Boundary-bulk relation in topological orders

TL;DR

This paper establishes that the bulk of an anomaly-free topological order is uniquely determined by its boundary phase and is mathematically equivalent to the center of the boundary, linking geometric and algebraic concepts.

Contribution

It proves that the bulk of an anomaly-free topological order is given by the center of its boundary phase, providing a universal property that connects geometry and algebra.

Findings

Bulk is uniquely determined by boundary phase

Bulk corresponds to the algebraic center of boundary

Provides a universal property linking bulk and boundary

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.