematical structure describing the bulk-boundary microscopics is a C-module category. Given C and module categories associated to (distinct) boundaries, we can now describe a topologically ordered ground state on manifolds with boundaries by the space of its cellulations modulo topological moves. The exact topological invariance allows us to work with a minimal reference cellulation and use moves to relate a state on a different cellulation to an equivalent one on the chosen reference cellulation. Figure 3: Via the folding trick, a C-C domain wall, classified by a C-C bimodule, is equivalent to a C ⊗ C module, defined by L symbols of the above form. In Eq. (33) we give the equation relating the data defining the bimodule to the boundary data after the fold.

Domain walls and the folding trick
Given consistent F and L symbols -a fusion category C and a C-module category C M -we have defined a topological fixed-point model for a topological phase with boundary. In fact, this data can also describe interfaces between topological models each of which is described by fusion categories C and C . Such interfaces are often called domain walls and are defined by a C-C -bimodule category [8,11,26]. A C-C -bimodule category is defined by a left C-, and a right C -action, each of which is equipped with L-, respectively R-symbols that are both compatible with fusion in C, respectively in C . Graphically, the associated moves can be depicted by where C-labeled strings are depicted in green and C -labeled strings in blue. Moreover, there are moves including strings on both sides of the domain wall. The associated linear maps are given by so-called C-symbols and can be graphically depicted as Again, the L, R and C symbols have to fulfill consistency conditions coming from different sequences of moves with the same overall effect. Another way of studying domain walls is by "folding" one side of the domain wall onto the other side. With this, the C-C domain wall becomes a boundary of a C ⊗ C model, where the bulk strings are labeled by tuples (a, a ) ∈ Obj(C) × Obj(C ) and the F -symbols are inherited from C and C but with the C F -symbols complex conjugated. Omitting the multiplicities in C and C individually, the folding trick can be depicted graphically as Note that the boundary vertex can get a multiplicity even if the two input fusion categories are multiplicity-free. This is less of a feature of the folding trick than a consequence of combining two connected tri-valent vertices into a single four-valent one. We will see that for the model class we are interested in this paper, there is no additional multiplicity coming in. The L-symbols of the folded model (see Fig. 3) are straight forwardly obtained by a combination of the (L, R, C) symbols using the correspondence above to resolve the L move of the folded model to a sequence of L, R and C moves in the unfolded model,

Topological gauge theory models
Let G be a finite group. For the rest of the section, we focus on C = Vec ω (G), the category of G-graded vector spaces. In Vec ω (G), the simple objects are group elements and the fusion multiplicities are given by the group multiplication, With that, the vertices take the following form Since there are just two free labels at any vertex, the F -moves are defined by a single function ω : In this case, the pentagon equation reduces to a 3-cocycle condition on ω, Moreover, we have a gauge freedom. Namely, we can multiply the F -symbols by product of local unitaries. In our model these unitaries are simple phases for every vertex, respectively face in the dual picture. In general this phase can depend on the local configuration around the vertex, given by a pair of group elements. Hence, any function η : G × G → U (1) defines a gauge transformation. Applying the corresponding gauge in (36) maps the associated 3-cocycle to . (38) In fact, the product of ηs with which the 3-cocycle gets multiplied is exactly the coboundary of the 2-cochain η. This shows that topological lattice models based on Vec ω (G) are classified by the third cohomology group H 3 (G, U (1)), see App. B. Boundaries in topological gauge theory models correspond to module categories of Vec ω (G) and have been studied in detail in the mathematical literature and classified by a (possibly twisted) group algebra of a subgroup H ⊂ G on which the input 3-cocycle ω is cohomologically trivial [8,15]. In this section, we see how topological boundaries are classified in our model and connect it to microscopically different, but equivalent, classifications in the literature. In particular, we explain how to model a boundary associated to a twisted group algebra where a ∈ α is a representative of the coset α.
Since every vertex is multiplicity-free and only two of the labels of its incident edges are independent, the L-symbols only have three open indices. They are defined by ψ : The phase function has to satisfy the mixed pentagon equation in Eq. (29). In this case, it simplifies to whereδ is the twisted coboundary operator (for more details, see App. B). This means ω has to be the twisted coboundary of ψ α if we interpret ω as a twisted cochain which is constant in α. By setting α to the trivial coset H and restricting all arguments to the subgroup H, we see that ω has to be a (untwisted) coboundary when restricted to H. Given any solution ψ to the equation above and a twisted cocycle α (that is,δα = 0), ψ = ψ · α is also a solution. Furthermore, two different boundaries ψ can be considered equivalent if they differ by on-site unitary gauge on the boundary fusion vertices. In our case, those fusion spaces are 1-dimensional (or 0dimensional if the fusion rules are not obeyed), so such an isomorphism is defined by a phase ξ : G/H ×G → U (1) depending on the labels of the strings adjacent to a boundary fusion vertex. Since ψ contains three boundary fusion vertices (two at its input and one at its output), the isomorphism acts as We see that the gauge corresponds to a multiplication of a coboundary with the same coboundary operator as in Eq. (41) but acting on the 1-cochains {ξ α }. In total, we see that although the different ψ are not twisted 2-cocycles, their differences are, whereas ψs differing by 2-coboundaries are considered equivalent. Hence, the gauge equivalence classes of boundary models cannot be directly identified with the twisted cohomology group H 2 (G, (U (1) G/H ) G ), but are equipped with a regular action of the latter. Here, the subscript G indicates the non-trivial right action of G. That is, the set of equivalence classes forms a torsor over the second twisted cohomology group.
In App. C and E, we show that this classification indeed coincides with known results, i.e., In particular, this isomorphism holds for every cohomology group H n for n ≥ 1 and is induced by the map m (n) : G/H × G ×n → H ×n that -to the best of our knowledge -has first been mentioned by T. Lawson in Ref. [27].

Models for Abelian phases
A topologically ordered phase is called Abelian if the fusion outcome of any pair of topological point defects -anyons -is deterministic. Note that not every Abelian fusion category gives rise to an Abelian phase (see Section 4.4). However, as we will see later, the Abelianess of the anyons can be traced back to properties of the input category if it is of the form Vec ω (G). In this case, the group has to be Abelian and the twisting 3-cocycle of a certain form. Any finite Abelian group is isomorphic to a product of cyclic groups, where p i is prime and m i positive integers for any i = 1, . . . , N . For the rest of this section it suffices to consider G having N independent cyclic factors. Inequivalent F -symbols are classified by 3-cocycle classes on the group above. In fact, any 3-cocycle class on such a product group can be represented as a product of so-called type-I, type-II and type-III cocycles (see App. B). The anyons in our model are only Abelian for a cocycle that is cohomologous to a product of type-I and type-II cocycles only, see Section 4.4. From the above decomposition into cyclic factors any subgroup H and the associated cosets G/H can be easily determined. Moreover, its second cohomology group decomposes similarly, see App.B. Given a subgroup H (with k factors), we can construct a non-trivial 2-cocycle by taking products of 2-cocycles on each pair where a i (a j ) is the component of a in Z hi (Z hj ) ⊂ H ij . Following the previous subscection, the associated L symbol is obtained by precomposition with m (2) ,

Examples
In this section, we give some examples of the input data for Vec ω (G) models with boundary.

Vec ω (Z N )
The cyclic group of order N , 3-cocycle classes each of which are generated from a single 3-cocycle class. The normalized 3-cocycle is a canonical representative for the nth cocycle class of H 3 (Z N , U (1)) = Z N [28]. We say that a 3-cocycle of that form, only supported on a single cyclic factor, is of type I. Using Z N and ω n I as an input gives an Abelian bulk theory.
Let us look at possible boundaries of such a bulk model. For simplicity, we take N = p prime. In this case, Z p only has two subgroups, H 0 = {0} and H 1 = Z p . Both H 1 and H 2 only have trivial 2-cocycles, i.e., only one potential boundary associated to either of them. In the untwisted case both give rise to a boundary and correspond to rough and smooth boundaries of Kitaev's toric code on qupits [9,17]. In the twisted case, for n = 0, ω n I becomes cohomologically trivial only on H 0 . Hence, these models only have one "standard" boundary.
I. Their normalized representatives decompose into a product of cocycles as in Eq. (47), each supported on one factor only. Additionally, there are non-trivial type-II cocycle classes represented by [28] ω n12 3-cocycles of type II are supported on two cyclic factors and are gauge inequivalent to any type I cocycle.
Together with the two subgroups generated by type I cocycles we have They In the untwisted case any of these subgroups defines a boundary model. Additionally, H 4 has non-trivial 2-cocycles of the form Since H 4 is the whole group, there is only one coset such that ψ only depends on group labels. Note that if H has two cyclic factors and is a proper subgroup of G the form of ψ will be different because the coset label α can be non-trivial, see Eq. (46). The 3-cocycle in Eq. (48) becomes cohomologically trivial on H 0 and the isomorphic subgroups H 1 , H 2 and H 3,l ∀l. All of them have no non-trivial 2-cocycle class which gives 3 + l inequivalent topological boundaries.

Vec ω (S 3 )
The smallest non-Abelian group is the permutation group of three elements S 3 = t, r | t 2 = r 3 = e, tr = r 2 t Z 3 Z 2 (e being the identiy element). Its third cohomology group is given by [28] Interestingly, the third cohomology group is the product of the cohomology groups of the two non-trivial subgroups of S 3 . However, since S 3 is a a semidirect product of the two, the 3-cocycles are not simple products of 3-cocycles of the subgroups. The 6 inequivalent classes can be represented by the normalized cocycles [28] where the elements in S 3 are represented by t A r a with A ∈ Z 2 , a ∈ Z 3 and ⊕ 3 denotes addition mod 3. When comparing the individual factors with a 3-cocycle of a cyclic group (see Eq. (47)), we note that the second factor, (−1) ABC , corresponds to the non trivial 3-cocycle of the (Abelian) subgroup Z 2 ⊂ S 3 and the first factor corresponds to the non-trivial 3-cocycle of the subgroup Z 3 ⊂ S 3 precomposed with the automorphism S 3 has four subgroups up to conjugation, H 0 = {e}, H t = t Z 2 , H r = r Z 3 and H G = S 3 . All of them have a trivial second cohomology group [8,29]. Hence, there is one boundary type associated to each of these three subgroups in the untwisted case. For a twisted bulk model, only the subgroups on which the 3-cocycle in Eq. (51) is trivial, define a consistent boundary. Specifically, ω p is cohomologically trivial on H r for p = 0, 3, and is trivial on H t for p = 0, 2, 4.

