I will discuss a novel construction of field theories based on the idea that one has only a half boson degree of freedom per  lattice site. Basically, instead of having a pair of canonical  conjugate commuting variables at each site, one has only one degree of freedom and the non-trivial commutators arise from connections to the nearest neighbors. The construction is very similar to staggered fermions and naturally produces gapless systems with interesting topological properties. When considering gauging  discrete translations on the phase space in one dimensional examples, one gets  interesting critical spin chains, examples of which include the critical Ising model in a transverse magnetic field and the 3-state Potts model at criticality. I will explain how these staggered boson variables are very natural for describing non-invertible symmetries.

These non-invertible symmetries are useful to describe the critical properties of these non-trivial spin chains.

Models in higher dimensions obtained this way can automatically produce dynamical systems of gapless fractons.