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Two-dimensional topological order and operator algebras

Yasuyuki Kawahigashi

Abstract

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to 2-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

Two-dimensional topological order and operator algebras

Abstract

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to 2-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

Paper Structure

This paper contains 6 sections, 3 theorems, 5 equations, 6 figures.

Table of Contents

  1. Introduction to operator algebras
  2. Topological phases of matter and tensor categories
  3. Algebraic quantum field theory
  4. Topological phases of matter and gapped domain walls
  5. The $\alpha$-induction and anyon condensation
  6. Tensor networks and matrix product operators

Key Result

Theorem 6.1

Ocneanu's tube algebra for the fusion category arising from a subfactor and the anyon algebra of Bultinck et al. arising from its flat connections are isomorphic. In particular, the two fusion rules are identical and the Verlinde formula also holds for the setting of anyon algebra.

Figures (6)

  • Figure 1: A vector $v_j$ and a matrix $a_{jk}$
  • Figure 2: A matrix product $(ab)_{jl}$
  • Figure 3: A matrix product state
  • Figure 4: A matrix product operator
  • Figure 5: A flat connection on the Dynkin diagram $A_n$
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 6.1
  • Theorem 6.2
  • Theorem 6.3