Covariant fractons and Weitzenböck torsion

The relation between covariant fracton gauge theory and Møller-Hayashi-Shirafuji theory of 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 the most general theory, whose action is quadratic in the Weitzenböck torsion. We show that the solutions of covariant fracton gauge theory describe a subsector of the space of solutions of Møller-Hayashi-Shirafuji theory, providing a new insight in the relation between covariant fractons and gravity, and elucidating the meaning of covariant fracton theory as a new type of gauge theory.