Bulk-to-boundary anyon fusion from microscopic models

Magdalena de la Fuente, Julio C.; Eisert, Jens; Bauer, Andreas

doi:10.1063/5.0147335

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 $Z_{2}$ as well as $Z_{3}$ 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.

I. 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 (QEC) 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 cellulation² or its dual formulation in terms of a triangulation of the same manifold.⁶ To achieve topological invariance, these models are also equipped with partial 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 three-dimensional cellulation of that space-time, and assigns a finite label set to any edge in that cellulation. The highest-dimensional 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. References 8–10 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. Reference 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. Reference 12 gives a formula to calculate the set of condensable anyons in gauge theory models with trivial three-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.^13–15

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 operators¹⁷ 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.^18–20 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 Sec. II, we give a general recipe to find the irreducible sub-spaces of twisted group algebras with action, which characterize both the bulk and boundary anyons in gauge theory models of a finite group. In Sec. III, 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 space-time 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 Secs. IV and V, 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 Sec. VI, 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 Sec. VII. For a reader interested in the technical details going into the derivation of the bimodule used in Sec. VI and tools to diagonalize the group algebras characterizing point defects, we refer to Appendixes A–E for further details.

II. 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. IV and V. 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 three-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,

g ▹ (h ▹ x) = (g h) ▹ x \forall g, h \in G, x \in X .

(1)

Given a G-set X, we define an algebra A over

C^{G \times X}

via the multiplication

(g, x) * (h, y) = δ_{x, h ▹ y} Ψ^{y} (g, h) (g h, y),

(2)

with

Ψ : G \times G \times X \to U (1) \subset C

⁠. For this multiplication to define an algebra it has to be associative. This imposes a non-trivial condition on the phase Ψ,

Ψ^{x} (h, k) Ψ^{x} (g, h k) = Ψ^{x} (g h, k) Ψ^{k ▹ x} (g, h) \forall g, h, k \in G, x \in X .

(3)

This can be seen as a two-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 \to C^{\times}

by (g, x) ↦ ξ(g)(g, x) to effectively map Ψ^x to a normalized two-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 Appendix 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. , A decomposes over transitive subsets of X, or equivalently, over G-orbits {X_i ⊆ X}, so we have an isomorphism

A ≃ ⨁_{i \in {X_{i}}} A_{i} .

(4)

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,

{S t a b}_{G} (x) ≔ {g \in G | g ▹ x = x},

(5)

are isomorphic. In particular, for x ∈ X, g ∈ G,

{S t a b}_{G} (g ▹ x) = g {S t a b}_{G} (x) g^{- 1} .

(6)

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 representative

{\hat{x}}_{i} \in X_{i}

and define

K_{i} ≔ {S t a b}_{G} ({\hat{x}}_{i}) \subseteq G .

(7)

Note that we can obtain the stabilizer group of any other element in X_i from K_i with Eq. .

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

Ψ^{{\hat{x}}_{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

{ρ^{{\hat{x}}_{i}}}

be the irreducible

Ψ^{{\hat{x}}_{i}}

-projective representations of K_i, defined via a representative

{\hat{x}}_{i} \in X_{i}

⁠. In particular, they fulfill

ρ^{{\hat{x}}_{i}} (k) ρ^{{\hat{x}}_{i}} (k^{'}) = Ψ^{{\hat{x}}_{i}} (k, k^{'}) ρ^{{\hat{x}}_{i}} (k k^{'}) \forall k, k^{'} \in K_{i},

(8)

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

ρ^{g ▹ x} (k) = \bar{Ψ^{x} (k g k^{' - 1}, k^{'})} Ψ^{x} (k, g) ρ^{x} (k^{'}),

(9)

which by construction fulfill

ρ^{y} (k) ρ^{y} (k^{'}) = Ψ^{y} (k, k^{'}) ρ^{y} (k k^{'}) \forall y \in X_{i}; k, k^{'} \in {S t a b}_{G} (y),

(10)

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. 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 representative ${\hat{x}}_{i} \in X_{i}$ ⁠.

