String condensation and topological holography for 2+1D gapless SPT

TL;DR

This paper explores string condensation in 3+1D topological orders and its duality with 2+1D gapless SPT phases, providing an algebraic classification and revealing physical structures of string condensations.

Contribution

It introduces a duality between non-Lagrangian string condensable algebras in 3+1D and 2+1D gapless SPT phases, with a new algebraic classification framework.

Findings

Classified condensable algebras in 3+1D G-gauge theories

Mapped 3+1D string condensations to 2+1D G-symmetric phases

Identified three classes of gapless SPTs and analyzed their properties

Abstract

The theory of anyon condensation is the foundation of the bulk-boundary relation and topological holography in 2+1D/1+1D. It is believed string condensation should replace anyon condensation in the 3+1D/2+1D topological holography theory. In this work we study string condensations in 3+1D topological orders and their relations to 2+1D phases. We find that a class of non-Lagrangian condensable algebras in 3+1D are exactly dual to a class of 2+1D symmetry enriched gapless phases known as gapless SPTs(gSPT). We show how topological properties of a gSPT can be fully extracted from the dual string condensation. We give an algebraic classification of this class of condensable algebras in 3+1D $G$-gauge theories that we call magnetic and simple. Through the topological holography dictionary, this maps to the classification of 2+1D $G$-symmetric phases with no topological order, including gapped and gapless SPTs. Utilizing the classification, we identify three classes of gSPTs and study their properties and gauging. Along the way, we reveal physical structures of string condensations.