FUSION STRUCTURE FROM EXCHANGE SYMMETRY IN (2+1)-DIMENSIONS

While the principles behind the exotic statistical behaviour of anyons are well-documented in the literature, a similar treatment for the origins of their fusion structure is often neglected. In this paper, we seek to clarify the matter. By considering the action of braiding on quasiparticles in two spatial dimensions, we describe the fusion structure amongst the superselection sectors of the system. Exchange symmetry is formulated in terms of the motion group and we recover the boson-fermion superselection rule in three spatial dimensions. We then adapt our formulation of exchange symmetry for the two-dimensonal case. Given a system of n quasiparticles, we see that the action of a specific n-braid βn uniquely specifies its superselection sectors. We prove several braid identities for βn that allow us to recover the fusion structure of anyons. Finally, we give an overview of the braiding and fusion structure of quasiparticles in the usual setting of braided 6j fusion systems and observe some R-matrix identities that follow from the action of βn. These identities give an ansatz for the form of the monodromy operator which agrees with the ribbon property.


Introduction
The study and classification of topological phases of matter is a pervasive theme of contemporary physics. Quasiparticles with exotic exchange statistics (called "anyons") are a hallmark of two-dimensional topological phases. Modular tensor categories and (2+1)-TQFTs provide the mathematical framework for studying the statistical behaviour of anyons. The experimental realisation and control of anyons is a much sought-after goal, owing especially to a proposed scheme for the robust processing of quantum information called "topological quantum computation" [1,2,3].
While the algebraic theory of anyons (of which various detailed accounts may be found [4,5,6]) is considered mature [7,8], the goal of this paper is to fill in an apparent gap in the literature concerning the origin of their fusion properties. It is well-understood that the statistical properties of anyons arise due the distinguished topology of exchange trajectories in two dimensions. In a given theory, anyons are distinguished by their "topological charges" which characterise their mutual statistics. However, it is further expected that these charges possess a fusion structure wherein the 'combination' (or fusion) of two anyons effectively results in a single anyon that may possibly exist in a superposition of topological charges. A treatment of the physical origins of the fusion structure of anyons is often neglected 1 . We thus seek to provide a ground-up construction of the braiding and fusion structure of anyons. "Quantum symmetries" is an umbrella term for some of the mathematical structures that are used to describe topological quantum matter. In this paper, one of our aims is to further clarify the connection between braided fusion categories and the elementary yet profound principle of exchange symmetry in quantum mechanics. Superselection sectors play a key role in our exposition.
A series of 'assumptions' or postulates A1-A7 are given throughout the text. The main narrative of this paper is presented in Section 4 where we show that the localisation and intrinsic entanglement assumptions A1 and A2 are sufficient to recover the the braiding and fusion structure of 2D quasiparticles. In Section 5, we outline the extra structure required to make contact with the standard theory of anyons; namely, the finiteness, duality, rigidity, twisting and nondegeneracy assumptions A3-A7.
1.1. Outline of paper. In Section 2, we recap the notion of superselection rules and identical particles. This is followed by a discussion of the difference between particle exchanges in two and three spatial dimensions.
In Section 3, we formulate exchange symmetry via the action of the motion group of a many-particle system (Eq. 3.1) and use this to recover the boson-fermion superselection rule for fundamental particles.
In Section 4, we consider the action of braiding on quasiparticle systems. Notably, the localisation postulate A1 means that this action is generally not given by a representation of the braid group; instead, it is given by a representation of the "coloured" braid groupoid. We give a detailed construction of this action in Remark 4.3(ii). The formulation of exchange symmetry from Section 3 is adapted accordingly (Eq. 4.13).
In spite of the localised nature of quasiparticles, the intrinsic entanglement postulate A2 means that subsystems of quasiparticles cannot belong to a superselection sector arising from the exchange symmetry amongst themselves: the superselection sector of an n-quasiparticle system is a global property of its constituents. Specifically, superselection sectors are shown to correspond to the eigenspaces under the action of a special n-braid β n which we call the superselection braid (Eq. 4.21, Theorem 4.5).
We then recover the fusion structure amongst these superselection sectors, showing that they exhibit the same statistical behaviour as quasiparticles which allows us to identify them as such (Theorem 4.13). The associativity and commutativity of fusion is deduced as a corollary (Corollary 4.15).
We prove several braid identities culminating in Theorem 4.22 which shows that the superselection braid encodes the strucure of all fusion trees for an n-quasiparticle fusion space. We finally show that β n is the unique braid (up to orientation) whose action specifies the superselection sectors of an n-quasiparticle system (Theorem 4.24).
In Section 5, we review the braiding and fusion structure from Section 4 within the usual framework of braided 6j fusion systems and give the additional postulates required to make contact with anyonic systems. In particular, we present some identities arising from consideration of the superselection braid (Remark 5.7) which in turn motivate the ansatz (5.32) foreshadowing the topological spin structure of anyons.
Examples of superselection observables 2 include spin, mass 3 and electric charge. Notably, the spin SSR concerns the superposition of integer and half-integer spins: by the spinstatistics theorem, this is equivalent to the boson-fermion SSR discussed in Section 3. These two equivalent SSRs are sometimes referred to as the univalence SSR.
The intrinsic properties of a particle correspond to quantum numbers with an associated SSR. Two particles are identical if all of their intrinsic properties match exactly e.g. all electrons are identical.

