  We present a dual formulation of the Cosserat theory of elasticity. In this
theory a local element of an elastic body is described in terms of local
displacement and local orientation. Upon the duality transformation these
degrees of freedom map onto a coupled theory of a vector-valued one-form gauge
field and an ordinary $U(1)$ gauge field. We discuss the degrees of freedom in
the corresponding gauge theories, the defect matter and coupling to the curved
space.
