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Boundary-bulk relation in topological orders

Liang Kong, Xiao-Gang Wen, Hao Zheng

TL;DR

The paper argues that for an anomaly-free n+1D bulk corresponding to a given nD boundary, the bulk is unique and is the center of the boundary phase. It formalizes morphisms between topological orders via gapped domain walls and constructs a canonical $\rho$ map to a restricted bulk, then proves that the bulk satisfies the universal center property, yielding bulk = center in all dimensions. This center-based bulk identification is independent of how boundary and bulk are described mathematically and leads to concrete physical predictions, including constraints on higher-dimensional lattice models like Walker-Wang. The results unify known 2+1D bulk-boundary relations with a general, dimension-agnostic center framework and point to a functorial perspective on boundary-bulk duality, with extensions to gapless boundaries.

Abstract

In this paper, we study the relation between an anomaly-free $n+$1D topological order, which are often called $n+$1D topological order in physics literature, and its $n$D gapped boundary phases. We argue that the $n+$1D bulk anomaly-free topological order for a given $n$D gapped boundary phase is unique. This uniqueness defines the notion of the "bulk" for a given gapped boundary phase. In this paper, we show that the $n+$1D "bulk" phase is given by the "center" of the $n$D boundary phase. In other words, the geometric notion of the "bulk" corresponds precisely to the algebraic notion of the "center". We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the "bulk" satisfies the same universal property as that of the "center" of an algebra in mathematics, i.e. "bulk = center". The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.

Boundary-bulk relation in topological orders

TL;DR

The paper argues that for an anomaly-free n+1D bulk corresponding to a given nD boundary, the bulk is unique and is the center of the boundary phase. It formalizes morphisms between topological orders via gapped domain walls and constructs a canonical map to a restricted bulk, then proves that the bulk satisfies the universal center property, yielding bulk = center in all dimensions. This center-based bulk identification is independent of how boundary and bulk are described mathematically and leads to concrete physical predictions, including constraints on higher-dimensional lattice models like Walker-Wang. The results unify known 2+1D bulk-boundary relations with a general, dimension-agnostic center framework and point to a functorial perspective on boundary-bulk duality, with extensions to gapless boundaries.

Abstract

In this paper, we study the relation between an anomaly-free 1D topological order, which are often called 1D topological order in physics literature, and its D gapped boundary phases. We argue that the 1D bulk anomaly-free topological order for a given D gapped boundary phase is unique. This uniqueness defines the notion of the "bulk" for a given gapped boundary phase. In this paper, we show that the 1D "bulk" phase is given by the "center" of the D boundary phase. In other words, the geometric notion of the "bulk" corresponds precisely to the algebraic notion of the "center". We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the "bulk" satisfies the same universal property as that of the "center" of an algebra in mathematics, i.e. "bulk = center". The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.

Paper Structure

This paper contains 5 sections, 1 theorem, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basics of topological orders
  3. The universal property of the center of an algebra
  4. A morphism between two topological orders
  5. bulk = center

Key Result

Theorem 5.1

The pair $(P_n(\mathfrak{Z}_n(\EuScript{C}_n)), \rho)$ satisfies the universal property of the center. More precisely, if $(\EuScript{X}_n, f)$ is another pair, where $\EuScript{X}_n$ is an $n$D topological order and $f: \EuScript{X}_n \boxtimes \EuScript{C}_n \to \EuScript{C}_n$ is a morphism and a

Figures (1)

  • Figure 1: $\EuScript{M}_n$ is an invertible domain wall between two anomaly-free $n+$1D topological orders $\EuScript{C}_{n+1}$ and $\EuScript{D}_{n+1}$, and $\EuScript{N}_n$ is its inverse. The "$\times$" in these pictures represents a non-trivial topological excitation in the $\EuScript{C}_{n+1}$-phase. Note that the non-trivialness requires "$\times$" to be at least 2-codimensional. These pictures depict a process of the excitation "$\times$" tunneling through the $\EuScript{M}_n$ wall. In particular, the second "$\rightsquigarrow$" is obtained by annihilating $\EuScript{N}_n$ with the $\EuScript{M}_n$ on the right side of $\EuScript{N}_n$. Similarly, there is a tunneling process from $\EuScript{D}_{n+1}$ to $\EuScript{C}_{n+1}$, which is inverse to it. This gives a way to identify the $\EuScript{C}_{n+1}$-phase with the $\EuScript{D}_{n+1}$-phase.

Theorems & Definitions (16)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Example 4.4
  • Example 4.5
  • Theorem 5.1
  • proof
  • ...and 6 more