Particle exchanges.
Consider the exchanges of n identical particles 4 on a connected m-manifold M for m ≥ 2. The homotopy classes of exchange trajectories in M form a group G n (M) ∼ = π 1 (U n (M)) under composition (the fundamental group of the nth unordered configuration space of M). We will call this the motion group.
We are interested in two cases for M. Firstly, we have G n (R d ) ∼ = S n (the symmetric group) for d ≥ 3. Here, a tangle 5 is homotopic to 0 tangles and exchanges are insensitive to orientation ( Figure 1). 2 SSRs for whichĴ is an observable. 3 Bargmann's mass SSR arises through demanding the Galilean covariance of the Schrödinger equation: this only pertains to nonrelativistic systems, since Galilean symmetry is superseded by Poincaré symmetry in special relativity. 4 It will be assumed that particles are point-like unless stated otherwise. 5 We call two successive exchanges of the same orientation on a pair of adjacent particles a tangle.
Secondly, for a surface S we have G n (S) ∼ = B n (S) (the surface braid group). Given any n points in (the interior of) S, we can take some disc D ⊂ S such that all n points lie inside D. Furthermore, we know that G n (D 2 ) ∼ = B n where D 2 is the 2-disc and (2.4) is the Artin braid group. We will denote the identity element by e. The braid relations for B n thus also hold in B n (S) [10]. When considering particle exchanges on a surface S, we henceforth restrict our attention to B n (D 2 ). Remark 2.1. In particular, this means that what we learn about the exchange statistics of particles on a disc is also applicable to particles on surfaces with arbitrary topology. Figure 3. A braid diagram with n strands will be interpreted as a worldline diagram for n particles on a disc. We will let the time axis run downwards. The above diagram depicts this for the 3-braid σ 2 σ 1 .

Exchange Symmetry in Three or More Spatial Dimensions
A permutation of n identical particles will be indistinguishable from the original configuration: this is called exchange symmetry and is expressed concisely by (3.1) [Ô, ρ(g)] = 0 ∀Ô, and ∀g ∈ G whereÔ is an observable on H (the n-particle Hilbert space), G is the motion group of the n particles and ρ : The eigenvalues of ρ(s i ) belong to a nonempty subset of {±1}. We respectively denote the corresponding eigenspaces (one of which is possibly zero-dimensional) by H Proof. Take arbitrary |ψ + ∈ H (i) We may write |ψ We must thus have β = 0. This tells us that H (i) . The result follows easily. Theorem 3.1 allows us to recover the familiar exchange operator 6P : H → H for a many-particle system, and (3.2) becomes (3.4) [Ô,P ] = 0 ∀Ô where H = H + ⊕ H − (the superscript indices are dropped following Theorem 3.1). That is, the wavefunction of n identical particles is either symmetric or antisymmetric under an exchange, whence all fundamental particles respectively fall into two distinct classes: bosons and fermions. This recovers the boson-fermion superselection rule.
Remark 3.2. For a system of n bosons or fermions, there is typically no subspace describing a subsystem of k < n particles. This is implicit in the structure of Fock space 7 (here H (k) (±) denotes the space of (anti)symmetric states for k identical particles): + . For instance, states such as 1 − do not describe a physical entanglement, since the subsystem for an individual particle is physically inaccessible [11]. This is in contrast to anyonic systems which have a welldefined description of state spaces for particle subsystems (since anyons are localised phenomena).

