Anyons in
2+1 dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by
F-symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The
F-symbols for many of the most common
2+1d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose
F-symbols cannot be made real. We call such
F-symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real
F-symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex
F-symbols:
Rep(Z7⋊Z3) and
Rep(Z5⋊Z4). Consequently, the corresponding
Z(Rep(Z7⋊Z3)) and
Z(Rep(Z5⋊Z4)) anyon models also have inherently complex
F-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.
Until recently, a careful derivation of the fusion structure of anyons from some underlying physical principles has been lacking. In Shi et al. (2020), the authors achieved this goal by starting from a conjectured form of entanglement area law for 2D gapped systems. In this work, we instead start with the principle of exchange symmetry, and determine the minimal prescription of additional postulates needed to make contact with unitary ribbon fusion categories as the appropriate algebraic framework for modelling anyons. Assuming that 2D quasiparticles are spatially localised, we build a functor from the coloured braid groupoid to the category of finite-dimensional Hilbert spaces. Using this functor, we construct a precise notion of exchange symmetry, allowing us to recover the core fusion properties of anyons. In particular, given a system of n quasiparticles, we show that the action of a certain n-braid βn uniquely specifies its superselection sectors. We then provide an overview of the braiding and fusion structure of anyons in the usual setting of braided 6j fusion systems. By positing the duality axiom of Kitaev (2006) and assuming that there are finitely many distinct topological charges, we arrive at the framework of ribbon categories.
We construct large families of mixed-state topological orders in (3+1)D, including intrinsically mixed-state topological orders. We realize these theories by relating the data parameterizing braided fusion 2-categories to the strong higher-form symmetries and their anomalies that characterize mixed-state topological order. We focus here on bosonic theories, which include theories which admit an emergent fermion in the spectrum of line operators. Our general framework is exemplified on the lattice, where we present concrete models and identify how the data of braided fusion 2-categories are physically encoded. We then formalize this data and show that every braided fusion 2-category arises from a general construction (decoherence followed by strong and weak gauging) as the category of topological operators of a (3+1)D mixed state, with intrinsically mixed-state topological orders distinguished by the presence of strong higher-form symmetries whose anomalies cannot be realized in any (3+1)D gapped ground state. Notably, we show that (3+1)D admits a new mechanism with no (2+1)D counterpart: intrinsically mixed-state order whose topological operators braid non-degenerately, but which carries a gravitational anomaly.
We investigate whether the braid group representation on
N anyons uniquely determines the modular data (
S and
T matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local
N-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global
S-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon
R-matrix with the asymptotic Hecke relation
σi2=1 (up to phase), which rigidly fixes the local
F and
R symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (
σ×σ=1+ψ), precluding alternative non-Abelian fusion rules. The
T-matrix is determined by topological spins, but the full
S-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory. ## 1. Introduction Topological quantum computation exploits the non-Abelian statistics of anyons—quasiparticle excitations in two-dimensional systems whose braiding operations are represented by non-commuting unitary matrices on a degenerate ground-state manifold [1]. A central question is whether the local braiding data of
N anyons suffices to reconstruct the full modular data of the underlying topological order (TO), or whether additional global information is required. This question has direct physical consequences for 2D topological superconductors (TSCs) hosting Majorana zero modes (MZMs), where the anyonic excitations are vortices binding MZMs and the fermionic quasiparticle is a transparent (local) fermion [2,3]. The modular data of a TO consists of the
S-matrix, which encodes the mutual braiding statistics of anyons, and the
T-matrix, which encodes the topological spins. Together, these matrices form a projective representation of t
Majorana zero modes (MZMs) localized on vortices of a two-dimensional topological superconductor carry a projective representation of the braid group, and the associated anyon theory is encoded in a modular tensor category whose fusion rules and modular data (
S and
T matrices) determine the topological order. We address two questions: whether the braid group representation on
N anyons uniquely determines the modular data of the underlying non-Abelian topological order, and whether a representation-theoretic framework can classify all possible fusion rules for MZMs in 2D topological superconductors. We develop a classification pipeline that (i) fixes the local MZM algebra as the Clifford algebra generated by
2n Majorana operators, (ii) imposes the pentagon and hexagon coherence equations on candidate braided fusion structures, and (iii) filters candidates by modular-data consistency conditions, including the Verlinde formula and Frobenius–Schur indicators. As a fully worked example we compute, with explicit arithmetic, the modular
S matrix and total quantum dimension for the Ising-type theory relevant to chiral
p-wave superconductors, obtaining
D=2 and verifying unitarity of
S to machine precision. We then analyze metaplectic categories
SO(N)2, which generalize the Ising/MZM fusion pattern, and show how Property F (finiteness of braid images) constrains the classification. We argue that the framework classifies MZM fusion rules up to gauging and extension operations, but that non-semisimple and non-unitary candidates escape the semisimple machinery; we state falsification criteria and open problems. ## 1. Introduction A two-dimensional topological superconductor (2D TSC) supports vortices that bind Majorana zero modes (MZMs): self-adjoint fermionic operators
γi=γi† obeying the Clifford algebra
{γi,γj}=2δij. Exchanging vortices
i and
j acts on the ground-state manifold by the operato Full text and updates: papers.qnfo.org/papers/braid-group-representations-modular-data-and-the-classification-of-majo-5099abb8/
A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this dissertation, we provide a comprehensive review of the fundamentals of logic design in topological quantum computing. The braid group and knot invariants in the skein theory are discussed. The physical insight behind the braiding is explained by the geometric phases and the gauge transformation. The mathematical models for the fusion and braiding are presented in terms of the category theory and the quantum deformation of the recoupling theory. The topological phases of matter are described by the topology of band structure. The wave function of quasiparticles in the quantum Hall effect and the theory of Majorana fermions in topological superconductors are also discussed. The dynamics of the charge-flux composites and their Hilbert space are expressed through the Chern-Simons theory and the two-dimensional topological quantum field theory. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. We reduced the quantum cost of the existing ternary quantum arithmetic gates and proposed that these gates can be implemented with the metaplectic anyons.