Bulk anyons: Tube algebra
In this section, we will revisit the question of how to add anyons to the fixed-point models from Section 3.1. Anyons are (irreducible) point-defects in the bulk of a topological phase. Their world-lines live in a threedimensional space-time and are equipped with compatible fusion and braiding structure. In mathematical terms, anyons form the simple objects in a (unitary) modular tensor category (U)MTC. In fact, all their defining data can be calculated with fixed-point models. In this manuscript, we mainly focus on finding the anyons themselves, i.e., the set of irreducible sub-spaces of the point-defects and comment shortly on how to derive the fusion and braiding data with similar methods in Section 4.2.

General fixed-point models
Anyons are point-like topological defects in the bulk, and are known to be characterized by string-nets on an annulus, respectively a "tube", (0, 1) × S 1 . In fact, the associated vector space can be equipped with a multiplicative action on itself, rendering it an algebra. The irreducible representations of this tube algebra [30][31][32] can be associated to the simple objects in the UMTC describing the bulk anyons. In this section, we will illustrate this concept by first calculating the irreducible sub-spaces of the tube algebra in topological gauge theory models and sketch how to obtain a consistent fusion and braiding on them.
In principle, one can choose any tesselation of the annulus for the upcoming analysis. It is beneficial to choose a simple representative. For the rest of this section, we define a basis element of the tube algebra T , labeled by (a, b, c, d) ∈ Obj(C) 4 (again, omitting the multiplicity labels at the vertices), via the following cellulation: (52) We can define an (associative) multiplication on the vector space spanned by the diagrams of the above form by gluing two tessellated tubes together and using topological moves to map back to the initial cellulation, see Fig. 4. Evaluating the space-time complex corresponding to the recellulation, we obtain

Fusion and braiding in the bulk
The anyons in topological fixed-point models form simple objects of a unitary modular tensor category (UMTC). This means they are equipped with additional data/topological quantum numbers related to their fusion and braiding. In fact, this data together is known as the Drinfeld center of the fusion category defining the lattice model. Although it is outside of the scope of this paper to compute the full center, i.e., the fusion and braiding data, we want to comment on how they can be calculated with similar techniques that we have already introduced. For related discussions we refer to App. D and Refs. [6,30]. Before deducing new quantities of the topological phase of the lattice model, let us take a step back and put the tube algebra into a larger context. The fact that the topological vector spaces associated to a tube, S 1 × (0, 1), forms an algebra, comes from the fact that gluing a tube onto a tube again gives a tube. Hence, we can define an action of T onto itself, i.e., there exist a map Figure 4: The multiplication of two tube algebra basis elements (a, b, c, d)T and (a , b , c , d )T is defined via gluing the two associated string diagrams together (left) and using F -moves to reduce it to the cellulation on the right. The phase acquired by the sequence of moves can be derived by evaluating the space-time complex that maps the two dual triangulations to each other, which is composed of three tetrahedra (middle). Note that the front-and the back-side edges of the space-time complex above are identified.
defining an (multiplicative) action of T onto itself. Similarly, when considering cellulations (modulo local moves) of other manifolds, one can define endomorphisms on the topological vector space from gluing operations that leave the manifold invariant. We will now turn our attention to how this allows us to calculate the fusion multiplicities in the UMTC formed by the anyons. Consider a fusion process in 2+1-dimensional space-time. It can represented as a vertex where three anyon world-lines meet. The boundary of the regular neighborhood of this vertex is a pair of pants -or three-punctured sphere. Just as before we can define a vector space (in terms of its basis) by tessellating this manifold with trivalent vertices from the input fusion category (see (18)). Let's call this space F . In fact, gluing a tube onto any of the three holes of the pair of pants does not change its topology. Hence, we can define a "tri-representation" As a representation, F decomposes into a direct sum of triples of irreducible representations of T , where a, b, c label the irreducible representations of T , i.e., the anyons of the bulk. The multiplicities in this decompositionÑ c ab coincide with the fusion multiplicities of the bulk anyons. In order to calculate them, we consider the tri-representation in Eq. (55) for a triple of central idempotents (P T a , P T b , P T c ) ∈ T ×3 yielding a projector P F , and then take the trace of P F . We will see that we use a similar approach to get the fusion multiplicities into the boundary, see Section 6. In App. D we give a formula forÑ c ab for an input category of the form Vec ω (G). We will use the same method to calculate the dimension of the "fusion" vector spaces for another type of space-time event, namely the partial condensation of an anyon in the bulk to an anyon in the boundary, see Section 6.
To complete the UMTC, we have to define a braiding and an associator on the fusion spaces of the irreducible representations of T . One way to fully define a braiding is via so-called R-symbols, defining a half-braiding on anyon world-lines, The associator on the other hand is defined by so-called F -symbols changing the order of fusion. Both R-and F -symbols have to obey consistency conditions known as pentagon and hexagon conditions [26]. Given this set of data, one can derive the topological quantum numbers like self-exchange statistics (topological spin θ a ) and the mutual-exchange statistics (S matrix) for any anyon with a simple calculation. For more details, see App. E of Ref. [3].
Topological fixed-point models allow for a direct calculation of both R and F -symbols. 2 In App. D, we give an explicit prescription of how to obtain the F -symbols and give a formula for the case of a Vec ω (G) model. Let us here shortly lay out how to obtain the R-symbols with microscopic models. For this we again consider the vector space F defined via a tessellated pair of pants. This space represents the fusion space of three bulk anyons. The half-braiding acts on this vertex by interchanging two of its legs as in Eq. (57). This induces a non-trivial action on the cellulation. In the same spirit as before we can use local moves to map it back to the original cellulation and thereby define an action on F . To express this action as a tensor in the basis of anyons, i.e., irreducible representation of T , we have to project the R-action onto this associated sub-spaces by precomposing it with a triple of central projectors (P T a , P T b , P T c ) ∈ T ×3 . Similarly, one can calculate the F -symbols from microscopic models by mapping between the two ways of decomposing a sphere with four holes into two pairs of pants.

Topological gauge theory models
For gauge theory models the input fusion category is Vec ω (G), each vertex is multiplicity-free. The basis state in Eq. (52) are parametrized by two group elements. We label them by (g, h) T = (hgh −1 , h, hg, g) T . Plugging in the defining data from Section 3.3, we obtain the tube algebra T for the group case as, defined over the vector space C G×G , where β g (h, h ) is the phase assigned to the sequence of moves from the left to the right hand side of Fig. 4. Evaluating the space-time complex representing the moves in Fig. 4, we get which can be seen as a slant product i g ω, see App. B. The tube algebra represents D ω (G), the twisted quantum double of G, introduced by Dijkgraaf et al in Ref. [33] and plays an important role in the study of (finite) gauge theory models for topological phases and its applications to topological stabilizer codes [6,30,34,35].
In fact, we find that Eq. (58) is of the form of Eq.
(2) with X = G acting on itself via conjugation, i.e., h g = hgh −1 . Indeed, β g plays the role of Ψ x and fulfills in analogy to Eq. (3). Choosing a normalized 3-cocycle ω makes β g normalized and with that we can use the algorithm from Section 2.
To construct the irreducible representations of T and the associated central idempotents we proceed as described in Section 2. First, the transitive subsets {X i } are the conjugacy classes {c}. For each conjugacy class c, we pick a representativeĉ. It is stabilized by its centralizer Z(ĉ) = {g ∈ G | gc = cg}. Following the general considerations from Section 2 the irreducible sub-spaces of the tube algebra are additionally labeled by irreducible βĉ-projective representation of Z(ĉ). Combined, we identify the irreducible sub-spaces of T with a pair (c, ρ c ), a conjugacy class and an irreducible β g -projective representation of Z(ĉ). The central idempotents are given by whereχ g ρc (h) := Tr(ρ g c (h)) denotes the projective character of the IPR ρ g c of Z(g). Note that the stabilizer group of any element in c is given by conjugating the centralizer of the chosen representative, In practice, finding the irreducible projective character functions is not a straight forward task. In the special case, where β g is a coboundary, i.e., the IPRs of Z(g) are in bijection with the irreducible linear representations. In particular, one can "gauge" away the twist β g with the cochain g and the projective characters are given bỹ where χ g ρ (h) is the linear (not projective) character of the irrep ρ of Z(g). With that, the central idempotents simplify to We will see in the following sections that the examples considered in this paper are all of the above form. A well-studied example where β g defines a non-trivial projective irrep is obtained from G = Z ×3 N and ω being cohomologous to a type-III 3-cocycle. In this case, there are higher-dimensional irreducible sub-spaces in the tube algebra even though the input group is Abelian, see, e.g., Refs. [6,28].

Abelian tube algebra
In an Abelian anyon model the fusion of two anyons a and b yields a unique anyon c, i.e.,Ñ c ab = 1 for exactly one c and 0 otherwise. In other words, the fusion space of any pair of anyons is one-dimensional. If we want our microscopic model to yield an Abelian anyon theory, we thus require |c| = dim(ρ c ) = 1 for all conjugacy classes and IPR ρ c . The fact that all conjugacy classes consist of a single element implies that G is Abelian. All linear irreducible representations of an Abelian group are one-dimensional and can be labeled by group elements. In order for the IPRs to be one-dimensional as well, β g has to be trivial in the sense of Eq. (62). In this case, the central idempotents simplify to With these |G| 2 independent central idempotents in the |G| 2 -dimensional algebra T we have found all irreducible sub-spaces. Each is labeled by a pair of group elements (g, k) and one-dimensional. The latter shows that the associated anyons are indeed Abelian and is directly related to the fact that β g is a coboundary. If it is a non-trivial 2-cocycle, the resulting anyon theory is non-Abelian [6,28].