Quasiparticles and braiding.
We begin by remarking that there are no fundamental particles in two spatial dimensions. However, it is well-known that various two-dimensional systems are theoretically capable of supporting localised excitations with fractional statistics [12,13,14,15]: these emergent phenomena are known as quasiparticles; they have no internal degrees of freedom and may thus be considered as identical. The localised nature of these two-dimensional excitations is instrumental in the emergence of fusion structure.
Anyonic statistics are expected to manifest in strongly correlated systems exhibiting long-range entanglements [9,17,18,19]. We translate this observation into the following "intrinsic entanglement" condition: A2. Generally, subsystems of quasiparticles are entangled with the rest of the system. Definition 4.1. Let a collection of quasiparticles be some n ≥ 2 adjacent quasiparticles. In light of A2, a system of quasiparticles is defined to be given by a collection of quasiparticles such that (i) it is not a subcollection entangled with a larger collection of quasiparticles; (ii) it cannot be further partitioned into subcollections of quasiparticles such that its Hilbert space is a tensor product of the Hilbert spaces for these subcollections. For instance, a collection of quasiparticles satisfying (i) but not (ii) comprises more than one system. Any subcollection or single quasiparticle subspace of a system defines a subsystem.
Recall that B n is the motion group of n particles on a disc. Then for a two quasiparticle system with Hilbert space V, the action of the motion group is is given by a unitary representation ρ : B 2 → U (V). In particular, the eigenvalues {e iu Q } Q of ρ(σ 1 ) lie in U (1), and we have the corresponding decomposition V = Q V Q (where eigenspaces V Q define superselection sectors by exchange symmetry). The possibly arbitrary exchange phase e iu Q is what earns anyons their namesake [16].
Remark 4.2. Mathematically, A1 permits us to consider the Hilbert space associated with a subsystem of adjacent quasiparticles. Consequently, the action of the motion subgroup on any such subsystem will be independent of the rest of the system. The description of the superselection sectors (and associated exchange statistics) given by the action of B 2 is thus a property of a given pair of quasiparticles.
Consider a 2-quasiparticle subsystem (of particles labelled q i and q i+1 located at the i th and i + 1 th positions respectively) of an n-quasiparticle system. We denote the Hilbert space of this subsystem by V {q i ,q i+1 } where {q i , q i+1 } is an unordered set. Following Remark 4.2, we have the fixed action and we write the eigenspace decomposition . Note that (4.1) makes no reference to the positions of q i and q i+1 .
We label the quasiparticles from 1 to n and let S {1,...,n} be the set whose elements are all possible permutations of the string 12 . . . n. Given some s ∈ S {1,...,n} we write s = q 1 . . . q n where q i is the i th character of string s. We denote the Hilbert space for quasiparticles q 1 . . . q n (in that order) by V q 1 ...qn or V s . E.g. V q 1 ...q i q i+1 ...qn and V q 1 ...q i+1 q i ...qn are the state spaces assigned to the system in the initial and final time-slices of Figure 4 respectively. Let ρ s V s (σ i ) be the unitary linear transformation describing the action of braid σ i ∈ B n on the n-quasiparticle system (as shown in Fig. 4). For n > 2, Remark 4.2 and A2 tell us that 8 , and the {V (s) Q } Q describe the rest of the system with which the subsystem is entangled. We let ρ s (σ ±1 i ) denote the action of (anti)clockwise exchanging q i and q i+1 , and so it is defined on any V u for which u ∈ S {1,...,n} contains the substring q i q i+1 or q i+1 q i . Following from (4.2), that is 9 It is thus evident that the action of B n on the system will generally depend upon the order of the quasiparticles for n > 2. E.g. the action of σ 1 ∈ B 3 on V 123 clearly differs from its action on V 231 . We must therefore distinguish between the spaces {V s } s∈S {1,...,n} in order to consider the action of braiding on the whole system. Q } Q are possibly zero-dimensional (but at least one must be nonzero). (ii) The above tells us that the right way to think about the action of braiding on an n-quasiparticle system is as follows: let {V s } s be defined as above and let b(s) be the obvious group action 10 on s for any b ∈ B n . We construct an action of the braids b ∈ B n as linear transformations between spaces {V s } s . This 8 We simply drop the tensor product in (4.2) for n = 2. 9 We could choose to permute the terms in the square brackets (so long as we write the decomposition of the space on which it acts in a consistent manner).
The action of the braiding can be thought of as a unitary linear representation of the braid groupoid for n distinctly coloured strands. Since s ∈ U s,b for all b ∈ B n , the above tells us that is a unitary linear transformation. By (B1) we always have s ∈ U s,b 1 ∩ U s,b 2 b 1 whence we may always apply (4.6) for u = s. (B3) tells us that u ∈ U s,b 1 ∩ U s,b 2 b 1 if and only if u ∈ U s,b 1 and b 1 (u) ∈ U b 1 (s),b 2 . (B4) tells us that u ∈ U s,b if and only if b(u) ∈ U b(s),b −1 . It also tells us that ρ s (b) is a diagonalisable, norm-preserving map. Writing s = q 1 . . . q n ∈ S {1,...,n} , (B5) tells us that the elements of U s,σ ±1 i are given by all u ∈ S {1,...,n} such that u contains the substring q i q i+1 or q i+1 q i . It also tells us that given any b whose group action is either (iii) Following (4.2) and the notation established above, we will write Q . We will also write . For n > 2, we note that the eigenspaces V in (4.11) will no longer constitute superselection sectors when Q runs over more than one nonvanishing summand: this is a direct consequence of A2. That is, the commutator [Ô , ρ {q i ,q i+1 } (σ 1 )] cannot be assumed to vanish for all observablesÔ on V {q i ,q i+1 } since these observables are entangled with the rest of the system. Crucially, this means that we can have a coherent superposition over the eigenspaces {V Similarly, an arbitrary entangled subsystem cannot give rise to superselection sectors (further discussed in Remark 4.7(ii)).

Superselection sectors.
The next task is to determine the superselection sectors of an n-quasiparticle system for n > 3. For n = 2, we know that they are given by the eigenspaces of the action of the σ 1 -braid. Before solving the general case, it will be instructive to consider n = 3.

Example 4.4. (3-quasiparticle system)
The superselection sectors of the system must be preserved under the action of braiding. Let β := σ 1 σ 2 σ 1 . Then we have eigenspace decomposition We observe that is the e iu Q -eigenspace of ρ 132 (β), and so we write Similarly, by consideration of for any s ∈ S {1,2,3} , and where the isomorphism V s Q ∼ − → V s Q is given by ρ s (b) for any b ∈ B 3 such that b(s) = s . Identifying these isomorphic eigenspaces, we have the decomposition In particular, this corresponds to a unitary representation and by exchange symmetry, we have (4.19) [ρ [3] (β),Ô] = 0 for all observablesÔ on V [3] . Thus, the spaces {V [3] Q } Q are superselection sectors of the system (from which it is clear that each superselection sector is indeed preserved under the action of braiding) and so V s Q defines a superselection sector for any (s, Q). We have shown that the superselection sectors of a 3-quasiparticle system are given by the eigenspaces of the action of the σ 1 σ 2 σ 1 -braid.
It will be convenient to define the following notation for braids: ..j ∀j ≥ 1 and b 0 := e A natural candidate for the braid that specifies the superselection sectors of an nquasiparticle system (we shall henceforth refer to this as the superselection braid and denote it by β n ) is one which exchanges each pair of quasiparticles once i.e.
In Theorem 4.5, we will show that the proposed braid (4.21) does indeed specify the superselection sectors; in fact, it does so uniquely (Theorem 4.24).
for any s ∈ S {1,...,n} , and where the isomorphism V s Following the same reasoning presented in Example 4.4, Theorem 4.5 tells us that we have the decomposition is a superselection sector. Thus, V s Q defines a superselection sector for any (s, Q). In conclusion, the superselection sectors of an n-quasiparticle system are given by the eigenspaces of the action of the β n -braid.
string s in reverse order). By Theorem 4.5, giving 12 and so (i) Definition 4.1 defines a system of quasiparticles as belonging to a fixed superselection sector, and having no partition into subcollections of particles that respectively belong to fixed superselection sectors. (ii) Following on from Remark 4.3(iii), take an n-quasiparticle system (n > 2) with ..n} ) in fixed superselection sector Q, and consider some k-quasiparticle subcollection q l q l+1 . . . q l+k−1 . Then we have the decomposition into superselection sectors . By A1 and A2, using the same format as in (4.11), is possibly zero-dimensional (though at least one of these spaces must be nonzero). Applying the same reasoning as in Remark 4.3(iii) (i.e. the entanglement of observables on the subsystem with the rest of the system), the eigenspaces V q l q l+1 ...q l+k−1 X will no longer constitute superselection sectors when X runs over more than one nonvanishing summand 13 (thus allowing for coherent superpositions over the eigenspaces).
In order to prove Theorem 4.5, we will need the braid identity in Lemma 4.10 (whose proof relies on Lemmas 4.8 and 4.9).  β n σ n−1 = σ 1 β n , n ≥ 2 Proof.
A subtlety: X could possibly index precisely one nonvanishing summand in the following special cases: (a) k = n − 1 (for n > 2) ; (b) k = n − 2 where the k-particle subcollection lies between the remaining two quasiparticles (for n > 3). In each case, the subsystem is clearly not entangled (has fixed superselection sector). Both cases follow from Definition 4.1. Aside from these possible outlier cases, any proper subcollection of particles of an n-quasiparticle system (n > 2) will be entangled with the rest of the system (including configurations (a) and (b)). Lemma 4.9.
The proof of Theorem 4.5 follows the same steps as in Example 4.4.
Proof. (Theorem 4.5) Consider the n-quasiparticle space V s for some fixed s ∈ S {1,...,n} . We have the where σ i (s) swaps the i th and (i + 1) th characters of s, and β n (s) will reverse the order of the characters in s.
. The result follows.

