Covariant Fractons and Weitzenb\"ock Torsion

The relation between covariant fracton gauge theory and Moller-Hayashi-Shirafuji extension of teleparallel gravity is investigated. The former is the gauge theory of a rank two symmetric tensor with gauge symmetry given by the double derivative of a scalar parameter; the latter is a class of theories which generalises Einstein gravity, whose action is quadratic in the Weitzenb\"ock torsion. We show that the solutions of covariant fracton gauge theory can be viewed as solutions of Moller-Hayashi-Shirafuji theory, providing a new insight in the relation between covariant fractons and gravity, and elucidating the meaning of covariant fracton theories as a new type of gauge theory.