  Mechanical properties of crystals on curved substrates mix elastic, geometric
and topological degrees of freedom. In order to elucidate properties of such
crystals we formulate the low-energy effective action that combines metric
degrees of freedom with displacement fields and defects. We propose new
dualities for elasticity coupled to curved geometry formulated in terms of
tensor gauge theories. We show that the metric degrees of freedom, evolving
akin to linearized gravity are mapped to tensors with three indices. When
coupled to crystals these degrees of freedom become gapped and, in the presence
of dislocations and disclinations, multivalued. The elastic degrees of freedom
remain gapless and mapped to symmetric gauge fields with two indices. In the
dual formulation, topological defects, which act as sources for the gauge
fields, are fractons or excitations with restricted mobility. We show that
mobility restrictions are manifest only when singularities in both displacement
fields and metric are taken into account.

High Energy Physics - Theory;Soft Condensed Matter;Strongly Correlated Electrons