Fusion structure.
A composite collection of quasiparticles will exhibit the same statistical behaviour as a single quasiparticle under exchanges: the scheme under which a collection of quasiparticles is considered as a composite is known as fusion. In this section, we will carefully show the emergence of this behaviour through consideration of the superselection braid.
Definition 4.11. We define t k,l to be the braid in B k+l that clockwise exchanges k strands with l strands. Similarly, we define t −1 k,l to be the braid in B k+l that anticlockwise exchanges k strands with l strands: take care to note that this is not the inverse braid of t k,l (which we instead write as (t k,l ) −1 ). Clearly, (t k,l ) −1 = t −1 l,k .
For any a ∈ N 0 , we have the homomorphism where r a 1 • r a 2 = r a 1 +a 2 . We also have the anti-automorphism which reverses the order of the generators in a braid word. Let and that ← − t k,l = t l,k .
Following on from Remark 4.7(ii), consider some n-quasiparticle system V s Q with fixed superselection sector Q for some s ∈ S {1,...,n} . Partition s into m 1 , . . . , m j i.e.
are possibly zero-dimensional (but at least one must be nonzero). The spaces V X 1 ...X k Q generically characterise the entanglement of the subsystems with the rest of the system. 14 If we have some m i such that |m i | = 1, we canonically identify V Xi Xi ∼ = C (more details follow in Section 5.1).
We now examine the consequences of the above theorems. Theorem 4.13 tells us that the k and l-quasiparticle composites m 1 and m 2 (in eigenstates of ρ m 1 (β k ) and ρ m 2 (β l ) respectively) behave identically to a pair of quasiparticles under exchange: if we fix eigenspaces V m 1 x and V m 2 y such that V xy Q is nonzero, then composites m 1 and m 2 behave as a pair of quasiparticles in superselection sector Q with exchange phase e i(u Q −ux−uy) . Therefore, the eigenspaces of ρ m 1 (β k ) and ρ m 2 (β l ) may be considered as representing different 'types' of quasiparticles (since the exchange phase depends on x and y). We will refer to the 'type' of a quasiparticle as its (topological) charge. If e.g. k > 1, we say that the subcollection of quasiparticles m 1 fuses to a quasiparticle of charge x. It follows that the possible (x, y) for which V xy Q is nonzero represent the distinct possible fusion outcomes here.
In particular, note that we can have a coherent superposition of different fusion outcomes on an entangled subsystem of quasiparticles. Furthermore, since the eigenspaces of any ρ Σ (β n ) (where Σ is an unordered set of quasiparticles of cardinality n) can be identified with quasiparticle charges, it follows that the superselection sector of a system can be identified with a (composite) quasiparticle of fixed charge. Definition 4.1 defines a system of quasiparticles as having fixed total charge (fusion outcome), and having no partition into subcollections of particles that respectively have fixed total charge. This also lends the hitherto abstract factor V xy Q in (4.35) a more concrete interpretation: Q is the space of states describing the process where collection m 1 fuses to (a quasiparticle of charge) x, collection m 2 fuses to y, and then x and y fuse to Q (see Figure 7(i)). The interpretation of any such tensor decomposition follows analogously. Such Hilbert spaces are thus known as fusion spaces and their constituent states are called fusion states. Proof. Commutativity follows from Theorem 4.5: the possible fusion outcomes for an n-quasiparticle system correspond to the eigenspaces of ρ [n] (β n ) on V [n] (whence the order of the n quasiparticles is irrelevant). Associativity follows from recursive application of Theorem 4.13 i.e. further partitioning m 1 and m 2 and so on until no further partitions can be made: we will view such a recursive choice of partitions as a full rooted binary tree with n leaves. This provides us with a fusion tree illustrating the order in which n quasiparticles are fused (see Figure 8). Since Q corresponds to an arbitrary eigenspace of ρ s (β n ), it follows that the set of possible fusion outcomes (i.e. the possible labels for the root) does not depend on the order of fusion. By the associativity and commutativity of fusion, the charge of an unordered collection Σ of quasiparticles can be thought of as a property of any connected region of the system in which solely the excitations in Σ are enclosed. This is one of the reasons that quasiparticle charge is called 'topological' (as opposed to e.g. electric charge which is defined geometrically via the charge density). Similarly to electric charge, we have seen that topological charge may correspond to a superselection rule of a system; but unlike electric charge, we may also observe a superposition of topological charges for an entangled subsystem. Remark 4.16. Take care to note that statistical phases of the form e iu Q are not a property of charge Q alone, but arise as eigenvalues of some ρ s (β n ) i.e. the phase also depends on the constituent charges fusing to Q.
As indicated by Theorem 4.13, fusion generally does not correspond to a physical process but rather describes how a collection of charges may be considered as a composite charge. Of course, the measurement of a fusion outcome is physically significant. Note that transporting quasiparticles between two separate systems will result in the merging of the systems: the superselection sector of the resulting system will be the combined topological charge of the original two systems. It is amusing to observe that evolutions where quasiparticles from one system wind around quasiparticles in another system (and then return to their own system) equates to having tangled worldlines between the two systems: in this sense, 'tangling' two systems results in their entanglement. This is a thought-provoking concept under "ER=EPR". 15 Figure 10. Exchange interactions between two systems results in the merging of their respective superselection sectors: here, a tangle results in the merging of Q 1 and Q 2 to superselection sector Q. The systems become entangled, merging to a single system.
In order to prove Theorems 4.12 and 4.13, we will need Lemma 4.20 and the key braid identities in Lemma 4.21 (whose proof relies on Lemmas 4.17,4.18 and 4.19).
from which we see that Proof.
(i) By Lemma 4.17, we have and whence it suffices to show that where the right-hand side of (4.44) is β l · t k,l . We prove (4.44) by induction.
First, we perform induction on l for fixed k. The base case (k, l) = (k, 1) is which is clearly true. Now suppose (4.44) holds for some l given fixed k. Then we want to show that (4.44) also holds for (k, l + 1) i.e. (4.46) Observe that and so the right-hand side of (4.46) is where the final equality follows by the induction hypothesis. Thus, in order to show (4.46), we must show that under the induction hypothesis. Lemma 4.9 tells us that b n σ i = σ i+1 b n for any n ≥ 2 and 1 ≤ i ≤ n − 1. Applying this result to the right-hand side of (4.47), we see that b k+l acts on each r j term by r 1 as it moves to its right, yielding the left-hand side. This completes the induction on l.
Next, we perform induction on k for fixed l. The base case (k, l) = (1, l) is which we show via repeated application of Lemma 4.18 on the right-hand side.
which proves the base case. Now suppose (4.44) holds for some k given fixed l. Then we want to show that (4.44) also holds for (k + 1, l) i.e. (4.49) Observe that t k+1,l = t k,l · r k ( ← − b l ), and so the right-hand side of (4.49) is where the second equality follows by the induction hypothesis. Thus, in order to show (4.49), we must show that under the induction hypothesis. For l = 1, (4.50) is which is clearly true.
Claim: Then, One can easily show that which we can recursively apply in (4.53) to get This proves the claim (4.52).
We recursively apply (4.52) to the right-hand side of (4.50) for i = 1, . . . , l − 1 (in increasing order): which is the left-hand side of (4.50). This completes the induction on k.
(ii) Applying the anti-automorphism χ to (i), we get where the second line follows by Lemma 4.19 and ← − t k,l = t l,k . It is clear that β l commutes with r l (β k ). The result follows.
We are now ready to prove Theorems 4.12 and 4.13.

