Title: Fracton Models on General Three-Dimensional Manifolds,
Abstract: Fracton models, a collection of exotic gapped lattice Hamiltonians recently
discovered in three spatial dimensions, contain some 'topological' features:
they support fractional bulk excitations (dubbed fractons), and a ground state
degeneracy that is robust to local perturbations. However, because previous
fracton models have only been defined and analyzed on a cubic lattice with
periodic boundary conditions, it is unclear to what extent a notion of topology
is applicable. In this paper, we demonstrate that the X-cube model, a
prototypical type-I fracton model, can be defined on general three-dimensional
manifolds. Our construction revolves around the notion of a singular compact
total foliation of the spatial manifold, which constructs a lattice from
intersecting stacks of parallel surfaces called leaves. We find that the ground
state degeneracy depends on the topology of the leaves and the pattern of leaf
intersections. We further show that such a dependence can be understood from a
renormalization group transformation for the X-cube model, wherein the system
size can be changed by adding or removing 2D layers of topological states. Our
results lead to an improved definition of fracton phase and bring to the fore
the topological nature of fracton orders.