  We investigate the rich quantum phase diagram of Wegner's theory of discrete
Ising gauge fields interacting with $U(1)$ symmetric single-component fermion
matter hopping on a two-dimensional square lattice. In particular limits the
model reduces to (i) pure $\mathbb{Z}_2$ even and odd gauge theories, (ii) free
fermions in a static background of deconfined $\mathbb{Z}_2$ gauge fields,
(iii) the kinetic Rokhsar-Kivelson quantum dimer model at a generic dimer
filling. We develop a local transformation that maps the lattice gauge theory
onto a model of $\mathbb{Z}_2$ gauge-invariant spin $1/2$ degrees of freedom.
Using the mapping, we perform numerical density matrix renormalization group
calculations that corroborate our understanding of the limits identified above.
Moreover, in the absence of the magnetic plaquette term, we reveal signatures
of topologically ordered Dirac semimetal and staggered Mott insulator phases at
half-filling. At strong coupling, the lattice gauge theory displays fracton
phenomenology with isolated fermions being completely frozen and dimers
exhibiting restricted mobility. In that limit, we predict that in the ground
state dimers form compact clusters, whose hopping is suppressed exponentially
in their size. We determine the band structure of the smallest clusters
numerically using exact diagonalization. The rich phenomenology discussed in
this paper can be probed in analog and digital quantum simulators of discrete
gauge theories and in Kitaev spin-orbital liquids.