Proof. (Theorem 4.12) Let
in the same sense as (4.12). We construct unitary braid actions Z 1 and Z 2 for subsets of braids S 1 , S 2 ⊆ B n respectively. For any g ∈ S 1 and g ∈ S 2 , we have either Given g, h ∈ B n such that g, hg ∈ S 1 , we have and similarly for Z 2 (h g ) given g , h ∈ B n such that g , h g ∈ S 2 . Note that S 1 and S 2 cannot be closed under braid composition.
Following Theorem 4.5, we have representations . This allows us to define representations Let X and Y be the index sets of X and Y respectively. Then we define Σ 1 ⊆ X × Y as the subset of indices (x, y) such that V xy Q is nonzero, and Σ 2 ⊆ X × Y as the subset of indices (x, y) such that V yx Q is nonzero. For (x, y) ∈ Σ 1 , we have the subrepresentations where for (x, y) ∈ Σ 2 we have the subrepresentations Our next goal will be to describe the action of t k,l and t l,k on spaces V where m 1 , m 2 := r k (σ i )(m 1 m 2 ) = m 1 , σ i (m 2 ). By Lemma 4.20(i), Let {|v e } e and {|w e } e be orthonormal bases for V and W respectively. Then we may write For k > 1 and 1 ≤ i < k, We similarly have Let t k,l ∈ S 1 and t l,k ∈ S 2 with (4.73) We now describe the action of β n on spaces V . This follows the same ideas as the proof of Theorem 4.5. Take arbitrary |ψ ∈ V m 1 ,m 2 Q . For a string v, we will denote the reverse string byṽ. By Lemma 4.10, Suppose k > 1 and let 1 ≤ i < k. Then (4.74) becomes . We may similarly show that We similarly have and by Lemma 4.21(i), Since [Z 2 (r l (β k )), Z 2 (β l )] = 0 we know that Z 1 (t k,l ) |ψ is in a simultaneous eigenspace of Z 2 (r l (β k )) and Z 2 (β k ) whence it is clear that The result follows.