Examples
In this section, we will first give two examples for Abelian models that are representative for any Abelian lattice model. To illustrate the generic procedure of explicitly finding the central idempotents also for non-Abelian models we further discuss twisted versions of a S 3 lattice model. For any Vec ω (G) model, the tube algebra is diagonalized by the irreducible β-projective representations of the centralizers of all the conjugacy classes of G.  )) and the multiplication in T is given by This tube algebra is Abelian in the sense described above since for any every a ∈ Z p , we can define the 1-cochain Together with the group character function χ a (b) = e 2πi p ab , the central idempotents of the tube algebra -labeled by (a, k) ∈ Z ×2 p -are given by For a type-I cocycle on either of the two factors of Z p × Z p , the central idempotents will have the same form as the ones in Eq. (68). In this example, we consider a type-II cocycle defined in Eq. (48). The multiplication in T is given by where we abbreaviated δ a1,a 1 δ a2,a 2 with δ a,a . Again, we can find a 1-cochain the central idempotents -labeled by (a, k) ∈ (Z p × Z p ) ×2 -are given by In this section, we will first consider the tube algebra of an untwisted Vec(S 3 ) model in detail and then sketch how the twisting by a 3-cocycle affects the tube algebra. Note that the untwisted model has been studied in various contexts in the past [8,29,36]. S 3 has two independent generators, r and t. They satisfy t 2 = r 3 = e, where e is the identity element, and tr = r 2 t = r −1 t. The subgroup Z 3 generated by r is normal. S 3 has three conjugacy classes with the respective centralizers To obtain the central idempotents of the tube algebra, we need to find the irreducible representations of the centralizers. Let us first consider the trivial conjugacy class A. Its centralizer is all of S 3 and has three irreducible representations, a trivial one Γ 0 , a one-dimensional one Γ 1 and a two-dimensional one Γ 2 . The character table is given by We obtain three anyons associated to the trivial conjugacy class. The corresponding idempotents read Note that (e, Γ 0 ) is the trivial anyon -or vacuum -subspace. The dimensions of the respective sub-spaces coincide with the dimensions of the irreducible representations Γ 0 , Γ 1 and Γ 2 respectively, i.e., d (e,Γ 0 ) = d (e,Γ 1 ) = 1 and d (e,Γ 2 ) = dim(Γ 2 ) = 2. Now consider the two non-trivial conjugacy classes. Their centralizers are cyclic groups, so their irreducible representations are one-dimensional and labeled by group elements. The character tables read Evaluating Eq. (64) for the two conjugacy classes and the character functions shown above gives the projectors The dimensions of the associated irreducible representations are So in total we have found 8 different anyons and the corresponding central idempotents of the tube algebra. Next, let us consider a non-trivial 3-cocycle from Eq. (51). This yields the phase (see Eq. (58)), in the multiplication in T , which turns out to be a (twisted) coboundary [28] The cochain t A r a defines the projective characters in terms of the linear ones, see Eq. (63). With that, the central idempotents of the ω p -twisted S 3 theory are given by Eq. (64). The number and dimensions of the irreducible sub-spaces are the same as in the untwisted case but the phases in the central idempotents are modified by p g . This in turn changes their fusion and braiding (see Refs. [6,28,37]).

Boundary anyons: Semi-tube algebra
After having modeled point-defects (anyons) in the bulk, we continue with point-defects at the boundary. Even though these defects are not equipped with a braiding structure (they can only move along a boundary), we call them boundary anyons. However, they form a fusion category. A framework to calculate the fusion multiplicities thereof was introduced in Refs. [8,10] as "vertical fusion". In fact, the boundary anyons have to form a Morita equivalent category to the one that defines the bulk strings [29].

General fixed-point models
Boundary anyons are determined by string-nets on a "semi-tube", a tube that is cut in half by the physical boundary. One might also think of one half being replaced by vacuum. Note that for a domain wall between two phases, modeled by fusion categories C and C , the other half would not be vacuum but another bulk for the fusion category C . However, as illustrated in Section 3.2.1, these domain wall diagrams can be related to an equivalent boundary diagram of the form considered in this section. This semi-tube can be tessellated by a string diagram as follows, which defines an orthonormal basis of the defect space S. In the following, we will omit the multiplicity labels at the vertices. Again, we can use gluing operations and topological moves to define a multiplication on S. In Eq. 5 we show the sequence of local moves that defines the multiplication in terms of its associated space-time complex. Evaluating the complex gives rise to the multiplication Similar structures have been already introduced in various places in the literature, e.g., in Refs. [8,11,38,39] as "module tube algebra", "module annular algebra" or "ladder category".

Topological gauge theory models
In the case where the bulk is defined via Vec ω (G), the boundary model is given by a subgroup H ⊆ G (and a 2-cocycle in Z 2 (H, U (1))), the boundary labels are cosets in G/H and every vertex is multiplicity-free, see Figure 5: The multiplication of two basis elements in S is defined via gluing the two associated semi-tube string diagrams together (left) and using L (and F ) moves to reduce it to the cellulation on the right. The associated phase can be derived by considering the space-time complex that maps from the initial to the final cellulation.
over the vector space C G/H×G/H×G . By analogy to the tube algebra, we call the resulting algebra semi-tube algebra.
The anyons on the boundary correspond to the irreducible sub-spaces of S and we again want to find the associated indecomposable central idempotents in S.
Again, S is an algebra of the form discussed in Section 2 where X = G/H × G/H with G acting via simultaneous left-translation: g (aH, bH) = (gaH, gbH). Using the (twisted) 2-cocycle condition of ψ we can easily see that is a twisted 2-cocycle fulfilling in analogy to Eq. (3). Following Section 2, we first have to find the G-orbits of G/H ×2 . Let α −1 ∈ H\G be the left inverse of α, i.e., α −1 α = H. Then we find that the double coset x := α −1 β ∈ H\ G/H , is invariant under the G action. Moreover, all the subsets S x := {(α, β) ∈ G/H | α −1 β = x} are transitive as we will show now.
Next, we find the stabilizer subgroup for any element in S x . In particular, we can pick a representativê x = (α,β), explicitly calculate its stabilizer subgroup and then use Eq. (6) to derive the stabilizer groups of the other elements in S x . Let (α,β) = (aH, bH) ∈ S x . By construction, (H, a −1 bH) will also be in S x . The stabilizer of this element, is particularly easy to compute and given by This formula holds for any element in S x , replacing a −1 b with the respective double-coset representative of x. From these general considerations above, we can see that K x simplifies in some special cases: • For the trivial double coset x = H, K x = H.
This result already appeared in Ref. [16] in the derivation of boundary defects in untwisted Quantum Double Models. In the twisted case, boundary anyons are labeled by irreducible Ψ α,β -projective representations of K x instead of linear ones. In the following, if not stated otherwise, we label these representations with κ x .
Combined, the irreducible sub-spaces of S are labeled by a doble-coset x ∈ H\ G/H and an irreducible Ψα ,β -projective representation of K x . The associated central idempotents are given by whereχ (α,β) κx is the projective character of the IPR κ x of Stab G ((α, β)).

Abelian semi-tube algebra
In this section, we will consider models for which both the fusion of the bulk and the boundary anyons is Abelian. As we have seen, the bulk is then defined by a finite Abelian group G with N cyclic factors and a 3-cocycle cohomologous to a product of type-I and type-II cocycles. The boundary is defined by a subgroup H on which the bulk 3-cocycle is cohomologically trivial. Since G is an Abelian group any subgroup H is normal and there is a one-to-one correspondence of double cosets and cosets. In particular, α −1 β = x implies that we can rewrite β = αx and the sum over the two coset labels in Eq. (86) reduces to one. Moreover, K x = H for all x such that we can omit the x-label from the irrep label κ x . For the fusion of the boundary anyons to turn out Abelian, every irreducible subspace has to be onedimensional. Hence, Ψ α,β = Ψ α,αx has to be cohomologous to a trivial 2-cocycle and we can express the irreducible projective characters in terms of linear ones, Here, η is the cochain that trivializes the 2-cocycle Ψ α,αx . Linear characters χ κ of (finite) Abelian groups are well known, see Section 4.5. Combining the observations above, the central idempotents read We believe that any boundary models to a bulk that leads to Abelian anyons only result in an Abelian semi-tube algebra.

Examples
In this subsection we will sketch the construction for some exemplary cases. First, we note how the form reduces in the two extreme cases where the subgroup is one of the two trivial subgroups. Then, we will focus on small Abelian groups and as a simple example for an non-Abelian model we will give a full description of how to obtain the central idempotents in the case of G = S 3 . In particular, we will see how Eq. (85) helps significantly in constructing the stabilizer group whose irreducible representations labels parts of the boundary anyons.

Trivial Subgroups
Before we consider specific groups (and their subgroups), let us take a closer look on how Eq. (86) simplifies in the case of the two trivial subgroups {1 G } and G. In the former case, we denote the resulting rectangle algebra S. The central idempotents are labeled by x ∈ {1 G }\ G/{1 G } G alone. All the prefactors in Eq. (86) evaluate to 1 resulting in In the latter case, where H = G, we denote the resulting semi-tube algebra by agaom by S. In this case the central idempotents are labeled by irreducible (linear) representations of G since there is only one double coset, namely the trivial one. Moreover, since H = G only defines a boundary in the case of a trivial 3-cocycle in the bulk, S is isomorphic to an untwisted group algebra. With that, only the G-character functions χ κ enter into the prefactor in Eq. (86) and the central idempotents take the form Note that with H = {1 G }, the fusion category of boundary anyons is the same as the input fusion category Vec(G), whereas for H = G, they form the Morita equivalent category Rep(G).

