  We introduce a generalization of conventional lattice gauge theory to
describe fracton topological phases, which are characterized by immobile,
point-like topological excitations, and sub-extensive topological degeneracy.
We demonstrate a duality between fracton topological order and interacting spin
systems with symmetries along extensive, lower-dimensional subsystems, which
may be used to systematically search for and characterize fracton topological
phases. Commutative algebra and elementary algebraic geometry provide an
effective mathematical toolset for our results. Our work paves the way for
identifying possible material realizations of fracton topological phases.