Theorem 4.22. (Superselection braid by recursion)
Let n ≥ 2. For any positive integers k, l such that k + l = n, β n is given by (iii) β l · t k,l · β k (iv) r l (β k ) · t k,l · r k (β l ) and β 1 := e. The terms enclosed in square brackets commute.
By choosing between forms (i)-(iv) at each iteration (and permuting the terms in square brackets if desired), Theorem 4.22 yields explicit braid words for β n . The expression (4.21) is recovered by repeated application of (ii) with l = 1. Note that β −1 n is given by (i)-(iv) but with a superscript '−1' on each t and β (this is easily seen by inverting (i)-(iv)).
Proof. (Theorem 4.22) Expressions (i) and (ii) were already proved in Lemma 4.21. From Lemma 4.20, it easily follows that for any positive integers 17 k, l, we have Expressions (iii) and (iv) are implied by (i) and (ii) using either one of (4.86a),(4.86b).
Given the fusion space V s = Q V s Q (where s = q 1 . . . q n ∈ S {1,...,n} and Q indexes the superselection sectors), fix a fusion tree: by Theorem 4.13, each of the n−1 fusion vertices 18 corresponds to an eigenspace of ρ s(v) (β |s(v)| ), where for a fusion vertex v we let s(v) denote the substring of s given by the leaves descending from v and |s(v)| the length of s(v). Note that 2 ≤ |s(v)| ≤ n. We thus label each fusion vertex v with an eigenspace of ρ s(v) (β |s(v)| ) (recall that such a label represents a fixed topological charge and is called a 'fusion outcome' in this context). Such a labelling is called admissible if the corresponding fusion subspace of V s has nonzero dimension. Note that the root label corresponds to the superselection sector of the system. Observe that fixing a fusion tree specifies a decomposition of V s in terms of the eigenspaces of {ρ s(v) (β |s(v)| )} v . We write such a decomposition in the form yielded by recursive application of (4.34) e.g. a fusion tree of the form illustrated in Figure 11 specifies the decomposition Figure 11. The labels x 1 , x 2 and q correspond to eigenspaces of ρ q 1 q 2 (β 2 ), ρ q 1 q 2 q 3 (β 3 ) and ρ q 1 q 2 q 3 q 4 (β 4 ) respectively. The triple (x 1 , x 2 , q) of charges is an admissible labelling of the tree i.e. the fusion subspace V q 1 q 2 Theorem 4.22 provides a method for parsing β n into a composition of braids of the form r d (t k,l ). Any such parsing involves making a choice of n − 1 partitions.
From any possible sequence of partitions, we can always recover a fusion tree with which the parsing of β n is compatible. By compatibility, we mean that it is readily apparent how the fusion tree will transform under the action of β n i.e. β n can be parsed into a sequence of braids that each have a well-defined action on the decomposed components of the system. The incoming branches of each fusion vertex in the tree are clockwise exchanged and so the initial fusion tree is sent to its mirror image. β n is thus compatible with all n-leaf fusion trees (as expected).  Given |ψ ∈ V s Q , we know that ρ s (β n ) |ψ = e iu Q |ψ . It is illuminating to examine how the phase e iu Q arises given a decomposition of V s Q . Consider any admissibly labelled fusion tree in V q 1 ...qn Q (whence the root has label Q). We know that ρ s (β n ) will clockwise exchange the incoming branches of every fusion vertex. For any fusion vertex, the clockwise exchange is given by where the phase evolution follows from Theorem 4.13. It is easy to see that the total phase evolution acquired by clockwise exchanging the incoming branches of every fusion vertex will be e i[u Q −(uq 1 +···+uq n )] (phases associated to internal nodes of the tree will cancel). Finally, observe that the u q i are zeroes (since they are arguments of eigenvalues under the action of β 1 = e). n are the unique braids under whose action the fusion space decomposes into the superselection sectors of an n-quasiparticle system.
A proof of Theorem 4.24 is outlined in Appendix B.

Theories of Anyons
This section primarily serves to connect our exposition in Section 4 with the usual formalism in the literature, and to outline the additional postulates required to make contact with anyonic systems. Our presentation therefore omits a detailed discussion of various details (including quantum dimensions, Frobenius-Schur indicators, gauge transformations, ribbon structure and modularity). For a more detailed treatment, we refer the reader to [4,5]. In relation to insights arising from consideration of the superselection braid, we highlight Remarks 5.7 and 5.9.

Finiteness and duality.
In any standard theory of anyons, it is assumed that there are finitely many distinct topological charges. A theory of anyons thus comes equipped with a finite set of labels L whose cardinality is called the rank of the theory. It is also assumed that the representation space in (4.1) is finite which immediately tells us that dim(V {a,b} c ) is finite for any a, b, c ∈ L (from which it easily follows that a fusion space for finitely many quasiparticles is finite-dimensional). We package these two assumptions into the finiteness assumption A3 below. Any label set will include the trivial label (which we will write as 0 ) which represents (the topological charge of) the vacuum: the fusion of any charge with the vacuum yields the original charge i.e. N 0q r ∝ δ qr for any q, r ∈ L. Since we always have the freedom to insert the trivial charge anywhere, we must have (5.4) dim(V ab c ) = dim(V a0b c ) = dim(V 0ab c ) = dim(V ab0 c ) Associativity and (5.4) tell us that N a0 a N ab c = N ab c N 0b b = N ab c and so N a0 a = N b0 b = 1 for all a, b ∈ L. Thus, (5.5) N q0 r = N 0q r = δ qr for any q, r ∈ L Following the presentation in [4], write V a0 a = span C {|α a } and V 0b b = span C {|β b }. The relation between the spaces in (5.4) is characterised by trivial isomorphisms Braiding with the vacuum must be trivial i.e. using the same notation as in (4.1), (5.7) ρ {q,0} (σ ±1 1 ) = 1 for all q ∈ L A4. For each charge in a theory of anyons, there exists a unique dual charge with which it may fuse to the vacuum (annihilate) in a unique way.
In terms of the fusion coefficients, duality assumption A4 says that (5.8) ∀q ∈ L ∃!q ∈ L : N qq 0 = 1 whereq denotes the dual charge for q. Together with associativity, A4 tells us that for any a, b, c ∈ L we have N ab c Nc c 0 = N aā 0 N bc a and so N ab c = N bc a . We thus have Corollary 5.2. Any topological charge q ∈ L may realise a superselection sector.
Proof. We know that it is possible for a fusion outcome to realise a superselection sector. Suppose there exists a charge q ∈ L such that it is not a fusion outcome for any pair of charges. For any charge b there exists a charge c such that Nq b c = 0. By (5.9) we have Nq b c = N bc q which gives a contradiction.
We see that the duality assumption permits any charges to realise a superselection sector. For this reason, labels are often called topological charges and superselection sectors interchangeably in the literature.

