fcc lattice, checkerboards, fractons, and quantum field theory

We consider XY-spin degrees of freedom on an fcc lattice, such that the system respects some subsystem global symmetry. We then gauge this global symmetry and study the corresponding $U(1)$ gauge theory on the fcc lattice. Surprisingly, this $U(1)$ gauge theory is dual to the original spin system. We also analyze a similar ${\mathbb{Z}}_{N}$ gauge theory on that lattice. All these systems are fractonic. The $U(1)$ theories are gapless and the ${\mathbb{Z}}_{N}$ theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics. Our ${\mathbb{Z}}_{2}$ fcc gauge theory is the continuum limit of the well-known checkerboard model of fractons. Our continuum analysis leads to a straightforward proof of the known fact that this theory is dual to two copies of the ${\mathbb{Z}}_{2}$ X-cube model. We find new models and new relations between known models. The ${\mathbb{Z}}_{N}$ fcc gauge theory can be realized by coupling three copies of an anisotropic model of lineons and planons to a certain exotic ${\mathbb{Z}}_{2}$ gauge theory. Also, although for $N=2$ this model is dual to two copies of the ${\mathbb{Z}}_{2}$ X-cube model, a similar statement is not true for higher $N$.