Foliated fracton order is a qualitatively new kind of phase of matter. It is similar to topological order, but with the fundamental difference that a layered structure, referred to as a foliation, plays an essential role and determines the mobility restrictions of the topological excitations. In this work, we introduce a new kind of field theory to describe these phases: a foliated field theory. We also introduce a new lattice model and string-membrane-net condensation picture of these phases, which is analogous to the string-net condensation picture of topological order.
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 $d\ensuremath{\ge}2$ and provides a step towards building an algebraic theory of
p-string condensation.