  We present a theoretical framework for a class of generalized $U(1)$ gauge
effective field theories. These theories are defined by specifying geometric
patterns of charge configurations that can be created by local operators, which
then lead to a class of generalized Gauss law constraints. The charge and
magnetic excitations in these theories have restricted, subdimensional
dynamics, providing a generalization of recently studied higher-rank symmetric
$U(1)$ gauge theories to the case where arbitrary spatial rotational symmetries
are broken. These theories can describe situations where charges exist at the
corners of fractal operators, thus providing a continuum effective field
theoretic description of Haah's code and Yoshida's Sierpinski prism model. We
also present a $3+1$-dimensional $U(1)$ theory that does not have a non-trivial
discrete $\mathbb{Z}_p$ counterpart.