Vec
where ⊕ p denotes addition modulo p. The full subgroup only defines a boundary for the trivial 3-cocycle n = 0. The central idempotents are then given as in Eq. (90). With that, the p 2 central idempotents are labeled by (x, κ) ∈ Z p × Z p and are given by

Vec ω (S 3 )
As an examplary case for non-Abelian models, consider G = S 3 . It has four conjugacy classes of subgroups, see for both p = 0 and 3. This shows that there are 6 inequivalent boundary anyons. In fact, when considering their fusion on the boundary, they form the fusion category Vec(S 3 ) which is the same as the input category for the bulk model and with that -as expected -Morita equivalent to it, compare Ref. [29].
Let us now consider the more subtle case of the non-normal subgroup H t Z 2 . It defines a boundary of a bulk model defined by the 3-cocycle ω p for p = 0, 2, 4 (see Eq. (51)). There are two double cosets x ∈ H t \ G/H t = {{e, t} = H t , {r, r 2 , tr, tr 2 } = H t rH t }. We obtain the stabilizer subgroup of both double cosets with Eq. (85).
For the trivial double coset it is given by H t itself whose irreducible representations are labeled by Z 2 = {0, 1}.
For the non-trivial double coset x = H t rH t we pick the representative r ∈ S 3 . Noting that rH t r −1 = tr only shares the identity element with H t , the stabilizer group for this double coset is the trivial subgroup {e}. Hence, we find only one boundary anyon for x = H t rH t . 6 Bulk-to-boundary fusion events After defining our model in the bulk of a spacial 2-manifold and on its boundary, we modeled point-like defects, namely anyons, both in the bulk and on the boundary. In this section, we will look at fusion events between bulk and boundary anyons, which in space-time can be interpreted as point defects on the boundary where a boundary anyon world-line meets a (bulk) anyon world-line. In space such a fusion event can be interpreted as the process of moving a bulk anyon to the boundary and turning it into a boundary anyon.

Bimodule construction in general fixed-point models
As we have seen in Section 4 and Section 5, the (boundary) anyons can be computed by considering stringnet diagrams on a (semi-)tube. Stacking a (semi-)tube to another one defines an action of the vector space of (semi-)tubes on that same vector space, which can be interpreted as an algebra, namely the (semi-)tube algebra. Since a bulk-to-boundary fusion event maps a bulk to a boundary anyon, it should be related to a 1-manifold which connects between tube segments and semi-tube segments. This can be achieved by an annulus where one boundary circle connects to tubes, and the other boundary circle half consists of the physical boundary and half connects to semi-tubes, A tube can be attached to the small gray circle in the center, and a semi-tube to the large gray half circle at the bottom. As shown, we choose a minimal string diagram to cover the 2-manifold. This 2-manifold defines a bimodule C with an action of the tube algebra T and an action of the semi-tube algebra S, such that the representations commute with each other. The vector space of C is spanned by labeled string diagrams of the above form. The T -action is defined by gluing a tube into the inner hole and using F moves to rearrange the string diagrams to the above form. Similarly, the S-action is defined by gluing a semi-tube segment to the exterior gray half circle and using F and L moves. The general actions are given by where we again omitted the multiplicity labels. In Fig. 6 we show the S-action as a space-time complex. The T -action is essentially given by the complex in Fig. 4 with two additional boundary labels at two of the vertices. As a bimodule, C is a equivalent to a representation of T ⊗ S, and thus decomposes into pairs (i, j) of irreducible representations i of T and j of S with multiplicities m ij , Figure 6: The S-action on C is defined by gluing a semi-tube element onto a C element from the top and using local moves to reduce the cellulation back to the reference one (95). These moves can be represented by the space-time complex shown above. It decomposes into three bulk tetrahedra and two boundary triangles.
where T i and S j are the irreducible representations of the tube, respectively rectangle, algebra. Physically, m ij the dimension of the vector space of the bulk-boundary fusion event "i fuses to j". Acting with the associated central idempotents of these irreducible representations from the left and right constructs a projector onto the respective sub-spaces of dimension m ij which we will denote with P C ij . The multiplicities are obtained by taking the trace of this projector,

Bulk-boundary fusion in gauge theory models
In gauge theory models the bulk strings are given by Vec ω (G) and the boundary strings by a Vec ω (G)-module category, classified by a subgroup H and a 2-cocyle class in H 2 (H, U (1)). The basis of C can by fully parameterized by two group and one coset element. We choose the following assignment: with g, h ∈ G and α ∈ G/H . The T action reduces to Similarly, the S-action is given by Given these two actions we can use the central idempotents derived in Sections 4.3 and 5.2 respectively their regular representation to project onto the associated irreducible representations of T and R within C. In combination, Eq. (98) defines a projector P C (c,ρc),(x,κx) ∈ End(C), projecting onto the pair of Irreps (c, ρ c ) and (x, κ x ) in C. With the T and S action above, the consecutive action of P T (c,ρc) and P S (x,κx) onto a basis element (g, h, α) C is given by where we have used Eq. (9) to relate the projective characters of different -but isomorphic -stabilizer groups. Taken together, where we have used that h ∈ Z(g) in every term of the sum above. The multiplicities m (c,ρc),(x,κx) can now be obtained using Eq. (98). Using the dimensions of the irreps, we obtain where we have used the twisted 2-cocycle condition for β g , Eq. (60), and Eq. (9) to simplify the trace to the expression in the last line. Eq. (105) describes the fusion events at the boundary of any topological twisted gauge theory model. The condensation formula in Ref. [12] can be seen as a special case where the bulk 3-cocycle is trivial and the boundary defect is set to the trivial one. Especially for non-Abelian models this formula gives insight into how non-Abelian bulk anyons split into boundary anyons when approaching the boundary.

Abelian Models
For Abelian models, the above expression simplifies significantly. First note, that each group element is its own conjugacy class and the double coset x corresponds to a unique left coset. Moreover, the projective representations are in one-to-one correspondence to the linear representations (see Section 4.4) which can be labeled by group elements. Similarly, the boundary anyon labels simplify (see Section 5.3). We get m (g,k),(x,κ) =δ g x, H 1 |G| where χ G k denotes an irreducible G-character and χ H κ an irreducible H-character.

Calculating Lagrangian algebras in gauge theory models
Macroscopically a (topological) boundary is defined via the set of anyons that can condense at the boundary, i.e., fuse to the trivial boundary charge. This set has to form consistency conditions involving their fusion and exchanges statistics which render the set of condensable anyons a Lagrangian algebra [13][14][15] in the UMTC describing the anyon theory. Our framework connects the microscopic description of the boundary as a lattice model to the macroscopic picture of anyon condensation. In particular, Eq. (105) allows us to explicitly calculate the Lagrangian algebra corresponding to a (H, ψ)-boundary by setting (x, κ x ) = (H, Γ 0 ), the trivial boundary defect. We get using that g as well as g are restricted to H. Hence, ψ α is a cocycle on H and does not depend on α, see App. C. Note that for a trivial 3-cocycle, this formula can be derived from Ref. [12]. In particular, we have shown that in twisted models the linear character gets replaced by a projective character.
In the Abelian case, it simplifies further to where χ G k denotes an irreducible G-character.

Vec ω (Z N )
As first example we consider the simplest class of Abelian models, when G = Z N , the cyclic group of order N .
Its character function reads χ k (g) = e 2πi N kg . The bulk can be twisted with a type-I 3-cocycle of the form of Eq. (47) giving rise to g (h) q = e 2πi N 2 qgh , q ∈ Z N .
For any q, the trivial subgroup H 0 = {0} defines a boundary. Plugging the defining data into Eq. (108) yields the associated Lagrangian subgroup i.e., the boundary condenses all the pure charges.
For an untwisted bulk, q = 0, the other trivial subgroup H 1 = Z N defines a boundary as well. Plugging in its defining data into Eq. (108) gives rise to the associated Lagrangian subgroup consisting of all the pure fluxes.
If N is not prime Z N has more subgroups. Depending on q, these might also define a boundary. For example, for N = 4n for n ∈ N and q = N/2 = 2n, the subgroup H 2 = N/2 Z 2 also defines a boundary, since ω N/2 I H2 ≡ 1. Using Eq. (108) yields the associated Lagrangian subgroup p with the character function χ (k1,k2) ((g 1 , g 2 )) = e 2πi p (k1g1+k2g2) and a bulk twist of type-II ω q from Eq. (48), giving rise to q (g1,g2) ((h 1 , h 2 )) = e 2πi p 2 g1h2 . A non-trivial subgroup defining a boundary is H 3 = (0, 1) on which ω q evaluates to 1. It only has a trivial 2-cocycle class, giving rise to the lagrangian subgroup independent of q.
We see that the non-trivial 2-cocycle on H 4 (m = 0) "couples" the flux with the charge that is condensed at the associated boundary. The explicit fusion multiplicities, also for non-trivial boundary boundary defects, can be found in Tab. 10.

Condensation domain walls for Abelian phases
It is known that any Abelian twisted quantum double of a finite group can be obtained from boson condensation in an untwisted quantum double of a group of larger cardinality [35,40]. The condensation process can be viewed as a non-invertible domain wall from the parent phase (an untwisted quantum double of an Abelian group) to the condensate (the twisted quantum double in question). In this section, we illustrate how such a domain wall is described on the microscopic level in two exemplary cases.

Condensation domain walls between type-I twisted and untwisted quantum doubles
The simplest examples of the above condensation is the double semion phase. On the one hand, it is realized by the twisted quantum double of Z 2 . On the other hand it can be described as the phase obtained when condensing e 2 m 2 in the Z 4 toric code, the (untwisted) quantum double of Z 4 . More generally, a Z N quantum double twisted by a type-I cocycle ω n I (see Eq. (47)) is equivalent to a condensate of a quantum double of N m N is condensed). Such a condensation process can be viewed as a non-invertible domain wall between the parent phase D (Vec(Z N 2 )) and the condensate D(Vec ω I (Z N )). In this section, we will see how these condensation domain walls can be obtained from our microscopic models. Interestingly, we see how the non-trivial type-I cocycle of Z N becomes trivial when considered as a cocycle of a larger subgroup of Z N × Z N 2 . Via the folding trick (see Section 3.2.1), a domain wall between an untwisted Z N 2 and a Z N quantum double twisted by a type-I cocycle ω n I (see Eq. (47)) corresponds to a boundary of a Vec ω n where ω n I is trivial on Z N 2 and of the form of Eq. (47) on Z N . In the following we show that the subgroup where we have used that addition of the integers in the exponent is modulo N 2 . This shows that ω n I is a coboundary on H I cond and with that, that this subgroup defines a valid boundary with L-symbols given by ψ n (see Eq. (41)). In Tab 1 we list the fusion multiplicities for N = 2 and n = 1 which align with what one would expect from the anyon condensation picture, a transition from the Z 4 toric code to the double-semion phase. In general, if H has non-trivial 2-cocycles, we can multiply the trivializing cochain with such a 2-cocycle to obtain a boundary of a different type.