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

c_{i, ρ_{i}} = \frac{\dim (ρ_{i})}{| K_{i} |} \sum_{x \in X_{i}} \sum_{g \in {S t a b}_{G} (x)} \bar{{\tilde{χ}}_{ρ_{i}}^{x} (g)} (g, x),

(11)

where

{\tilde{χ}}_{ρ_{i}}^{x} ≔ T r (ρ_{i}^{x}) : G \to C

is the irreducible Ψ^x-projective character derived from the IPR

ρ_{i}^{x}

⁠. Using unitarity and irreducibility within A_i,

\sum_{g} \bar{ρ_{i}^{x} {(g)}_{n}^{m}} ρ_{i}^{x} {(g)}_{n^{'}}^{m^{'}} = δ_{ρ, ρ^{'}} δ_{n, n^{'}} δ_{m, m^{'}} \frac{| K_{i} |}{\dim (ρ_{i})} \forall x \in X_{i},

(12)

we can show straightforwardly that the

c_{i, ρ_{i}}

s are idempotent,

c_{i, ρ_{i}} * c_{j, ρ_{j}^{'}} = δ_{i, j} δ_{ρ, ρ^{'}} c_{i, ρ_{i}},

(13)

and central,

c_{i, ρ_{i}} * (g, x) = (g, x) * c_{i, ρ_{i}} \forall (g, x) \in G \times X .

(14)

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, ρ_{i}}

s are not only central and idempotent but also indecomposable. To see this, consider the following set of smaller idempotents

d_{x_{i, ρ_{i}}} = \frac{\dim (ρ_{i})}{| K_{i} |} \sum_{g \in {S t a b}_{G} (x)} \bar{{\tilde{χ}}_{ρ_{i}}^{x} (g)} (g, x),

(15)

that are irreducible (by definition of

\tilde{χ}

⁠) 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, ρ_{i}}

irreducible. Furthermore, the isomorphism discussed in Appendix C provides a 1-1 mapping from irreducible representations of the twisted algebra with action in Eq. and

Ψ^{\hat{x}}

-twisted group algebras without action of (a collection of) subgroups. In Appendix 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 summarize this section by giving a recipe on how to find the irreducible representations of an algebra of the form of Eq. (2).

Decompose X into transitive G-orbits {X_i}.
For each X_i:
- Pick representative ${\hat{x}}_{i} \in X_{i}$ and calculate its stabilizer group $K_{i} = {S t a b}_{G} ({\hat{x}}_{i})$ ⁠.
- Find irreducible $Ψ^{{\hat{x}}_{i}}$ -projective representations of K_i, denote them with ρ_i.
- Use Eq. (9) to derive irreducible representations and character functions, ${\tilde{χ}}_{ρ_{i}}^{x} (g) = T r (ρ_{i}^{x} (g))$ for all x ∈ X_i.
The indecomposable central idempotents in A are labeled by pairs (i, ρ_i) ∈ (G-orbits of X, Ψ^x-projective Irreps of K_i) and given by
$c_{i, ρ_{i}} = \frac{\dim (ρ_{i})}{| K_{i} |} \sum_{x \in X_{i}} \sum_{g \in {S t a b}_{G} (x)} \bar{{\tilde{χ}}_{ρ_{i}}^{x} (g)} (g, x) .$
(16)
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.

III. 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 space-time 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 Appendix 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.

A. 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

O b j (C) = {1_{C}, i, j, \dots, k}

and fusion multiplicities

N_{ij}^{k} \in Z_{\geq 0}

⁠. These define a fusion-operation

i \times j = \sum_{k} N_{ij}^{k} k with a \times 1_{C} = 1_{C} \times a = a \forall a \in O b j (C) .

(17)

A fusion category is called Abelian if for every pair

(i, j) \in O b j (C)

there exists only one

k \in O b j (C)

such that

N_{ij}^{k} > 0

⁠. In other words, the fusion of i and j is unique. Note that the term Abelian does not refer to the commutativity of the fusion operation. 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 trivalent graph (tessellating some two-dimensional manifold) with local Hilbert spaces $H_{i} = s p a n (O b j (C))$ on each edge. 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).

The total Hilbert space is the tensor product space of all the local spaces,

H_{tot} = \otimes_{i} H_{i}

⁠.

O b j (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} \subset 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_{ij}^{k} \neq 0

⁠. For

N_{ij}^{k} > 1

the vertex itself carries in additional degree of freedom,

s p a n ({0, 1, \dots, N_{ij}^{k} - 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

(18)

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. 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 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

(19)

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_{cdf,νρ}^{aabe,μη}}

⁠,

(20)

These F-symbols are part of the input category

C

⁠. Any sequence of topological moves can be represented by a three-dimensional triangulation derived from the dual cellulation indicated in gray in Eqs. and . In the multiplicity-free case, where μ, ν, ρ, η are fixed, the F-move above, for example, is represented by a tetrahedron,

(21)

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. . The remaining four edges correspond to the outgoing/incoming edges in Eq. . 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 Appendix 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 two 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,

F_{fdc}^{eba} F_{fgd}^{hai} = \sum_{k} F_{cgd}^{hek} F_{fgc}^{kbi} F_{ihk}^{eba}

(22)

for all a, b, c, d, e, f, g, h, i which can be depicted as

(23)

On the left hand side the diamond formed by the five 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.

