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

TL;DR

The paper advances topological holography from 2+1D gapped orders to 3+1D topological orders by developing and classifying string condensations in 3+1D $G$-gauge theories and their 2+1D dual gSPTs. It introduces a complete algebraic framework for magnetic simple condensations, parameterized by a normal subgroup $N\lhd G$ and a quadruple $(\alpha,\phi,\sigma,\beta)$ in $H^3_{qa}[(G,N),\mathbb{C}^\times]$, with Lagrangian cases $N=G$ reproducing the familiar $H^3[G,\mathbb{C}^\times]$ classification of 2+1D SPTs. The work establishes a precise SymTO/gSPT correspondence, uses the sandwich and club sandwich constructions to relate bulk string condensations to boundary gSPTs, and identifies three non-Lagrangian gSPT classes (type I–III) whose IR anomalies are captured by twists in post-condensation theories. Gauging gSPTs is analyzed via equivariantization, yielding pre-modular categories and clarifying when gauged theories fail to be modular. Together, these results extend topological holography to gapless phases, provide practical tools for constructing and classifying gSPTs, and deepen understanding of higher-categorical condensations and their physical content.

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.

Figures (2)

• Figure 1: The sandwich construction. Generalized charges of $\EuScript{S}$ correspond to topological excitations of the bulk $\mathcal{Z}[\EuScript{S}]$. The energy gap of the sandwich is infinite everywhere except in a neighbourhood of $\mathsf{B^{phys}}$. If $Q_1$ is condensed on $\mathsf{B^{sym}}$, then there exists a $p+1$-d operator connecting the two boundary, such that a single $Q_1$ is created on $\mathsf{B^{phys}}$(red). After compactifying the vertical direction, this operator becomes a $p$-d operator that creates a $p$-d charge. If $Q_2$ is not condensed on $\mathsf{B^{sym}}$, then its creation operator(blue) can not end on $\mathsf{B^{sym}}$, instead it continues to extend on $\mathsf{B^{sym}}$. After dimension reduction, this maps to a $p+1$-d half-infinite operator that creates a $p$-d charge on its boundary. By definition, nontrivial topological operators on $\mathsf{B^{sym}}$ are the symmetry operators. Therefore $Q_2$ is a generalized charge in the twisted sector.
• Figure 2: (a).The club sandwich construction for a gapless state. The bottom half of the sandwich is obtained by condensing a non-Lagrangian condensable algebra of $\mathcal{Z}[\EuScript{S}]$, and takes the form $\mathcal{Z}[\EuScript{S}']$. The bottom boundary of $\mathcal{Z}[\EuScript{S}']$ has nothing condensed, and has energy gap 0. The domain wall in the middle has a finite energy gap $\Delta_s$, which hosts the gapped symmetry charges of the system. Everywhere else in the sandwich the gap is infinite. A confined charge(red) can only be created on the domain wall with a finite energy gap, while a deconfined charge(yellow) can be created on the bottom boundary with zero energy gap. (b). After taking $\Delta_s\to \infty$ and shrinking the top half of the sandwich, we obtain a standard sandwich construction for a gapless state with $\EuScript{S}'$ symmetry. This shows the gapless degrees of freedom of the sandwich has an effective symmetry $\EuScript{S}'$, and has full categorical symmetry $\mathcal{Z}[\EuScript{S}']$ unbroken.