Condensation domain walls between fully twisted and untwisted quantum doubles
As a second example we consider the "fully twisted" quantum double of Z N × Z N , i.e., we consider a 3-cocycle cohomologous to a product of type-I cocycles on both Z N factors, see Eq. (47), and the type-II cocycle on the pair, see Eq. (48), with n 1 = n 2 = n 12 =: n. For N = 2, it describes the six-semion phase where all elementary fluxes have semionic self-exchange statistics, see Ref. [35]. This twisted quantum double can also be reppresented as a non-trivial condensate of the untwisted quantum double of Z N 2 × Z N 2 .
Via the folding trick, the domain wall in question is equivalent to a boundary of a Vec ω (Z ×2 with ω being a product of type-I cocycles (see Eq. (47)) and a type-II cocycle (see Eq. (48)) on both Z N factors, ω n ((a 1 , a 2 , A 1 , A 2 ), (b 1 , b 2 , B 1 , B 2 ), (c 1 , c 2 , C 1 , The domain wall that models the condensation transition is associated to the subgroup H I,II cond = (1, 0, 1, 0), (0, 1, 0, 1) ⊂ Z ×2 N 2 × Z ×2 N . We parametrize it by two Z N 2 variables, H I,II cond = {(a 1 , a 2 , a 1 mod N, a 2 Table 1: Fusion multiplicities at the the domain wall implementing the condensation from a Z4 toric code to the double-semion phase. We see that e 2 m 2 is condensed since it fuses with the vacuum charge in the double-semion phase. Moreover, for a fixed double-semion anyon, the two toric-code anyons with which it has non-zero fusion multiplicity differ by e 2 m 2 . An all-zero row shows that the associated toric-code anyon is confined to one side of the domain wall, i.e., there is no valid fusion event with any of the double-semion anyons.  and reverting the fold, we recover that the associated domain wall indeed implements the condensation from one phase to the other, compare Ref. [35]. In Tab. 2 we printed a snippet of the condensation table for the domain wall implementing the transition from two copies of the Z 4 toric code to the six-semion phase. In particular, we find that the correct toric code anyons either 1) condense, i.e., fuse to 1 in the double-semion phase, 2) confine, i.e., do not fuse to any anyon in the double-semion phase and 3) tunnel to the expected anyon in the double-semion phase.

Non-Abelian islands in Abelian phases
Topological stabilizer-based quantum error correction (QEC) is believed to be a promising framework to protect a logical qubit against (local) noise. Any Abelian phase can be used to construct a topological stabilizer code [34,35]. In such codes, the logical Pauli operators can be associated with certain (non-trivial) loops of anyon ribbon operators. However, one cannot topologically protect a universal gate set with an Abelian phase alone [41,42]. One approach to achieve a topologically protected universal gate set is to go beyond stabilizer-based topological QEC and use non-Abelian phases. However, even though first analysis show that -in principlenon-Abelian QEC is possible [43,44], it does not appear to be the best approach to simply store a qubit and perform simple operations, like Clifford gates. In Ref. [36] Laubscher et al. have introduced the concept of non-Abelian islands within an Abelian phase. The authors construct a protocol that allows to teleport a qubit encoded within punctures of an untwisted Z 2 model into a puncture-encoded qubit within an untwisted S 3 model (and vice versa). This allows to perform a non-Clifford gate within the S 3 phase that can be teleported back into the Z 2 phase. In this section, we show that the same method can be used to interface twisted models on a microscopic level by constructing the associated tunneling tables from Eq. (107). cond and trivial 2-cocycle on it (top) and m = 1, 2, 3 cocycle on it (from top to bottom) and m = 3, see Eq. (49) with p = N 2 . m = 2 seems to have the right condensation pattern of e 2 2 m 2 2 , m 2 1 e 2 1 e 2 2 → 1 and two inequivalent semions m2e1e2 and m1e1 get mapped to valid generators of the 6-semion model. In the following, consider puncture-encoded qubits within a topological phase, as in Ref. [36]. To be able to teleport logical information from one phase to the other, one has to interface the two codes with a suitable domain wall. It has to tunnel exactly the right anyons whose ribbon operators span the logical Pauli group. To connect the macroscopic physics described by the tunneling of anyons with a microscopic model, Eq. (105) is of essence. Starting from a microscopic description in terms of a subgroup and a 2-cocycle on it, it allows to check which domain walls can be used to teleport logical information from side of the domain wall to the other. In this section, we show how twisted non-Abelian models can be interfaced with twisted Abelian models to teleport a qubit, respectively qudit. In particular, we use Eq. (107) to construct tunneling tables that show which anyons can be transported though a given domain wall with local operations. We illustrate the concept via two examples, namely how different twisted S 3 models can be interfaced with a twisted Z 2 (double-semion) model and a twisted Z 3 model. After that we comment on how to find suitable microscopic models for more general phases that allow for teleportation of logical information from an Abelian to a non-Abelian phase.
Imagine we want to interface a non-Abelian model, for example Vec ω (S 3 ), with an Abelian one, for example Vec ω (Z N ). Any domain wall between such two models is equivalent to a boundary of a stacked model, for example Vec ωω (S 3 × Z N ), see Section 3.2.1. In the following, we will illustrate how this folding trick is used to construct the tunneling tables in two examples, where Vec ω (S 3 ) is interfaced with Vec ω (Z N ) for N = 2, 3. In particular, the 3-cocycles ω on S 3 have to be in different cohomology classes to tunnel anyons non-trivially through the domain wall. Note that the generator of H is supported on S 3 as well as on Z 2 which in turn makes the domain wall (partly) transparent.
Similarly, one can interface an S 3 and a Z 3 model non-trivially. In this example, we consider a ω 4 -twisted S 3 model (see Eq. (51)) and a type-I twisted Z 3 model. After folding, the boundary in question is a boundary of a S 3 × Z 3 model twisted by a 3-cocycle of the form ω ((A, a, x), (B, b, y), (C, c, z)) = e
the results via unfolding, we obtain the tunneling table in Tab. 4. Comparing the two tunneling tables Tab. 3 and 4, we see that -in a wayr and t have changed roles.

Conclusion and outlook
In this manuscript, we have explored bulk-to-boundary anyon fusion events in non-chiral topologically ordered quantum models, aimed at understanding how anyonic defects interact with external defects such as boundaries or domain walls. It has been motivated by considerations both in the study of quantum phases of matter and quantum information theory. Specifically, we have calculated bulk-to-boundary fusion multiplicities in topological fixed-point models in 2+1 space-time dimensions. Our framework allows for a step-by-step calculation of the fusion multiplicities in all such models. Apart from the calculation of projective irreducible representations of the G-subgroups which define the action in the tube and semi-tube algebras, the calculation only involves the evaluation of linear expressions. The fusion multiplicities allow to characterize the behavior of anyonic bulk excitations when approaching a boundary and -via the folding trick -to calculate the effect an anyon has on a domain wall when moved through it. At the core of our construction lies a bimodule that is a representation of the tube algebra defining the bulk anyons as well as the semi-tube algebra defining the boundary anyons. We have defined this bimodule for any fixed-point model and explicitly derived a closed formula for the fusion multiplicities in the subclass of topological lattice gauge theories, where the fusion category defining the bulk is given by Vec ω (G). We have used this formula to calculate Lagrangian algebras in various gauge theory models without the need to explicitly solve the consistency conditions defining a Lagrangian algebra. This is particularly useful for more involved non-Abelian topological phases where finding Lagrangian algebras is more intricate. We showcase this in the derivation of the Lagrangian algebras for Vec ω (S 3 ) models with non-trivial 3-cocycles. Moreover, using the folding trick we can use bulk-to-boundary fusion events to study the tunneling of anyons through domain walls. In particular, our formula allows to keep track of which anyon is left behind at a domain wall when moving a bulk anyon from one side to the other. As a proof of principle, we have calculated the fusion multiplicities of a special class of non-invertible domain walls which implement anyon condensation from Abelian untwisted to twisted quantum doubles. This is a well-known transition on the level of the anyon models [35,40] but as far as we know we are the first to give a microscopic description of the corresponding domain walls in space-time. Moreover, we have shown how to interface Abelian twisted quantum double phases with non-Abelian ones on a microscopic level.
In our construction, we have observed that both bulk and boundary anyons are characterized by a special type of algebra which we call twisted group algebra with action. We give a step-by-step recipe to derive the irreducible representations of such algebras from simpler projective representations of the subgroups stabilizing the action. In fact, any sort of line-like defects in space-time in a Vec ω (G) state-sum model will be classified by irreducible representations of a certain group algebra with action. Hence, our techniques can be used to study these defects and their interaction with membrane-like defects in 3+1-dimensional models. Generalizing this recipe to groupoid-like algebras (see Ref. [31]) and thereby extending the methodology beyond twisted quantum double models might be an interesting avenue for further research.
Together with previous work [8][9][10][11]45] the fusion multiplicities calculated in this paper contributes to the algebraic description of the anyons in topological models with boundaries and domain walls. An understanding of similar depth of defects in higher-dimensional models is lacking. Further research directions can include the application of the techniques used in this paper to line-defects in 3+1-dimensional models and extending the techniques to further understand the interaction of defects of different (co-)dimensions in higher-dimensional models.
Again, our work is not only interesting from the perspective of the mathematically minded study of topological phases of matter but it has also important applications in different areas of physics. Above all, any practical topological quantum error correction (QEC) scheme involves boundaries or domain walls in one way of the other. On the one hand, stabilizer-based topological QEC can be understood as storing a qudit in the ground space of an Abelian topological phase modeled by a Vec ω (G) fixed-point model [34,35].
On the other hand, going beyond stabilizer-based approaches allows to natively perform universal topological quantum computation [43]. Given the overhead in resources of protocols that uplift non-universal stabilizerbased approaches of quantum computing to universal ones by means of magic-state distillation [4,46], such an avenue may well have its benefits. In both cases a thorough understanding of the interaction of anyons with other types of defects is important. For example, finding the logical operators in a given planar code including boundaries and domain walls reduces to characterizing which anyons can condense at which boundary. Our work shows how to calculate these quantities in the most general case, in particular we extend the results of Ref. [12] to twisted quantum doubles. The bulk-to-boundary fusion multiplicities can also be used to study computational protocols including boundaries and domain walls [18,36,47]. This includes lattice surgery [20,48] schemes in Abelian topological codes and allows for systematic study of the computational possibilities of a given code via anyon condensation [18]. Moreover, domain walls between non-Abelian phases und Abelian ones can be used to generalize the scheme presented in Ref. [36] where a partly-transparent domain wall between a Vec(S 3 ) and a Vec(Z 2 ) phase is used to teleport a topologically encoded qubit from one model to the other. Starting there, it will be interesting to investigate universal computing schemes based on twisted quantum doubles, particularly when combined with a Pauli-based description of the Abelian phase from Ref. [35].
Lastly, to fully understand the computational capabilities of non-Abelian quantum error correction [43,44] we want to investigate domain walls between non-Abelian phases to see how external defects can extend non-Abelian codes. This is again partially motivated by the quest to find schemes for quantum computing without the need for magic state distillation. These few examples should illustrate the wide applicability of bulk-toboundary fusion events, especially in topological QEC. We hope our work sparks inspiration to develop new QEC and computing protocols based on more exotic topological phases. On a higher level, this work is aimed at contributing to building new interfaces between quantum information theory and mathematical condensed matter physics which seems a mutually inspiring intersection.