FIG. 1.

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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.

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.

B. 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_{a, α}^{β} \in Z_{\geq 0}

⁠,

a ▹ α = \sum_{β} M_{a α}^{β} β, a \in O b j (C), α, β \in O b j ({}_{C}M) .

(24)

This action has to be compatible with the fusion operation in

C

⁠, i.e.,

(a ▹ (b ▹ α)) ≃ (a \times b) ▹ α \forall a, b, α .

(25)

This gives consistency conditions on

{M_{a α}^{β}}

in terms of

{N_{a, b}^{c}}

⁠.

Graphically, the

C

-action can be represented by trivalent boundary vertices

(26)

where μ is non-trivial for

M_{a α}^{β} > 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 are linear maps corresponding to deformations of the boundary, whose entries we refer to as L-symbols,

(27)

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

(28)

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

(29)

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. 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.

FIG. 2.

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Different sequences of L- and 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 L- and F-symbols is called boundary pentagon equation.

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.

1. 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

(30)

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

(31)

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 \otimes \bar{C^{'}}

model, where the bulk strings are labeled by tuples

(a, a^{'}) \in O b j (C) \times O b j (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

(32)

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 straightforwardly 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,

{\tilde{L}}_{α (a, a^{'}), (b, b^{'})}^{β (γ, β^{'}, α^{'}) (c, c^{'}) γ^{'}} = C_{α^{'}, b, a^{'}}^{β^{'} γ γ^{'}} L_{γ^{'} a b}^{β β^{'} c} \bar{R_{α a^{'} b^{'}}^{γ^{'} α^{'} c^{'}}} .

(33)

FIG. 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.

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Via the folding trick, a $C$ - $C^{'}$ domain wall, classified by a $C$ - $C^{'}$ bimodule, is equivalent to a $C \otimes C^{'}$ 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.

C. Topological gauge theory models

Let G be a finite group. For the rest of the section, we focus on

C = {V e c}^{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,

N_{g h}^{k} = δ_{g h, k} .

(34)

With that, the vertices take the following form

(35)

Since there are just two free labels at any vertex, the F-moves are defined by a single function ω: G × G × G → U(1),

(36)

In this case, the pentagon equation reduces to a three-cocycle condition on ω,

ω (h, k, l) ω (g, h k, l) ω (g, h, k) = ω (g h, k, l) ω (g, h, k l) \forall g, h, k, l \in G .

(37)

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 maps the associated three-cocycle to

ω (g, h, k) \mapsto ω (g, h, k) (δ η) (g, h, k) = ω (g, h, k) \frac{η (h, k) η (g, h k)}{η (g h, k) η (g, h)} .

(38)

In fact, the product of ηs with which the three-cocycle gets multiplied is exactly the coboundary of the two-cochain η. This shows that topological lattice models based on Vec^ω(G) are classified by the third cohomology group H³(G, U(1)), see Appendix 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 three-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 $C^{ψ} [H]$ – a subgroup H and a two-cocycle ψ on it – in our calculations.

Given a subgroup H, the labels at the boundary are cosets in G/H = {aH|a ∈ G}. The bulk G-action is then defined by left action on the coset,

g ▹ α = g ▹ (a H) ≔ (g a) H,

(39)

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: G/H × G × G → U(1) with

(40)

The phase function has to satisfy the mixed pentagon equation in Eq. . In this case, it simplifies to

ω (g, h, k) = \frac{ψ^{α} (h, k) ψ^{α} (g, h k)}{ψ^{α} (g h, k) ψ^{k ▹ α} (g, h)} = (\tilde{δ} ψ^{α}) (g, h, k),

(41)

where

\tilde{δ}

is the twisted coboundary operator (for more details, see Appendix 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,

\tilde{δ} α = 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 one-dimensional (or zero-dimensional 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

ψ^{α} (g, h) \mapsto ψ^{α} (g, h) \frac{ξ^{α} (h) ξ^{h ▹ α} (g)}{ξ^{α} (g h)} = ψ^{α} (g, h) (\tilde{δ} ξ^{α}) (g, h) .

(42)

We see that the gauge corresponds to a multiplication of a coboundary with the same coboundary operator as in Eq. but acting on the one-cochains {ξ^α}. In total, we see that although the different ψ are not twisted two-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 Appendixes C and E, we show that this classification indeed coincides with known results, i.e.,

H^{2} (G, {(U {(1)}^{| G / H |})}_{G}) ≃ H^{2} (H, U (1)) .

