String membrane nets from higher-form gauging: An alternate route to p-string condensation

Abstract

We present a unique perspective on the 𝑝-string condensation procedure for constructing (3+1)-dimensional [(3+1)D] fracton phases by implementing this process via the gauging of higher-form symmetries. Specifically, we show that gauging a 1-form symmetry in (3+1)D that is generated by Abelian anyons in isotropic stacks of (2+1)-dimensional [(2+1)D] topological orders naturally results in a (3+1)D 𝑝-string condensed phase, providing a controlled nonperturbative construction that realizes fracton orders. This approach clarifies the symmetry principles underlying 𝑝-string condensation and generalizes the familiar connection between anyon condensation and 1-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 ℤ𝑁 X-cube model by gauging a higher-form symmetry in stacks of (2+1)D ℤ𝑁 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≥2 and provides a step towards building an algebraic theory of 𝑝-string condensation.

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