arXiv:2607.10181

Anyons and Inherently Complex F-symbols

Matthew Buican, Peter Huston, Jiannis K. Pachos

cond-mat.str-elhep-thmath-phmath.QAquant-phSubmitted Jul 11, 2026• Announced Jul 14, 2026• 0 citations• Revised Aug 23, 2026 (v2)

arXiv HTML PDF

Abstract

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 $2+1$d 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: ${\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)$ and ${\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)$. Consequently, the corresponding $\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))$ and $\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))$ anyon models also have inherently complex $F$-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.

Comments: 25 pages; 1 figure; v2: typos fixed and minor clarifications added;

Related papers

Reality and Complexity of $F$-symbols in $2+1$d Topological Phases

Matthew Buican, Peter Huston, Jiannis K. Pachos

arXiv:2608.23693Aug 24, 20260 citations

Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors

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Sep 7, 20260 citations

Braid Group Representations and Modular Data: Non-Uniqueness, Finite Images, and the Ising Test Case

Rowan Brad Quni-Gudzinas

Oct 1, 20260 citations

Metaplectic anyons, Majorana zero modes, and their computational power

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Universal Gates from Braiding and Fusing Anyons on Quantum Hardware

Chiu Fan Bowen Lo, Anasuya Lyons, Dan Gresh +8 more

quant-pharXiv:2601.20956NatureJan 30, 20260 citations

Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors

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Oct 1, 20260 citations

arXiv:2607.10181

Anyons and Inherently Complex F-symbols

Matthew Buican, Peter Huston, Jiannis K. Pachos

cond-mat.str-elhep-thmath-phmath.QAquant-phSubmitted Jul 11, 2026• Announced Jul 14, 2026• 0 citations• Revised Aug 23, 2026 (v2)

arXiv HTML PDF

Abstract

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+1

d 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:

{\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)

and

{\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)

. Consequently, the corresponding

\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))

and

\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))

anyon models also have inherently complex

F

-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.

Comments: 25 pages; 1 figure; v2: typos fixed and minor clarifications added;

Related papers

Reality and Complexity of $F$ -symbols in $2+1$ d Topological PhasesReality and Complexity of $F$-symbols in $2+1$d Topological Phases

Matthew Buican, Peter Huston, Jiannis K. Pachos

The $F$ -symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the $F$ -symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all $F$ -symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated $F$ -symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,'' or equivalently when the associated "twisted'' Frobenius-Schur (or "generalized'' Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose $F$ -symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex $F$ -symbols.The $F$-symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the $F$-symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all $F$-symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated $F$-symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,'' or equivalently when the associated "twisted'' Frobenius-Schur (or "generalized'' Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose $F$-symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex $F$-symbols.

arXiv:2608.23693Aug 24, 20260 citations

Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors

Rowan Brad Quni-Gudzinas

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 $\sigma_i^2 = 1$ (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 ( $\sigma \times \sigma = 1 + \psi$ ), 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 tWe 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 $\sigma_i^2 = 1$ (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 ($\sigma \times \sigma = 1 + \psi$), 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

Sep 7, 20260 citations

Braid Group Representations and Modular Data: Non-Uniqueness, Finite Images, and the Ising Test Case

Rowan Brad Quni-Gudzinas

A central question in the theory of topological order asks whether the braid group representation $B_N$ carried by $N$ non-Abelian anyons uniquely determines the modular data — the $S$ and $T$ matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map $\Phi$ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the $SO(N)_2$ / $O(N)_2$ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins $\theta_\sigma = e^{i\pi/8}$ , $\theta_\psi = -1$ , the total quantum dimension $\mathcal{D}=2$ , and the full $S$ matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation $\mathrm{Ising}(7)$ with identical fusion rules but distinct $S$ matrix, and verify that its braid representation is inequivalent to the Ising representation on $B_3$ by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules $N_{ab}^c$ specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/A central question in the theory of topological order asks whether the braid group representation $B_N$ carried by $N$ non-Abelian anyons uniquely determines the modular data — the $S$ and $T$ matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map $\Phi$ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the $SO(N)_2$/$O(N)_2$ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins $\theta_\sigma = e^{i\pi/8}$, $\theta_\psi = -1$, the total quantum dimension $\mathcal{D}=2$, and the full $S$ matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation $\mathrm{Ising}(7)$ with identical fusion rules but distinct $S$ matrix, and verify that its braid representation is inequivalent to the Ising representation on $B_3$ by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules $N_{ab}^c$ specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/