(43)

In particular, this isomorphism holds for every cohomology group Hⁿ 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 Lawson in Ref. 27.

D. Models for Abelian phases

A topologically ordered phase is called Abelian if the fusion outcome of any pair of topological point defects – anyons – is unique, i.e. they form an Abelian fusion category. Importantly, not every Abelian fusion category

C

gives rise to an Abelian phase

D (C)

(see Sec. IV D). 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 three-cocycle of a certain form. Any finite Abelian group is isomorphic to a product of cyclic groups,

G ≃ Z_{p_{1}^{m_{1}}} \times Z_{p_{2}^{m_{2}}} \times \dots \times Z_{p_{N}^{m_{N}}},

(44)

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 three-cocycle classes on the group above. In fact, any three-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 Appendix 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 Sec. IV D.

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 Appendix B. Given a subgroup H (with k factors), we can construct a non-trivial two-cocycle by taking products of two-cocycles on each pair

H_{ij} = Z_{h_{i}} \times Z_{h_{j}}

in the factor decomposition of H. In particular, there are gcd(h_i, h_j) inequivalent two-cocycle classes on H_ij represented by normalized two-cocycles of the form

Ω {(a, b)}^{q_{ij}} = e^{\frac{2 π i}{\gcd (h_{i}, h_{j})} q_{ij} a_{i} b_{j}} with q_{ij} \in {0, 1, \dots, \gcd (h_{i}, h_{j})},

(45)

where a_i(a_j) is the component of a in

Z_{h_{i}} (Z_{h_{j}}) \subset H_{ij}

⁠. Following the previous subsection, the associated L symbol is obtained by precomposition with m⁽²⁾,

ψ^{α} {(a, b)}^{q_{ij}} = (Ω ◦ m^{(2)}) {(α, a, b)}^{q_{ij}} .

(46)

E. Examples

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

1. ${V e c}^{ω} (Z_{N})$

The cyclic group of order N,

Z_{N} = {0, 1, \dots, N - 1}

with group operation a ⊕_p b ≔ a + b mod N has N three-cocycle classes each of which are generated from a single three-cocycle class. The normalized three-cocycle

ω_{I}^{n} (a, b, c) = e^{\frac{2 π i}{N} n a (b + c - b \oplus c) / N}, n \in Z_{N},

(47)

is a canonical representative for the nth cocycle class of

H^{3} (Z_{N}, U (1)) = Z_{N}

⁠.²⁸ We say that a three-cocycle of that form, only supported on a single cyclic factor, is of type I. Using

Z_{N}

and

ω_{I}^{n}

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, H₀ = {0} and $H_{1} = Z_{p}$ ⁠. Both H₁ and H₂ only have trivial two-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, $ω_{I}^{n}$ becomes cohomologically trivial only on H₀. Hence, these models only have one “standard” boundary.

2. ${V e c}^{ω} (Z_{N} \times Z_{M})$

An Abelian group with two cyclic factors

Z_{N} \times Z_{M} = {(a, b) | a \in Z_{p}, b \in Z_{q}}

has NM three-cocycle classes of type I. Their normalized representatives decompose into a product of cocycles as in Eq. , each supported on one factor only. Additionally, there are non-trivial type-II cocycle classes represented by²⁸

ω_{I I}^{n_{12}} (a, b, c) = e^{\frac{2 π i}{\gcd (N, M)} n_{12} a_{1} (b_{2} + c_{2} - b_{2} \oplus c_{2}) / M}, n_{12} \in Z_{\gcd (N, M)} .

(48)

Three-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

H^{3} (Z_{N} \times Z_{M}, U (1)) = Z_{N} \times Z_{M} \times Z_{\gcd (N, M)}

⁠. Using

Z_{N} \times Z_{M}

and a cocycle of type I or II as in input gives an Abelian bulk model.

For simplicity, consider N = M = p prime. The possible boundary models are given by subgroups of

Z_{p} \times Z_{p}

⁠. They are

H_{0} = {0}, H_{1} = ⟨ (0, 1) ⟩ ≃ Z_{p}, H_{2} = ⟨ (1, 0) ⟩ ≃ Z_{p}, H_{3, l} = ⟨ (1, l) ⟩ ≃ Z_{p}

and

H_{4} = Z_{p} \times Z_{p}

⁠. In the untwisted case any of these subgroups defines a boundary model. Additionally, H₄ has non-trivial two-cocycles of the form

Ω {((a_{1}, a_{2}), (b_{1}, b_{2}))}^{m} = e^{\frac{2 π i}{p} m a_{1} b_{2}}, m \in Z_{p} .

