Anyon fusion is the process by which multiple anyons behave as one larger composite anyon. Anyon fusion is essential to understanding the physics of non-abelian anyons and how they can be used in quantum information.
Abelian anyons.
If formula_1 identical abelian anyons each with individual statistics formula_2 (that is, the system picks up a phase formula_3 when two individual anyons undergo adiabatic counterclockwise exchange) all fuse together, they together have statistics formula_4. This can be seen by noting that upon counterclockwise rotation of two composite anyons about each other, there are formula_5 pairs of individual anyons (one in the first composite anyon, one in the second composite anyon) that each contribute a phase formula_3. An analogous analysis applies to the fusion of non-identical abelian anyons. The statistics of the composite anyon is uniquely determined by the statistics of its components.
Non-abelian anyon fusion rules.
Non-abelian anyons have more complicated fusion relations. As a rule, in a system with non-abelian anyons, there is a composite particle whose statistics label is not uniquely determined by the statistics labels of its components, but rather exists as a quantum superposition (this is completely analogous to how two fermions known to each have spin 1/2 and 3/2 are together in quantum superposition of total spin 1 and 2). If the overall statistics of the fusion of all of several anyons is known, there is still ambiguity in the fusion of some subsets of those anyons, and each possibility is a unique quantum state. These multiple states provide a Hilbert space on which quantum computation can be done.
Specifically, two non-abelian anyons labeled formula_7 and formula_8 have a fusion rule given by formula_9, where the formal sum over formula_10 goes over all labels of possible anyon types in the system (as well as the trivial label formula_11 denoting no particles), and each formula_12 is a nonnegative integer which denotes how many distinct quantum states there are in which formula_13 and formula_14 fuse into formula_15 (This is true in the abelian case as well, except in that case, for each formula_13 and formula_14, there is one type of anyon formula_15 for which formula_19 and for all other formula_15, formula_21 .) Each anyon type formula_13 should also have a conjugate antiparticle formula_23 among the list of possible anyon types, such that formula_24, i.e. it can annihilate with its antiparticle. The anyon type label does not specify all of the information about the anyon, but the information that it does indicate is topologically invariant under local perturbations.
For example, the Fibonacci anyon system, one of the simplest, consists of labels formula_25 and formula_26 (formula_26 denotes a Fibonacci anyon), which satisfy fusion rule formula_28 (corresponding to formula_29) as well as the trivial rules formula_30 and formula_31 (corresponding to formula_32).
The Ising anyon system consists of labels formula_25 , formula_34 and formula_35 , which satisfy fusion rules formula_36, formula_37, and the trivial rules.
The formula_38 operation is commutative and associative, as it must be to physically make sense with fused anyons. Furthermore, it is possible to view the formula_39 coefficients as matrix entries formula_40 of a matrix with row and column indices formula_41 and formula_42; then the largest eigenvalue of this matrix is known as the quantum dimension formula_43 of anyon type formula_13.
Fusion rules can also be generalized to consider in how many ways formula_45 a collection formula_46 can be fused to a final anyon type formula_10.
Hilbert spaces of fusion processes.
The fusion process where formula_13 and formula_14 fuse into formula_15 corresponds to a formula_39 dimensional complex vector space formula_52, consisting of all the distinct orthonormal quantum states in which formula_13 and formula_14 fuse into formula_15. This forms a Hilbert space. When formula_56, such as in the Ising and Fibonacci examples, formula_57 is at most just a one dimensional space with one state. The direct sum formula_58 is a decomposition of formula_59 the tensor product of the Hilbert space of individual anyon formula_13 and the Hilbert space of individual anyon formula_8. In topological quantum field theory, formula_62 is the vector space associated with the pair of pants with waist labeled formula_15 and legs formula_13 and formula_14.
More complicated Hilbert spaces can be constructed corresponding to the fusion of three or more particles, i.e. for the quantum systems where it is known that the formula_46 fuse into final anyon type formula_10. This Hilbert space formula_68 would describe, for example, the quantum system formed by starting with a quasiparticle formula_10 and, via some local physical procedure, splitting up that quasiparticle into quasiparticles formula_46 (because in such a system all the anyons must necessarily fuse back into formula_15 by topological invariance). There is an isomorphism between formula_72 and formula_73 for any formula_74. As mentioned in the previous section, the permutations of the labels are also isomorphic. 
One can understand the structure of formula_68 by considering fusion processes one pair of anyons at a time. There are many arbitrary ways one can do this, each of which can be used to derive a different decomposition of formula_76 into pairs of pants. One possible choice is to first fuse formula_77 and formula_78 into formula_79, then fuse formula_80 and formula_81 into formula_82, and so on. This approach shows us that formula_83, and correspondingly formula_84 where formula_85 is the matrix defined in the previous section.
This decomposition manifestly indicates a choice of basis for the Hilbert space. Different arbitrary choices of the order in which to fuse anyons will correspond to different choices of basis.