| Summary: | We introduce a notion of homogeneous topological order,
which is obeyed by most, if not all, known examples of topological order
including fracton phases on quantum spins (qudits).
The notion is a condition on the ground state subspace, rather than on the Hamiltonian,
and demands that given a collection of ball-like regions,
any linear transformation on the ground space be
realized by an operator that avoids the ball-like regions.
We derive a bound on the ground state degeneracy $\mathcal D$
for systems with homogeneous topological order
on an arbitrary closed Riemannian manifold of dimension $d$,
which reads \[ \log \mathcal D \le c \mu (L/a)^{d-2}.\]
Here, $L$ is the diameter of the system, $a$ is the lattice spacing,
and $c$ is a constant that only depends on the isometry class of the manifold,
and $\mu$ is a constant that only depends on the density of degrees of freedom.
If $d=2$, the constant $c$ is the (demi)genus of the space manifold.
This bound is saturated up to constants by known examples.
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