Two-dimensional topologically ordered states such as fractional quantum Hall fluids host anyonic excitations, which are relevant for realizing fault-tolerant topological quantum computers. Classification and characterization of topological orders have been intensely pursued in both the condensed matter and mathematics literature. These topological orders can be bosonic or fermionic depending on whether the system hosts fundamental fermionic excitations or not. In particular, emergent topological orders in usual solid state systems are fermionic topological orders because the electron is a fermion. Recently, bosonic topological orders have been extensively completely classified up to rank 6 using representation theory. Inspired by their method, we provide in this paper a systematic method to classify the fermionic topological orders by explicitly building their modular data, which encodes the self and mutual statistics between anyons. Our construction of the modular data relies on the fact that the modular data of a fermionic topological order forms a projective representation of the ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\theta}}$ subgroup of the modular group ${\mathrm{SL}}_{2}(\mathbb{Z})$. We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod $\frac{1}{2}$) via a novel method, which does not require the explicit computation of modular extensions.
Two-dimensional topologically ordered states such as fractional quantum Hall fluids host anyonic excitations, which are relevant for realizing fault-tolerant topological quantum computers. Classification and characterization of topological orders have been intensely pursued in both the condensed matter and mathematics literature. These topological orders can be bosonic or fermionic depending on whether the system hosts fundamental fermionic excitations or not. In particular, emergent topological orders in usual solid state systems are fermionic topological orders because the electron is a fermion. Recently, bosonic topological orders have been extensively completely classified up to rank 6 using representation theory. Inspired by their method, we provide in this paper a systematic method to classify the fermionic topological orders by explicitly building their modular data, which encodes the self and mutual statistics between anyons. Our construction of the modular data relies on the fact that the modular data of a fermionic topological order forms a projective representation of the \ensuremathΓ\ensuremathθ subgroup of the modular group SL2(Z). We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod 21) via a novel method, which does not require the explicit computation of modular extensions.
We present an efficient method to compute the modular extension of both fermionic topological orders and Z2-symmetric bosonic topological orders in two spatial dimensions, from congruence representations of SL2(Z) and its subgroups. To demonstrate the validity of our approach, we provide explicit calculations for topological orders with ranks up to 10 for the fermionic cases and up to 6 for the bosonic cases. Along the way, we clarify the relation between fermionic rational conformal field theories, which live on the boundary of the corresponding fermionic topological orders, and modular extensions. We show that the SL2(Z) representation of the Ramond-Ramond sector can be determined from the Nuveu-Schwarz-Nuveu-Schwarz sector using the modular extensions.
We classify possible topological quantum field theories in 2+1 dimensions with up to 22 anyons in the bosonic case and up to 26 in the fermionic case, through the modular data of modular and supermodular tensor categories. We construct such data via congruence representations of SL(2,Z) using a numerical optimization algorithm that scales to high rank. We chart the relations between the resulting theories, including anyon condensations and Chern-Simons realizations. We find examples of modular data that do not appear to arise from any Chern-Simons theory by any sequence of condensations of anyons or gaugings of ordinary global symmetries. These examples give qualitatively new topological quantum field theories, which go beyond any notion of gauge theory. For examples in this class with 11 and 12 anyon types, we solve the pentagon and hexagon equations explicitly, showing that these modular data are realized by full unitary topological quantum field theories. We also introduce https://www.AnyonAtlas.org, an online resource for exploring the data.
We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category E to describe the symmetry. Here, E=sRep(Z2f) describing particles in a fermionic product state without symmetry, or E=sRep(Gf)[E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry E are classified by nondegenerate unitary braided fusion categories over E, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p+ip fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c=41, whose only topological excitation has non-Abelian statistics with spin s=41 and quantum dimension d=1+2.
The existence of gapped boundaries of bosonic topological orders can be tested in terms of the vanishing of higher central charges, which can be easily computed in terms of the modular data. For fermionic topological orders, even the chiral central charge admits no simple expression in terms of the modular data. Using the congruence property of representations formed by the modular data, we develop a method that tests whether the higher central charges of a fermionic topological order, including the chiral central charge, vanish. The test can be carried out entirely in terms of the modular data of the super-modular tensor category describing the fermionic topological order, and does not require explicit computation of modular extensions. This leads to a stringent set of easily computable necessary conditions for a fermionic topological order to admit a gapped boundary. We apply our test to known examples of fermionic topological orders to determine which of them potentially admit a gapped boundary.
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/