A central question in the theory of topological order asks whether the braid group representation
BN 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
Φ 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
θσ=eiπ/8,
θψ=−1, the total quantum dimension
D=2, and the full
S matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation
Ising(7) with identical fusion rules but distinct
S matrix, and verify that its braid representation is inequivalent to the Ising representation on
B3 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
Nabc specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/
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/
[CORRECTION NOTE, June 2026: This companion paper inherits the corrected classification: the object 1 + ((1,1);1) (d_A = 4) is NOT an algebra (its generator fails fusion closure); the correct algebras are the Z3 simple-current algebra (d_A = 3) and the maximal algebra (d_A = 6). The condensed phase is the nu=11 Ising-type category of the sixteen-fold way - the modular tensor category of SO(11)_1 (h_sigma = 11/16), fixed by the preserved central charge c = 11/2 mod 8 - which shares its fusion rules with Ising and its fusion category with SU(2)_2 but is equivalent to neither as a braided category (the claim 'matching the Ising MTC (equivalently SU(2)_2)' below is corrected accordingly). See DOI 10.5281/zenodo.20618695 (concept 10.5281/zenodo.20128951). The text below is the original and is retained for the record.] We compute the local module category ClocA for the condensable algebra A = 1 ⊕ ((1,1);1) in the product modular tensor category SU(3)3 ⊗ SU(2)2. Starting from the rank-30 parent theory with total quantum dimension D2 = 144, we identify the unique modular invariant Z commuting with the modular S- and T-matrices, extract a 30×3 branching matrix, and compute the condensed S-matrix via Sloc = (1/dA)bTSb. The result is a rank-3 modular tensor category with quantum dimensions (1, 1, √2), total quantum dimension D2 = 4, and modular data matching the Ising MTC (equivalently SU(2)2) to machine precision. The D2 discrepancy between intrinsic (= 4) and extrinsic (= 9) values is resolved by the embedding dimension factor (d0embed)2 = 9/4. This is a companion paper to 'Anyon condensation in SU(3)3 ⊗ SU(2)2: condensable algebras, monodromy spectrum, and the Z6 fusion ring' (Bazih, 2026), resolving its Remark 3.4 on the structure of ClocA.