String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to $p$-String Condensation

TL;DR

This paper introduces a new approach to constructing 3+1D fracton phases by gauging higher-form symmetries, connecting topological order, string-net models, and higher-form gauge theories.

Contribution

It provides a controlled non-perturbative framework for $p$-string condensation via higher-form symmetry gauging, generalizing known 2D anyon condensation principles.

Findings

Derives foliated field theory for $ Z_N$ X-Cube model

Shows lattice gauging leads to string-membrane-net models

Generalizes to higher spatial dimensions $d \\geq 2$

Abstract

We present a new perspective on the $p$-string condensation procedure for constructing 3+1D fracton phases by implementing this process via the gauging of higher-form symmetries. Specifically, we show that gauging a 1-form symmetry in 3+1D that is generated by Abelian anyons in isotropic stacks of 2+1D topological orders naturally results in a 3+1D $p$-string condensed phase, providing a controlled non-perturbative construction that realizes fracton orders. This approach clarifies the symmetry principles underlying $p$-string condensation and generalizes the familiar connection between anyon condensation and one-form gauging in two spatial dimensions. We demonstrate this correspondence explicitly in both field theories and lattice models: in field theory, we derive the foliated field theory description of the $\mathbb{Z}_N$ X-Cube model by gauging a higher-form symmetry in stacks of 2+1D $\mathbb{Z}_N$ gauge theories; on the lattice, we show how gauging a diagonal 1-form symmetry in isotropic stacks of $G$-graded string-net models leads to string-membrane-nets hosting restricted mobility excitations. This perspective naturally generalizes to spatial dimensions $d \geq 2$ and provides a step towards building an algebraic theory of $p$-string condensation.