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

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 ${\mathbb{Z}}_N$ X-cube model by gauging a higher-form symmetry in stacks of (2+1)D ${\mathbb{Z}}_N$ 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 $d\ensuremath{\ge}2$ and provides a step towards building an algebraic theory of $p$-string condensation.