Abstract
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 U(1) vector-valued one-form gauge field and an ordinary U(1) gauge field. We discuss the degrees of freedom in the corresponding gauge theories, relation to symmetric tensor gauge theories, the defect matter and coupling to the curved space.
Highlights
It turns out that the dual theory is an Abelian gauge theory of tensor gauge field. Such theories have recently emerged in condensed matter physics in the study of algebraic spin liquids [12,13,14], and, later, of gapless fracton phases [15,16,17]
Upon choosing θ = 0 and picking a preferred frame of reference, that decouples the antisymmetric part stemming from non-zero orbital angular momentum, the Cosserat theory reduces to the symmetric elasticity. φ = θ corresponds to a special limit of Cosserat elasticity, in which the local rotation or spin is completely frozen, known as the couple-stress theory [45, 52]
It was found that the Cosserat theory is dual to a theory of general tensor gauge field coupled to an ordinary, non-propagating, U(1) gauge field
Summary
Duality is a powerful tool that allows one to access non-perturbative physics of interacting systems. It turns out that the dual theory is an Abelian gauge theory of (symmetric) tensor gauge field Such theories have recently emerged in condensed matter physics in the study of algebraic spin liquids [12,13,14], and, later, of gapless fracton phases [15,16,17]. Dislocations have to satisfy the so-called glide constraint, which forces them to move along their Burgers vector, provided that total number of lattice sites is conserved, while disclinations cannot move without creating dislocations This parallels the phenomena in the physics of type-I gapless fracton phases where certain fractons can only move via creating other fractons. Appendix A is devoted to the inversion of elasticity coefficients and Appendix B to the Stückelberg mechanism of the massive mode in the gauge theory dual to Cosserat elasticity
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