Chiral Topological Elasticity and Fracton Order

We analyze the "higher rank" gauge theories, that capture some of the phenomenology of the Fracton order. It is shown that these theories lose gauge invariance when arbitrarily weak and smooth curvature is introduced. We propose a resolution to this problem by introducing a theory invariant under area-preserving diffeomorphisms, which reduce to the "higher rank" gauge transformations upon linearization around a flat background. The proposed theory is geometric in nature and is interpreted as a theory of chiral topological elasticity. This theory exhibits some of the Fracton phenomenology. We explore the conservation laws, topological excitations, linear response, various kinematical constraints, and canonical structure of the theory. Finally, we emphasize that the very structure of Riemann-Cartan geometry, which we use to formulate the theory, encodes some of the Fracton phenomenology, suggesting that the Fracton order itself is geometric in nature.