5.2.
Braided 6j fusion systems. We write orthonormal bases of fusion states given any a, b, c ∈ L, and 1 ≤ µ ≤ N ab c for N ab c = 0. Figure 13. A graphical depiction of the fusion state |ab → c; µ up to some normalisation: fusion vertices are normalised using the 'quantum dimensions' of the incident charges. We implicitly assume that vertices carry the appropriate normalisation. Details may be found in [4].
The R-matrices of a theory are given by a matrix representation of the unitary operators from (4.1) in an eigenbasis (for any pairs q i , q i+1 ∈ L): given any a, b ∈ L we have the eigenspace decomposition It is clear that R ab = R ba here. Following (5.7), we have (5.14) R q0 q = R 0q q = 1 for all q ∈ L. We let (R −1 ) ab denote the anticlockwise exchange i.e.
For an n-quasiparticle fusion space V q 1 ...qn (where q 1 , . . . , q n ∈ L) let D 1 and D 2 be decompositions of this space corresponding to distinct fusion trees. By associativity, we have an isomorphism Fixing a basis of fusion states, we see that F ∈ Aut(V q 1 ...qn ) is a change of basis matrix. Observe that F is given by any sequence of so-called F-moves that transform between decompositions of the form This change of basis is graphically expressed as Distinct fusion trees specify distinct bases on the fusion space and are therefore also called fusion bases. Since R ab is defined for an eigenbasis of V ab , we must fix a fusion basis such that the factors {V ab Q } Q∈L appear in the decomposition of the fusion space: for any such fusion basis, we say that 'a and b are in a direct fusion channel '. That is, R-matrices can only act on two charges in a direct fusion channel. Figure 14. Charges a and b are in a direct fusion channel with outcome Q. The above is a graphical expression of the equation

Remark 5.3. (Gauge freedom)
There is generally some redundancy amongst the F and R symbols 22 of a theory: this arises from the U (N ab c ) freedom when fixing an orthonormal basis on the spaces {V ab c } a,b,c∈L . A change of basis 23 is called a gauge transformation. We can only attach physical significance to gauge-invariant quantities. Although R-symbols are generally gauge-variant, gauge transformations are defined to respect the triviality of braiding with the vacuum (i.e. (5.14) is gauge-invariant by construction). A monodromy is a composition It is easy to show that monodromies are gauge-invariant, whence it follows that the action of any pure braid is gauge-invariant. We implicitly fixed a gauge where R ab = R ba for all a, b ∈ L in our construction: we will call this the symmetric gauge. R-matrices are not necessarily diagonal and symmetric in their upper indices outside of this gauge. Nonetheless, considering (5.22) in the symmetric gauge shows that monodromy matrices are always diagonal and symmetric in their upper indices (by gauge-invariance of monodromies).

Remark 5.4. (Coherence conditions)
Isomorphisms between fusion spaces must be 'compatible' with one another. That is, distinct sequences of isomorphisms (F-moves, R-moves and isomorphisms α and β from (5.6)) between two given spaces should correspond to the same isomorphism. Such compatibility requirements are called coherence conditions. Remarkably, all coherence conditions are fulfilled if the triangle, pentagon and hexagon equations are satisfied. Some additional details are provided in Appendix C.
(i) All isomorphisms α and β from (5.6) must be compatible with associativity (F-moves). This coherence condition is fulfilled if the triangle equations (C.1) are satisfied. (ii) Recall the isomorphism F from (5.16). It may be possible that multiple distinct sequences of F-moves realise F. Given some basis, the matrix representation of F must be the same for all such sequences. This coherence condition is fulfilled if all F -symbols satisfy the pentagon equation (C.2). (iii) Consider n-quasiparticle space V q 1 ...qn where q 1 , . . . , q n ∈ L and n ≥ 3. Let s and s be any two distinct permutations of the string q 1 . . . q n . Let D and D be any decomposition of V s and V s respectively. It may be possible that multiple distinct sequences of F and R moves realise the isomorphism B : D → D . Given some basis, the matrix representation of B must be the same for all such sequences. This coherence condition is fulfilled if all F and R symbols satisfy the hexagon equations (C.7).
The dual space of a fusion space has natural interpretation as a 'splitting space' i.e. for any a, b, c ∈ L. Fusion coefficients may thus also be thought of 'splitting' coefficients. Given an orthonormal basis, we can use the graphical calculus to express the inner product and completeness relation on V ab : For a, b, c ∈ L we define linear maps K ab c and L ab c , These are clearly linear isomorphisms (whence N ab c = Nā c b = N cb a ). Let T denote the Hermitian-conjugation operator. We have (5.26) where (i) corresponds to symmetries (5.9) and the composition of (i) and (ii) tells us that N ab c = Nbā c . Together with (5.1), these identities generate all symmetries of the fusion coefficients. Summarising these, for all a, b, c ∈ L we have  Definition 5.8. Let ZB be a free Z-module with finite basis B = {b i } i∈I . We equip ZB with a binary product · : ZB × ZB → ZB such that the following hold: (i) There exists an element 1 : For any i, j ∈ I there exists at least one k ∈ I such that c ij k > 0 (iv) There exists an involution i → i * of I such that c ij 0 = c ji 0 = δ i * j ∀i, j ∈ I The unital, associative Z-algebra A = (ZB, · ) satisfying the above is called a fusion algebra. If we also have The quantities {c ij k } i,j,k∈I act as the structure constants of a fusion algebra. We can also express properties (i),(ii) and (v) in terms of these constants: The structure constants clearly have symmetries of the same form as in (5.27b) (and (5.27a) for a commutative algebra). Observing that the * -involution may be extended to an anti-automorphism of A, it easily follows that the structure constants also have symmetry of the form (5.27c).
A commutative fusion algebra A admits a categorification if there exists a braided 6j fusion system with label set L and a bijection φ : B → L such that c ij k = N φ(i)φ(j) φ(k) for all i, j, k ∈ B. It is possible for a given A to admit more than one categorification, although finitely many (up to equivalence of categories) by "Ocneanu rigidity". The categorification of A yields a braided fusion category (whose skeletal data is given by the braided 6j fusion system). Of course, from a quantum-mechanical perspective we are only interested in categories for which (there exists a choice of gauge where) all associated F and R symbols are unitary; namely, unitary braided fusion categories.