A Examples of fusion multiplicities
In this appendix, we summarize the fusion multiplicities at boundaries of an untwisted Vec(Z 2 × Z 2 ) and Vec ω (S 3 ) models with different 3-cocycles.
A.1 Vec(Z 2 × Z 2 ) For the Z 2 ×Z 2 model, we first show the table of fusion multiplicities at the trivial boundary defined by the order by the subgroups (0, 1) and (1, 1) . The only subgroup with non-trivial 2-cocycles is H = G = Z 2 × Z 2 . It has two 2-cocycle classes and hence two boundaries associated to it. The corresponding fusion multiplicities can be found in Tab. 8. All these fusion events can either be seen as fusions into the boundary of a Z 2 × Z 2 model (which is in the phase as the topological color code [49]) or -equivalently -as domain wall tunneling events between two Vec(Z 2 ) models (toric code phase).     The bulk-to-boundary fusion multiplicities in a model where the bulk is defined by G = Z2 × Z2 and a trivial 3-cocycle and the boundary by the non-trivial subgroup H = (1, 1) . The bulk anyons (rows) are labeled by G ×2 and the boundary anyons (columns) by G/H × G. The coset label marked with an overline.

A.2 Vec ω (S 3 )
Depending on which 3-cocycles we use in Vec ω (S 3 ) different subgroups define topological boundaries. In Tab. 9 we give the fusion multiplicities for the cases when H r defines a boundary. In Tab. 10 we contrast it with the fusion multiplicities for the cases where H t defines a boundary.  9: All the fusion multiplicities for a model where the bulk is defined by G = S3 and the boundary by the non-trivial subgroup Hr = r (see Section 3.5.3). The bulk anyons (rows) are labeled as in Section 4.

and the boundary anyons by
Hr\G/Hr × Z3. We show the fusion multiplicites for the 3-cocycles ω p (see Eq. (51)) with p = 0, 3. Where they differ we give the value for p = 3 in brackets. These are all models where Hr defines a valid boundary.

B Cohomology of finite groups
In this section, we provide an algebraic definition of cohomology theory for finite groups. For a more detailed background, see, for example, Refs. [22,23].

Definition 1 ((left) G-module).
Consider a finite group (G, ·). An Abelian group (M, * ) together with a (left) group action : G × M → M, (g, a) → g a that ∀ a, b ∈ M , g, h ∈ G fulfills • g (a * b) = (g a) * (g b) and • (g · h) a = g (h a), is called a left G-module. Analogously, a right G-module is defined with a G-action from the right which we denote by . An Abelian group M equippped with both a G-action from the left and a G -action from the right is called G-G -bimodule.
Remark. Any Abelian group can be made into a G-module with a trivial group action by defining g a = a ∀a ∈ M, g ∈ G.
Definition 2 (n-cochain). Let (M, * ) be a G-module. A map η n : G ×n → M is called n-cochain (of G over M ). We denote the space of all such n-cochains by C n (G, M ). In fact, C n (G, M ) is a group with the group multiplication inherited from M .
is called n-coboundary operator.
Most importantly, for any n and η ∈ C n (G, M ), (δ n+1 • δ n )(η) = 1 C n (G,M ) , where 1 C n (G,M ) denotes the trivial (constant) map from G ×n to M . We say that the coboundary of a coboundary is trivial. Often, we will shortly write δ = n δ n which can act on any (combination of) cochain(s).
Remark. In this work we encounter two sorts of G-modules. First, U (1) with trivial action, and second, U (1) A for a finite set A whose action is determined by a permutation action of G on A. We will distinguish the latter twisted coboundary operator by explicitly writingδ.
is called nth cohomology group of G over M .
Remark. If two n-cocycles are in the same equivalence class in H n (G, M ) we call them cohomologous.
In any n-cocycle class we can find a representative ω ∈ Z n (G, M ) that fulfills We call such a cocycle normalized. η n (g 1 , . . . , g k , g −1 k · · · g −1 1 xg 1 · · · g k , g k+1 , . . . , g n−1 ) is called slant product. Again, we write i x = n i n x to abbreviate the collection of all slant products.
Remark. i n g induces a homomorphism on the cohomology groups H n (G, M ) → H n−1 (G, M G ) where M G is a G-module whose action is a combination of the action on M plus conjugation of the G-label. Indeed, one can check that (δ n−1 • i n ) = (i n+1 • δ n ). This explains why β g in Section 4 is a twisted 2-cocycle.
In the following, we will give some examples for cohomology groups of Abelian and non-Abelian groups over U (1) as a G-module with trivial action.
where r i is the chosen representative for coset (g i g i−1 · · · g 0 ) α and r the representative of α. The converse, δρ = δ ρ follows simply from the fact that the G-action on U (1) G/H becomes trivial when every argument of the cochain is restricted to H.
To complete the proof, we have to show that• and• induced by m (n) and m (n) are inverses of each other up to coboundaries and thereby preserve the cohomology class. One direction is easy to see from To show the converse, we explicitly construct a n − 1-cochain Ω : . . , g 1 , α))) = (δΩ) α (g n , . . . , g 1 ), namely with r 0 being the representative for α and r i for (g n−1 · · · g 1 ) α. This can be seen explicitly by rewriting ψ α (g n , . . . , g 1 ) with (δψ) α (g n , g n−1 , . . . , g 1 , r 0 ) = 1 in terms of ψ H . This isomorphism is of particular importance for our work. It not only defines the L-symbols of boundaries in Section 3.3 but also enters in the algebra diagonalization in Section 2. Moreover, Eq. (145) shows that the twisted 2-cocycle defining the semi-tube algebra Ψ α,αx is cohomologous for all α when restricted to H. This is implicitly used in the central idempotents since we sum over α. In App. E, we give a geometric interpretation of this isomoprhism as an invertible domain wall on a boundary state-sum.

D State-sum picture
In this appendix we give a concise and systematic rederivation of what is discussed in the main text, using the language of state-sum models in space-time. We will make use of the notions of extended manifolds/cellulations as defined in Appendix B of Ref. [50]. Roughly, an extended manifold is a composite of manifolds-withboundary called regions of different dimension attached to each other in different ways. The link of a region is the intersection of a small-enough -sphere around a point within the space normal to that region, and has to be the same for all points of that region. For example, if we have a 1-dimensional region embedded into a 3-dimensional one, the normal space at a point is a plane, so the link of the 1-dimensional region is a circle. To get an extended cellulation, we triangulate the Cartesian product of each region with its link. This triangulation is identified with the boundary triangulation of higher-dimensional regions.
We now associate state-sum variables and weights to different cells of the different regions, and demand their invariance under topology-preserving moves such as Pachner moves. The highest-dimensional region then defines the bulk of a state-sum model, and the lower-dimensional regions define topological boundaries, anyon world-lines, domain walls, and other sorts of defects. In the following, we will present the twisted quantum double model and its boundaries, anyons, etc. in this language.

D.1 Bulk
The bulk of the twisted quantum double model is a state-sum path integral on 3-dimensional space-time triangulations with a branching structure consisting of tetrahedra, At every edge of the triangulation there is a state-sum variable (such as a, b, and c above) taking values in the group G. On every triangle, the group labels a, b, and c have to satisfy the constraint At every tetrahedron with labels as in Eq. (147), we have a weight ω (a, b, c) .
Note that the three labels a, b, and c determine all other labels of the tetrahedron via the triangle constraint.
If the tetrahedron has the opposite orientation from Eq. (147), we instead associate the complex conjugate ω. This is how unitarity/Hermiticity is implemented in the state-sum language. Retriangulation invariance is then imposed via Pachner moves as shown in Eq. (23) in the main text. This makes ω a group 3-cocycle representing an element of H 3 (G, U (1)).

D.2 Boundary
Next we include boundaries. The normal space to a point in the boundary is a half-line, and an -sphere restricted to that half line is a single point. So the link of the boundary is a point, and an extended cellulation is essentially just a 3-dimensional triangulation with boundary. However, it better to "thicken" the boundary triangles into 3-cells which look like triangle prisms, The bottom triangle formed by the three edges labeled a, b, c is attached to a bulk tetrahedron, the three rectangles on the sides are attached to other boundary 3-cells, and the top triangle formed by the three dotted edges corresponds to the actual boundary. By replacing boundary triangles with boundary 3-cells, we can also make sense of two boundary triangles separated by an "infinitely thin" bulk as for example later in Eq. (173). At every boundary vertex (or more precisely, every of the short thick blue lines in Eq. (151)), there is a state-sum variable taking values in some finite set A, which is equipped with a right action of G, At each "thickened" boundary edge, such as at the three sides of the boundary triangle in Eq. (151), the labels fulfil the constraint To each triangle as in Eq. (151), we associate a weight noting that the labels c, β, and γ are determined from α, a, and b via the constraints. If the triangle has opposite orientation from Eq. (151), we associate the complex conjugate instead, just as for the bulk tetrahedra. Note that this Hermiticity condition holds generally for all weights/cells in a state-sum, and we will in the following always assume it without explicitly saying so. Topological invariance can be imposed by a move attaching/removing a tetrahedron to the boundary, The same move was depicted slightly differently in Eq. (29) in the main text. On the left side, there are two boundary triangles only, drawn with the bulk at the back such that the dashed blue lines are in front of the black lines with arrows. On the right hand side, we have two boundary triangles, and a bulk tetrahedron attached to both at the back. The boundary triangles on both sides alone are related by a 2-2 Pachner move. So ψ has to fulfil ψ α (a, b)ψ α (ab, c) = ω(a, b, c)ψ α (a, bc)ψ α a (b, c) , which is the same (apart from slightly different conventions) as in Eq. (41) in the main text.

