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On the structure of DHR bimodules of abstract spin chains

Lucas Hataishi, David Jaklitsch, Corey Jones, Makoto Yamashita

TL;DR

The paper analyzes the tensor categorical structure of DHR bimodules $\mathrm{DHR}(A_{\bullet})$ for abstract spin chains, identifying modularity under charge-transporter generation and algebraic Haag duality. It builds a half-line categorical framework $\mathcal{C}_-$ and $\mathcal{C}_+$, showing that under rationality these form fusion categories and, with locality, modular tensor categories; Haag duality ties $\mathrm{DHR}(A_{\bullet})$ to the Drinfeld center $\mathcal{Z}(\mathcal{C}_-)$, yielding Witt triviality. The work provides a fully faithful embedding of DHR into $\mathcal{Z}(\mathcal{C})$, nondegeneracy results, and an essentially surjective correspondence under suitable hypotheses, culminating in an equivalence with the Drinfeld center in favorable cases. Moreover, a Longo–Roberts style reconstruction from C$^*$-2-categories demonstrates how to realize abstract spin chains (not necessarily translation invariant) with controlled center and duality properties, generalizing fusion spin chains to broader categorical inputs and offering a holographic view on non-chiral local topological order.

Abstract

Abstract spin chains axiomatize the structure of local observables on the 1D lattice which are invariant under a global symmetry, and arise at the physical boundary of 2+1D topologically ordered spin systems. In this paper, we study tensor categorical properties of DHR bimodules over abstract spin chains. Assuming that the charge transporters generate the algebra of observables, we prove that the associated category has a structure of modular tensor category with respect to the natural braiding. Under an additional assumption of algebraic Haag duality, this category becomes the Drinfeld center of the half-line fusion category.

On the structure of DHR bimodules of abstract spin chains

TL;DR

The paper analyzes the tensor categorical structure of DHR bimodules for abstract spin chains, identifying modularity under charge-transporter generation and algebraic Haag duality. It builds a half-line categorical framework and , showing that under rationality these form fusion categories and, with locality, modular tensor categories; Haag duality ties to the Drinfeld center , yielding Witt triviality. The work provides a fully faithful embedding of DHR into , nondegeneracy results, and an essentially surjective correspondence under suitable hypotheses, culminating in an equivalence with the Drinfeld center in favorable cases. Moreover, a Longo–Roberts style reconstruction from C-2-categories demonstrates how to realize abstract spin chains (not necessarily translation invariant) with controlled center and duality properties, generalizing fusion spin chains to broader categorical inputs and offering a holographic view on non-chiral local topological order.

Abstract

Abstract spin chains axiomatize the structure of local observables on the 1D lattice which are invariant under a global symmetry, and arise at the physical boundary of 2+1D topologically ordered spin systems. In this paper, we study tensor categorical properties of DHR bimodules over abstract spin chains. Assuming that the charge transporters generate the algebra of observables, we prove that the associated category has a structure of modular tensor category with respect to the natural braiding. Under an additional assumption of algebraic Haag duality, this category becomes the Drinfeld center of the half-line fusion category.

Paper Structure

This paper contains 25 sections, 45 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Abstract spin chains
  4. DHR bimodules
  5. Charge transporters
  6. Distinguished algebra objects
  7. Lagrangian algebra in the Drinfeld center
  8. Half-line algebras and Jones's basic extension
  9. Half-line categories
  10. General setting
  11. Subalgebra from change transporters
  12. Finiteness of the DHR category
  13. Equivalence between C+ and C- under algebraic Haag duality
  14. Comparison with the Drinfeld center
  15. Fully faithful embedding into the center
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $A_{\bullet}$ be an abstract spin chain with charge transporter generation (see Condition cond:reduced-net). Then

Figures (1)

  • Figure 1: Effect of $F(Z^\mathrm{reg}) \to F'(Z^\mathrm{reg})$ on the image of $a \otimes b$

Theorems & Definitions (100)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • Proposition 2.9
  • proof
  • Proposition 3.1
  • proof
  • ...and 90 more