A6.
A clockwise 2π-rotation of an anyon q results in the phase evolution ϑ q ∈ U (1) (called the topological spin of q) satisfying ribbon property (5.34).
The topological spins are roots of unity [4,21] and are gauge-invariant. The ribbon property allows us to promote quasiparticle worldlines to worldribbons, or equivalently tells us how to evaluate type-I reidemeister moves on worldlines ( Figure 15).
Remark 5.9. It is known that a unitary braided fusion category admits a unique ribbon structure [22,23]. This tells us that given a unitary braided 6j fusion system, the ansatz (5.32) is correct and has a unique set of solutions. In the categorical setting, the structure imposed by assumptions A1-A6 corresponds to a unitary ribbon fusion category (also called a unitary premodular category). A7. A theory of anyons has nondegenerate braiding. That is, if monodromy operator M xq is the identity for all charges q then x must be the trivial charge.
Altogether, A1-A7 characterise anyonic systems. Nondegeneracy assumption A7 enforces the condition that the Müger center 24 of the category is trivial, yielding a unitary modular tensor category. A theory of anyons has all of its data contained in a unitary modular tensor category and is determined (up to gauge equivalence) by its skeletal data (fusion coefficients, F -symbols and R-symbols). The underlying fusion algebra is called the fusion rule of the theory. 25 The rank-finiteness theorem for modular categories [24] tells us that there are finitely many theories of anyons of any given rank. The objective of classifying theories of anyons thus motivates pursuing the classification of unitary modular tensor categories; we refer the reader to [25] for a classification up to rank four. The deduction in Remark 2.1 is verified, for example, by the toric code modular tensor category which describes quasiparticles on a torus.

Concluding Remarks and Outlook
The majority of this paper is devoted to considering the action of braiding on quasiparticle systems. To this end, the "superselection braid" proved to be central to our exposition. We saw that its action specified the superselection sectors of a system, illuminated the fusion structure possessed by topological charges and suggested the ribbon property.
A motion group may be defined in a more general context than that found in Section 2.2 in order to describe the 'motions' of a (typically disconnected) nonempty submanifold N in manifold M [27]. If M = R 3 and N is given by n disjoint loops then the motion group is the loop braid group LB n . Physically, we expect LB n to play a similar role in describing the exchange statistics of loop excitations in (3 + 1)-dimensions to that of the braid group for point-like excitations in (2 + 1)-dimensions [28]. The next possible generalisation could be to consider the statistics of knotted loops. The representation theory of motion groups and their relation to higher-dimensional TQFTs and topological phases of matter is an active area of research. In the case of loop excitations, various inroads have been made [29,30,31,32,33,34].
By formulating exchange symmetry via motion groups, the methods presented in this paper might be extended by adapting them to the setting of higher-dimensional excitations.
Illustrating the fusion trees in (C.2), The pentagon equation (C.2) may be written Fixing the fusion states in the initial and terminal fusion basis, we obtain an entry-wise form of (C.3) which is useful for direct calculations. Fix initial state |ab → p; α |pc → q; β |qd → e; λ and terminal state |as → e; ρ |br → s; δ |cd → r; γ . This gives us We have the hexagon equations 30 : commute for all a, b, c, d ∈ L.
(C.7) Figure 16. An illustration of the fusion trees in (C.7). 30 We roughly sketch the origin of the hexagon equations. Consider the set F n of n-leaf fusion trees. Let F n be the set whose elements are given by those in F n but with all possible permutations of the string q 1 . . . q n labelling the leaves (so that |F n | = n! · |F n |). We define a digraph KR n to have vertex set F n and edges given by all F and (identically oriented) R moves transforming between the elements of F n Any pair of adjacent vertices will share precisely one edge. In order to have compatibility between all F and R moves, it suffices to demand that the Yang-Baxter equation is satisfied: we thus only need to consider subgraphs of the form KR 3 i.e. the Franklin graph. This graph may be drawn as a dodecagon containing six hexagons and three (automatically commutative) quadrilaterals. The Yang-Baxter equation holds if the dodecagon commutes: imposing the hexagon equations ensures that the hexagons commute, and consequently that the dodecagon commutes. We remark that by restricting the edges of KR n to only permit R-moves acting on two leaves in a direct fusion channel, we obtain the graph corresponding to the n th permutoassociahedron [26].