D.3 Anyons
Anyons in the state-sum can be represented by considering triangulations of 3-manifolds with an embedded 1-manifold representing the anyon world-line. The link of an embedded 1-manifold is a circle, which can be triangulated with a single "looping" edge whose end vertices coincide, .
Since the 1-manifold itself is triangulated with edges, we need to trianglate an edge times the looping edge. We pick a triangulation of this "tube segment" with two triangles, Even though such a tube segment is formally something two-dimensional, it is good to think of it as a cylinderlike 3-cell, with a red shaded 1-gon at the top and bottom. An extended cellulation is a 3-manifold triangulation into which we embed sequences of tube segments as above. The two red shaded 1-gons attach to other tube segments, whereas the two triangles wrapping round the side attach to bulk tetrahedra. Those triangles may also be directly attached to the triangles of other tube segments with an infinitely thin bulk in between.
The state-sum associates one additional variable to each red shaded 1-gon where two tube segments meet, The dimension of this variable µ may to depend on the group label a at the corresponding loop edge. To each tube segment with labels as in Eq. (159), we associate a weight The topological invariance is again imposed by combinatorial moves. This time, we have two tube segments stacked on one side, and one tube segment on the other side. The one tube segment needs to be surrounded by bulk tetrahedra such that the two sides become compatible. In order to concisely state the axioms giving rise to topological invariance, and to classify the different anyons, it is useful to apply a dimensional reduction, more specifically a compactification. To this end, consider a mapping from two-dimensional triangulations to 3-dimensional ones by taking the cartesian product with the circle. Pulling back the 3-dimensional state-sum along this mapping we obtain a two-dimensional state-sum. On a combinatorial level, we take the cartesian product of a two-dimensional triangulation with the triangulation of the circle link, i.e., the looping edge in Eq. (158). Every triangle then becomes a "triangle prism" with top and bottom identified, We need to choose a simple 3-dimensional triangulation for this triangle prism which can be done with 3 tetrahedra as shown. We then redistribute the (independent) 3-dimensional state-sum variables onto the twodimensional triangulation as shown. This way, we get a two-dimensional state-sum with one G-variable at each edge and one at each vertex. At each edge, we have a constraint y = c −1 xc. The state-sum weight at each triangle is −1 xa, b)ω(x, a, b) .
One of the tetrahedra above has a different orientation as in Eq. (147), thus the according weight ω is complex conjugated. Up to differences in conventions, this is Eq. (58) from the main text. By construction, the topological invariance of the two-dimensional state-sum follows from the topological invariance of the 3-dimensional statesum. The resulting state-sum is a thick state-sum as discussed in Appendix E, whose vertex label set is G, with G-action x g = g −1 xg .
Accordingly, β is a G 2-cocycle with module U (1) G , such that g ∈ G acts on φ ∈ U (1) G by g φ(x) = φ(x g −1 ). Next, we extend the above mapping to a mapping from 2-manifolds with boundary to 3-manifolds with anyon world-lines. We do this by mapping the 1-dimensional boundary to a 1-dimensional anyon world-line. Combinatorially, we map a "thickened" boundary edge to an anyon tube segment. Pulling back the anyon state-sum along this mapping yields a two-dimensional boundary state-sum, Now, the boundary retriangulation invariance of the compactified boundary, is equivalent to the retriangulation invariance of the anyon world-lines. Thus, we see that line-like defects are in one-to-one correspondence with the boundaries of the compactified two-dimensional state-sum. So the weights (ρ a (c)) ν µ are determined by the equation Note that there is a finite number of irreducible ρ, which are what is commonly understood by anyons. To see this we notice that a two-dimensional state-sum with variables only on the edges is an associative (in fact, †-Frobenius-) algebra, and is a direct sum of full matrix algebras. We can copy the vertex variables of our state-sum onto each of the adjacent edges, such that at every edge we have a triple (x, a, y). The triples have to satisfy y = x a and are thus fully specified by (x, a). The resulting algebra equals the tube algebra discussed in the literature and in the main text in Section 4. Its irreducible matrix blocks, or the according irreducible representations, are the anyons.

D.4 Boundary anyons
To describe anyons within the boundary we need to define the state-sum on triangulations of manifolds with a 1-manifold embedded into the boundary. The normal space to the 1-manifold is a half-plane, and an -sphere within this half-plane is an interval. Thus the link is an interval, and it can be triangulated with only two (thickened) boundary vertices separated by an infinitely thin bulk, .
The tick on the right short blue line is in order to remove the symmetries of the link. Next we take the product of that link with a single edge, It is again helpful to think of this as a 3-cell which looks like a little triangle wedge, by imagining another rectangle face between the two dotted lines at the back, such that there is a red shaded triangle on the bottom and top. Then an extended cellulation contains sequences of such "semi-tube segments". The red shaded triangles are attached to other semi-tube segments, whereas the two rectangular faces on the left and right are attached to the sides of nearby boundary triangles. At every red shaded triangle where two semi-tube segments meet, canonical filling would be given by a cone with a singularity in the center. A combinatorial extended manifold is thus a 3-manifold with some torus 3-cells. At every torus 3-cell there is a weight We want the 0-dimensional defects defined by the torus 3-cell to be topological. Roughly, this means that a layer of bulk padding the torus 3-cell can be absorbed into it. To concisely impose this topological invariance, we again use compactification. We pull back the state-sum along a mapping from 1-manifolds to 3-manifolds. Combinatorially, an edge is mapped to an edge times the torus link, where front/back and left/right are identified. We thus get a 1-dimensional state-sum with one G-variable at every edge and two G-variables at every vertex, subject to the constraints d = x −1 bx and c = x −1 ax at every edge with labels as above. The weight associated to every edge is which can be obtain from a triangulation of Eq. (180) with 6 tetrahedra which is not shown. The topological invariance of this state-sum follows by construction from the topological invariance of the original state-sum. This is a thick 1-dimensional state-sum with vertex labels in G × G and right action Accordingly, P is a G 1-cocycle with module U (1) G×G , and left action determined by the above. We extend the mapping to a mapping from 1-manifolds with boundary to 3-manifolds with torus-link point defects, by mapping a boundary point to a torus 3-cell, such that topological invariance ab cd x x = ab (184) of the 1-dimensional state-sum boundary is equivalent to that of the torus-link point defects. Spelled out in letters, both are given by which can be rephrased as dS = P , if we interpret S as a twisted 0-cochain. This equation is linear in S, thus the ground states form a vector space, which is a feature of all 0-dimensional defects. 1-dimensional state-sums with variables only on the vertices are given by projectors, and our state-sum can be brought into such a form by simply summing over x in P a,b (x). Ground states S a,b are then vectors in the support of this projector.

D.6 Bulk fusion events
The link of a 0-dimensional defect might itself have defects. For example, take as link a 2-sphere with three embedded points, .
Such defects correspond to points in space-time where three anyon world-lines meet. The points in the link are just the intersections of a sphere around the 0-dimensional defect with those three anyon world-lines. At the same time, those point-like defects are ground states on a sphere with three anyons 3 . Here we will look at the fusion vertices between three different anyon world-lines which weights ρ 0 , ρ 1 and ρ 2 , in which case it would be natural to color the points differently. This sphere link can be triangulated by one triangle (formed on the "outside" enclosed by all three edges in the following picture) with three red shaded 1-gons, Those three 1-gons are attached to three tube segments corresponding to the anyon world-lines, and the outside triangle is attached to a bulk tetrahedron. As usual we may think of this as a "junction" 3-cell. So extended cellulations are cellulations including sequences of tube segments (of three different colors) and junction 3-cells where three tube segments meet. To each junction 3-cell, we associate a weight As usual, we consider the 1-dimensional state-sum arising from compactification with the link above. Combinatorially, we take the triangulation in Eq. (187) times an edge, and choose a 3-dimensional triangulation for the resulting volume, On the right-hand side, the three edges labeled x are identified, such that the edges a, b, c, d, e, and f form loops, which are the start and end loops of three tube segments as in Eq. (159). The labels α, β, γ, δ, , and φ are the labels at the bottom and top of the three tube segments, whose red 1-gons have not been drawn.. The resulting 1-dimensional state sum has one G-label at every edge, and two G-labels and three X-labels at every vertex, subject to the constraints d = x −1 ax, e = x −1 bx at every edge. The weight associated to each edge is coming from the cellulation using three tetrahedra and three tube segments. The resulting 1-dimensional statesum has vertex labels in G × G with action as in Eq. (182), as well as three free vertex labels. The fusion events are in one-to-one correspondence with boundaries of this 1-dimensional state-sum. The topological invariance is thus determined by As in the previous section, summing over x yields a projector, and S are vectors in the support of this projector. The dimension of this support vector space is the ground space dimension on the sphere with three anyons, or equivalently the fusion multiplicity N ρ0ρ1 ρ2 . It can be calculated by taking the trace of the projector, or another compactification to a 0-dimensional state-sum via the cartesian product with the circle in Eq. (158). Plugging Eq. (189) into this compactification means summing and identifying a and d, b and e, c and f , α and δ, β and , γ and φ, yielding, a, b, x)ω(a, x, b)ω(x, a, b) Tr(ρ a 0 (x)) Tr(ρ b 1 (x)) Tr(ρ ab 2 (x)) . (192)

D.7 Bulk-boundary fusion events
Next, let us consider fusion events at the boundary, between a bulk anyon and a boundary anyon. Those are 0-dimensional defects with link .

