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Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers

Liang Kong, Xiao-Gang Wen, Hao Zheng

TL;DR

This work develops a higher-categorical framework to study the boundary-bulk relation for local topological orders across dimensions, introducing a unique-bulk hypothesis and proving that the bulk coincides with the mathematical center under a natural notion of morphisms. It constructs categories of topological orders, walls, and their higher morphisms, notably TO_n^{closed-wall} and TO_n^{fun}, and demonstrates the universal center property of the bulk, together with functoriality phenomena for Z_n across walls and codimensions. The approach provides a macroscopic classification of n+1D topological orders through unitary multi-fusion n-categories, connects to Morita/Witt theory, and points toward a higher-dimensional condensation framework, potentially unifying both physical and mathematical center concepts. The results lay groundwork for a systematic category-theoretic study of local topological orders and their boundary-bulk relations, with implications for condensation theory and symmetry-enriched higher-dimensional topological phases.

Abstract

In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of "bulk" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition) of $n+$1D local topological orders by unitary multi-fusion $n$-categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal $n$-functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of "bulk = center". In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical structure which can be summarized by the functoriality of the bulk (or center). This functoriality also provides the physical meanings of some well-known mathematical results on fusion 1-categories. This work can also be viewed as the first step towards a systematic study of the category of local topological orders, and the boundary-bulk relation actually provides a useful tool for this study.

Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers

TL;DR

This work develops a higher-categorical framework to study the boundary-bulk relation for local topological orders across dimensions, introducing a unique-bulk hypothesis and proving that the bulk coincides with the mathematical center under a natural notion of morphisms. It constructs categories of topological orders, walls, and their higher morphisms, notably TO_n^{closed-wall} and TO_n^{fun}, and demonstrates the universal center property of the bulk, together with functoriality phenomena for Z_n across walls and codimensions. The approach provides a macroscopic classification of n+1D topological orders through unitary multi-fusion n-categories, connects to Morita/Witt theory, and points toward a higher-dimensional condensation framework, potentially unifying both physical and mathematical center concepts. The results lay groundwork for a systematic category-theoretic study of local topological orders and their boundary-bulk relations, with implications for condensation theory and symmetry-enriched higher-dimensional topological phases.

Abstract

In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of "bulk" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition) of 1D local topological orders by unitary multi-fusion -categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal -functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of "bulk = center". In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical structure which can be summarized by the functoriality of the bulk (or center). This functoriality also provides the physical meanings of some well-known mathematical results on fusion 1-categories. This work can also be viewed as the first step towards a systematic study of the category of local topological orders, and the boundary-bulk relation actually provides a useful tool for this study.

Paper Structure

This paper contains 25 sections, 7 theorems, 46 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Definitions of topological orders
  3. A microscopic definition of a topological order
  4. Unitary multi-fusion $n$-categories
  5. A macroscopic definition of a topological order
  6. The category $\EuScript{TO}_n^{closed-wall}$ of topological orders
  7. Time-reverse of a topological order
  8. Dimensional reductions and tensor products
  9. Closed domain walls and the category $\EuScript{TO}_n^{closed-wall}$
  10. The category $\EuScript{TO}_n^{fun}$ of topological orders
  11. The unique-bulk hypothesis
  12. A morphism between two topological orders
  13. Higher morphisms and the $(n+1,1)$-category $\EuScript{TO}_n^{fun}$
  14. The universal property of the bulk
  15. The universal property of the bulk
  16. ...and 10 more sections

Key Result

Lemma 4.8

The composition of morphisms is associative and unital, i.e. $h\circ (g \circ f) = (h\circ g) \circ f$ and $\mathrm{id}_{\EuScript{C}_n} \circ f = f = f\circ \mathrm{id}_{\EuScript{D}_n}$ for all composable $f,g,h$.

Figures (12)

  • Figure 1:
  • Figure 2: Consider a bosonic Hamiltonian lattice model defined on the closed $n$-disk $\overline{D}^{n+1}$ and an open $(n-1)$-disk $L$ in $\partial \overline{D}^{n+1}=S^n$. The disk $L$ is depicted as an open line segment between $a$ and $b$ on $\partial \overline{D}^{n+1}$, where $a \cup b = \partial L = S^{n-1}$. The lattice model in a neighborhood of $L$ determines a topological order on the open $n$-disk $L$ (see Def. \ref{['def:TO']}).
  • Figure 3: A 1-codimensional excitation $x$ in an $n+$1D simple topological order $\EuScript{C}_{n+1}$ can be viewed as an $n$D topological order, denoted by $(P_n(\EuScript{C}_{n+1}), x)$.
  • Figure 5: Above figures illustrate the dimensional reduction process in a Levin-Wen model discussed in Example \ref{['expl:lw-mod']}. $\EuScript{E}_2$ is given by (\ref{['eq:lw-mod-dim-red']}).
  • Figure 6: A closed gapped domain wall (or simply a domain wall) $\EuScript{M}_{n-1}$ between the $\EuScript{C}_n$-phase and the $\EuScript{D}_n$-phase sharing the same bulk $\EuScript{A}_{n+1}$.
  • ...and 7 more figures

Theorems & Definitions (108)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • ...and 98 more