Abstract In this chapter we begin the detailed structure of anyon theories, or TQFTs. We introduce the concept of fusion and explain how it is related to the Hilbert space of the anyon systems. We emphasize the importance of locality as a principle for understanding anyons. Associativity is a strong constraint on possible particle fusion rules. As examples, we introduce both Fibonacci anyons and Ising anyons. We explain how the fusion rules of an anyon theory can be used to calculate the degeneracy of the ground state on arbitrary 2-manifolds.
To the working physicist, anyon theory is meant to describe certain quasi-particle excitations occurring in two dimensional topologically ordered systems. A typical calculation using this theory will involve operations such as
⊗ to combine anyons,
Fdabc to re-associate such combinations, and
Rcab to commute or braid these anyons. Although there is a powerful string-diagram notation that greatly assists these manipulations, we still appear to be operating on particles arranged on a one-dimensional line, algebraically ordered from left to right. The obvious question is, where is the other dimension? The topological framework for considering these anyons as truly living in a two dimensional space is known as a modular functor, or topological quantum field theory. In this work we show how the apparently one-dimensional algebraic anyon theory is secretly the theory of anyons living in a fully two-dimensional system. The mathematical literature covering this secret is vast, and we try to distill this down into something more manageable.
The term ‘TQC' means the topological quantum computation, which is one of remarkable theories to realize quantum computers. This theory is related to various fields; theory of quantum groups, Lie algebras, Hopf algebras and conformal field or string theory. But there are few textbooks which readers of other fields can read easily. We wrote a survay[10]easy to observe from the basics of TQC theory to basic quantum gates for quantum computers. But we cannot explain modular S-matrices that is one of the keys in TQC in the preceding survey. So, in this paper, we try to express total basic issue including theories around modular S-matrix. In the many papers for TQC, there are some subtle mistakes and we correct them in this paper. At first we introduce formulae for modular S-matrix in the axiomatic (categorical) approach. Secondly we give an important model for TQC; spin network model based on the Temperly-Lieb-Jones algebra. Finally we mention another important model suk(2). In each levels, Fibannaci anyon is a basic example to explain them.
Abstract Ideas similar to that of the Kauffman bracket introduced in chapter 2 can be used to generate valid anyon theories. We begin with a planar diagram algebra defined only by the value of a loop (the loop weight). One wants to build new particle types by grouping together multiple strands of loops. However, to do so one needs so-called Jones-Wenzl projectors to orthogonalize between species resulting in the so-called Temperly-Lieb Algebra. We demonstrate this construction by building a planar diagram algebra corresponding to the Z2 loop gas (bosons, fermions, semions), and then Ising Anyons. We show how this generalizes to a much bigger set of possible theories. We show how these theories can be made unitary, and how the F-matrices may be calculated. Finally we discuss how the R-matrices can be simply obtained by returning to the Kauffman bracket rules.
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/
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.