(193)
A simple triangulation of this link consists of only one (thickened) boundary edge, an anyon 1-gon and a boundary-anyon triangle, Combinatorial extended manifolds are 3-manifolds with bulk tube segments and boundary wedge segments, meeting at "junction" 3-cells like above. At every such junction, we have a weight noting that α is determined by α = α a.
As usual, we pull back the 3-dimensional state-sum along the cartesian product with the link in Eq. (193). Concretely, an edge of a 1-dimensional triangulation is mapped to its cartesian product with Eq.
The right hand side consists of one tube segment, one semi-tube segment, as well as one thickened boundary 4-gon, which can be triangulated with two boundary triangles. The resulting 1-dimensional state-sum has one G-label at each edge, and one G-label, one A-label, one anyon-space label, and one boundary-anyon-space label at each vertex. Those labels are subject to the constraints β = α c and b = c −1 ac. For an anyon ρ and boundary anyon κ, the weight associated to an edge is

D.8 Bulk anyon F -symbol
The F -symbol of the resulting anyon theory does not correspond to a defect like in all the previous sections, but simply to the evaluation of the state-sum on a specific extended cellulation. The corresponding extended manifold is a 3-sphere with 6 anyon world-lines meeting at 4 3-valent fusion point defects, forming the 1-skeleton of an embedded tetrahedron. Now, the F -symbol is not a single state-sum evaluation, but the collection of such evaluations for all different anyons and fusion point defects. For the anyons, we choose one irreducible representation of the tube algebra from each such isomorphism class. For the fusion point defects, we use an isomorphism from some abstract fusion vector space into the vector space of all point defects. I.e., instead of a single S a,b αβγ as in Eq. (188) fulfilling Eq. (191), we consider More precisely, we can choose S[ρ a , ρ b , ρ c ] to be an isometry (and we should do so in practice), such that the dimension of χ is the rank of P , namely (203) Above we only drew the bulk tetrahedron. All four vertices are identified such that all 6 edges form loops. Each of the triangles is the triangle in one of the 4 junction 3-cells shown in Eq. (187). With this, the evaluation is given by F

E Thin versus thick boundaries
In this appendix, we describe the space-time picture for a different microscopic way of defining a boundary. We will call this the thin boundary state-sum as opposed to the thick boundary described in the main text and in Appendix D. We will first discuss the thin boundary in 1 + 1 dimensions and then in 2 + 1 dimensions, such that the generalization to arbitrary space-time dimensions will be straight-forward.

E.1 1+1 dimensions
Recall that a two-dimensional group-cohomology state-sum is defined on two-dimensional branching-structure triangulations with one group label at every edge, and one constraint ab = c and one weight ω(a, b) at every triangle as in Eq. (148). The thin-boundary state-sum is defined for a subgroup H ⊂ G and a H 1-cochain ψ such that dψ = ω| H , or more explicitly Let us now discuss the relation between the thick and thin boundary. Note that in Appendix D, a thick boundary was defined with values in an arbitrary right G-set A at the vertices. In 1 + 1 dimensions, there is a constraint β = α g and a weight ψ α (a) at every thickened edge as in Eq. (153). Every set with right G action is isomorphic to a disjoint union of left coset sets H\G for different H. Each H is determined up to conjugation, and the right action on H\G is transitive and given by multiplication with the G-element from the right. Note that physically, boundaries with a transitive action are those which are irreducible, that is, robust to perturbations.
An irreducible thick boundaryψ (i.e., one with transitive action) can be obtained from a thin boundary ψ with the same subgroup H as follows. We start by padding each thin boundary edge with little bulk rectangle, and flatten those rectangles to get a generalized thick boundary, which has labels in G instead of H\G at the boundary vertices. It is easy to see that the moves for the thick boundary such as Eq. (167) follow from the thin boundary moves such as in Eq. (207) and Eq. (208), after we plug in Eq. (226). We say "generalized" because the label y is not determined by x and a via some action, but only constrained through h := xay −1 ∈ H.
In order to obtain a proper thick boundaryψ, we realize that the state-sum on the right of Eq. (209) has a gauge symmetry acting on the edges adjacent to a boundary vertex, for all γ ∈ H, where the equality is for the weights at the surrounding triangles and boundary edges. On the left of Eq. (209), the same gauge symmetry is only acting on a single vertex label, Thus, we remove the gauge freedom by replacing the vertex labels x, y ∈ G by left cosets in α, β ∈ H\G. The constraint β = α a of the thick boundary then directly follows from xa = hy and h ∈ H on the right. The weightψ after gauge fixing can be obtained from the weight before gauge fixing by choosing a standard representative R(α) ∈ G for each coset α, and using x = R(α), y = R(β) = R(α a) .
The weight can then be read off from the right-hand side of Eq. (209), ψ α (a) := ω(x, a)ω(xay −1 , y)ψ(xay −1 ) , using the shortcut in Eq. (212). Vice versa, a thin boundaryψ can be obtained from an irreducible thick boundary ψ by realizing that the thick boundary has the following gauge symmetry around a boundary vertex: Change every ingoing adjacent edge by a → aγ, every outgoing adjacent edge by a → γ −1 a, and the coset label at the vertex itself by α → α γ, for any γ ∈ G. Since every such gauge symmetry only affects a single boundary vertex, we can use it to fix all the coset labels to the trivial coset H. Soψ is obtained from ψ by simply setting the coset label to H, which automatically restricts the group label to the subgroup H, for a ∈ H. It is easy to see that if we first transform a thin boundary into a thick one, and go back to a thin one, we end up with the same thin boundary again,ψ α (g) = ψ α (g) .
However, going from a thick to a thin and then back to a thick boundary does not yield the same boundary, but one that is in the same phase/cohomology class. In order to see this, we consider the following domain wall η between the thick boundaries ψ andψ, That is, ψ andψ are twisted 1-cochains which differ by a twisted 1-coboundary dη.
Again, we restrict the group labels at the boundary edges to H, and associate to every boundary triangle, This is because all involved edges labels are constrained to H and so this is just Eq. (220). However, the thick boundary is invariant under additional moves, such as = . (224) This move consists of one boundary triangle and one bulk tetrahedron on the left, and three boundary triangles only on the right. This move does not hold for the thin boundary, since the three edges adjacent to the central vertex are constrained to H on the right, but can take values in all of G on the left. This can be fixed by padding the right-hand side with three bulk tetrahedra. Another drawback of the thin boundary is that it is not directly compatible with the more general way of defining boundaries in terms of F and L symbols. That is, there is no analogue of a thin boundary for non-group-cocycle F -symbols or L-symbols.
Let us discuss the relation between the thin and thick boundary. We can construct a transitive thick boundarŷ ψ from a thin boundary ψ as follows. Analogous to the 1 + 1-dimensional case, we start with a generalized thick boundary obtained by padding each thin boundary triangle with a triangle prism, It is easy to see that the moves for the thick boundary in Eq. (156) and Eq. (224) follows form the thin boundary moves such as in Eq. (223), after we plug in Eq. (225).
The H boundary group labels on the right are determined by the other labels α, a, b ∈ G so they do not appear on the left. Just as in the 1 + 1-dimensional case, we have a gauge symmetry on the right involving all edges adjacent to a fixed boundary vertex, which becomes a gauge symmetry acting on a single boundary vertex label on the left. Again, this allows us to replace the G-elements x, y, z on the left by left cosets α, β, γ. Then using a triangulation of the above prism similar to Eq. (162), we find where R denotes a choice of standard representative of every coset as in the previous paragraph. Vice versa, we can construct a thin boundaryψ from a thick boundary ψ by realizing that the thick boundary has a gauge symmetry acting on a single boundary vertex label. This gauge freedom can be fixed by setting all the cosets to the trivial coset H, yieldingψ for a, b ∈ H. As in the 1 + 1-dimensional case, we havẽ but not vice versa. Similar to the 1 + 1-dimensional case,ψ and ψ are separated by a 1-dimensional domain wall which associates a weight η to the according edges. This weight comes from flattening the vertical side of a "step" representing the domain wall. While in Eq.
using Eq. (227). The relation between ψ andψ is noŵ The generalization of thick versus thin boundaries to higher dimensions is straight-forward. To map a thin to a thick boundary of an n-dimensional space-time bulk analogous to Eq. (209), pad the n − 1-simplex with the n − 1-simplex times an edge. The resulting n-cell can be triangulated using n n-simplices. The edge labels of the edges perpendicular to the boundary are set to standard representatives of the cosets of the thick boundary. Evaluation of the space-time volume yields an expression with n bulk weigths ω and one thin-boundary weight ψ. Moreover, the domain wall η between ψ andψ analogous to Eq. (231) is obtained from triangulations of a boundary n − 2-simplex times an edge, yielding a formula with n − 1 times ψ.

E.3 Thin versus thick bulk
When the bulk ω is trivial, then the thick boundary ψ gives rise to a state-sum on its own, which we will refer to as a thick state-sum. Such a thick state-sum is a two-dimensional state-sum with vertex labels equipped with a right G-action, and G-elements on the edges. Examples for such state-sums-with-action arose in the where x and y are chosen representatives of α and β as in Eq. (212).
Note that the two-dimensional state-sums arising from compactifications in Appendix D are thick statesums, as they have labels on the vertices that are acted on by G. The (boundary) anyons are in one-to-one correspondence with the (irreducible) generalized thick boundaries of those thick compactified state-sums. The formula above provides a way to obtain such boundaries from boundaries of a thin state-sum. The computation of the latter is simpler in practice as it takes place on smaller vector spaces. We follow the following steps, which are also discussed in a more algebraic way in Section 2 in the main text.
• Decompose the thick compactified state-sum into transitive ones, with vertex label set H\G for different subgroups H.
• For each transitive part, calculate the corresponding thin state-sum, that is, the corresponding H 2-cocycle ω(a, b).
• For each transitive part, find the irreducible generalized thick boundaries of this thin state-sum. This corresponds to finding the irreducible representations ψ(h) ν µ of the ω-twisted group algebra, or in other words, the projective irreducible representations of H with 2-cocycle ω.
• Use Eq. (241) to obtain the irreducible generalized thick boundaries of the thick state-sum.