(49)

Since H₄ 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. .

The three-cocycle in Eq. (48) becomes cohomologically trivial on H₀ and the isomorphic subgroups H₁, H₂ and H_3,l ∀l. All of them have no non-trivial two-cocycle class which gives 3 + l inequivalent topological boundaries.

3. Vec^ω(S₃)

The smallest non-Abelian group is the permutation group of three elements

S_{3} = ⟨ t, r | t^{2} = r^{3} = e, t r = r^{2} t ⟩ ≃ Z_{3} ⋊ Z_{2}

(e being the identiy element). Its third cohomology group is given by²⁸

H^{3} (S_{3}, U (1)) = Z_{3} \times Z_{2} ≃ Z_{6} .

(50)

Interestingly, the third cohomology group is the product of the cohomology groups of the two non-trivial subgroups of S₃. However, since S₃ is a a semidirect product of the two, the three-cocycles are not simple products of three-cocycles of the subgroups. The six inequivalent classes can be represented by the normalized cocycles²⁸

ω^{p} (t^{A} r^{a}, t^{B} r^{b}, t^{C} r^{c}) = e^{\frac{2 π i}{3} p {(- 1)}^{B + C} a ({(- 1)}^{C} b + c - [{(- 1)}^{C} b \oplus c]) / 3} {(- 1)}^{p A B C}, p \in Z_{6},

(51)

where the elements in S₃ are represented by t^Ar^a with

A \in Z_{2}

⁠,

a \in Z_{3}

and ⊕ denotes addition mod 3. When comparing the individual factors with a three-cocycle of a cyclic group [see Eq. ], we note that the second factor, (−1)^ABC, corresponds to the non trivial three-cocycle of the (Abelian) subgroup

Z_{2} \subset S_{3}

and the first factor corresponds to the non-trivial three-cocycle of the subgroup

Z_{3} \subset S_{3}

precomposed with the automorphism

ρ_{A, B, C} \in A u t (Z_{3}^{\times 3})

defined by ρ_A,B,C(a, b, c) = ((−1)^B+Ca, (−1)^Cb, c) for any

(A, B, C) \in Z_{2}^{\times 3}

⁠.

S₃ has four subgroups up to conjugation, H₀ = {e}, $H_{t} = ⟨ t ⟩ ≃ Z_{2}$ ⁠, $H_{r} = ⟨ r ⟩ ≃ Z_{3}$ and H_G = S₃. 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 three-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.

IV. BULK ANYONS: TUBE ALGEBRA

In this section, we will revisit the question of how to add anyons to the fixed-point models from Sec. III A. Anyons are (irreducible) point-defects in the bulk of a topological phase. Their world-lines live in a three-dimensional 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 Sec. IV B.

A. 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) × S₁. 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^30–32 can be associated to the simple objects in the unitary modular tensor category (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) \in O b j {(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

{(a, b, c, d)}_{T} * {(a^{'}, b^{'}, c^{'}, d^{'})}_{T} = δ_{d, a^{'}} \sum_{x, y} F_{yac}^{b b^{'} x} F_{ybx}^{b^{'} d^{'} c^{'}} \bar{F_{ybc}^{a^{'} b^{'} c^{'}}} {(a, x, y, d^{'})}_{T} .

(53)

FIG. 4.

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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.

B. 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 Appendix D and Refs. 6 and 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, S₁ × (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

T \to E n d (T),

(54)

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 ]. 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”

T \times T \times T \to E n d (F) .

(55)

As a representation, F decomposes into a direct sum of triples of irreducible representations of T,

F = ⨁_{a, b, c} {\tilde{N}}_{a b}^{c} a \otimes b \otimes c,

(56)

where a, b, c label the irreducible representations of T, i.e., the anyons of the bulk. The multiplicities in this decomposition

{\tilde{N}}_{a b}^{c}

coincide with the fusion multiplicities of the bulk anyons. In order to calculate them, we consider the tri-representation in Eq. for a triple of central idempotents

(P_{a}^{T}, P_{b}^{T}, P_{c}^{T}) \in T^{\times 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 Sec. VI. In Appendix D we give a formula for

{\tilde{N}}_{a b}^{c}

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 Sec. VI.

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,

(57)

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 Appendix E of Ref. 3.

Topological fixed-point models allow for a direct calculation of both R and F-symbols.³³ In Appendix 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 $(P_{a}^{T}, P_{b}^{T}, P_{c}^{T}) \in T^{\times 3}$ ⁠. 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.