Oct 1, 20260 citations

Metaplectic anyons, Majorana zero modes, and their computational power

Matthew B. Hastings, Chetan Nayak, Zhenghan Wang

We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with $\mathit{\text{SO}}{(m)}_{2}$ Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticle types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of $2n$ fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})\ensuremath{\ltimes}H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$, where $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the symplectic group over the finite field ${\mathbb{F}}_{m}$ and $H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the extra special group (also called the $(2n\ensuremath{-}1)$-dimensional Heisenberg group) over ${\mathbb{F}}_{m}$ . Moreover, the braiding of fundamental quasiparticles combined with a restricted set of measurements can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle followed by fusion into the identity is $#P$-hard. It is not universal for quantum computation because it has a finite braid group image. This is a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $#P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with $\mathit{\text{SO}}{(m)}_{2}$ Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticle types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of $2n$ fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})\ensuremath{\ltimes}H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$, where $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the symplectic group over the finite field ${\mathbb{F}}_{m}$ and $H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the extra special group (also called the $(2n\ensuremath{-}1)$-dimensional Heisenberg group) over ${\mathbb{F}}_{m}$. Moreover, the braiding of fundamental quasiparticles combined with a restricted set of measurements can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle followed by fusion into the identity is $#P$-hard. It is not universal for quantum computation because it has a finite braid group image. This is a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $#P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.

arXiv:1210.5477Physical Review BApr 11, 201352 citations

Universal Gates from Braiding and Fusing Anyons on Quantum Hardware

Chiu Fan Bowen Lo, Anasuya Lyons, Dan Gresh +8 more

Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$ , on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

quant-pharXiv:2601.20956NatureJan 30, 20260 citations

Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors

Rowan Brad Quni-Gudzinas

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 $\mathcal{D} = 2$ 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 $\gamma_i = \gamma_i^\dagger$ obeying the Clifford algebra $\{\gamma_i,\gamma_j\} = 2\delta_{ij}$ . 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/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 $\mathcal{D} = 2$ 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 $\gamma_i = \gamma_i^\dagger$ obeying the Clifford algebra $\{\gamma_i,\gamma_j\} = 2\delta_{ij}$. 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/

Oct 1, 20260 citations

Anyons in

2+1

F

F

-symbols for many of the most common

2+1

F

-symbols cannot be made real. We call such

F

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:

{\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)

and

{\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)

. Consequently, the corresponding

\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))

and

\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))

anyon models also have inherently complex

F

-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.

Reality and Complexity of

F

-symbols in

2+1

d Topological Phases

The

F

-symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the

F

-symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all

F

-symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated

F

-symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,'' or equivalently when the associated "twisted'' Frobenius-Schur (or "generalized'' Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose

F

-symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex

F

-symbols.

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

\sigma_i^2 = 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 (

\sigma \times \sigma = 1 + \psi

), 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

A central question in the theory of topological order asks whether the braid group representation

B_N

carried by

N

non-Abelian anyons uniquely determines the modular data — the

S

and

T

matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map

\Phi

from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the

SO(N)_2

O(N)_2

braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins

\theta_\sigma = e^{i\pi/8}

\theta_\psi = -1

, the total quantum dimension

\mathcal{D}=2

, and the full

S

matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation

\mathrm{Ising}(7)

with identical fusion rules but distinct

S

matrix, and verify that its braid representation is inequivalent to the Ising representation on

B_3

by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules

N_{ab}^c

specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/

\mathit{\text{SO}}{(m)}_{2}

Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticle types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of

2n

fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})\ensuremath{\ltimes}H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$, where $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the symplectic group over the finite field

{\mathbb{F}}_{m}

and $H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the extra special group (also called the $(2n\ensuremath{-}1)$-dimensional Heisenberg group) over

{\mathbb{F}}_{m}

. Moreover, the braiding of fundamental quasiparticles combined with a restricted set of measurements can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle followed by fusion into the identity is $#P$-hard. It is not universal for quantum computation because it has a finite braid group image. This is a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $#P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.

S_3

, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes

S_3

TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

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

\mathcal{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

\gamma_i = \gamma_i^\dagger

obeying the Clifford algebra

\{\gamma_i,\gamma_j\} = 2\delta_{ij}

. 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/