Title: Fractional Topological Elasticity and Fracton Order,
Abstract: We analyze the "higher rank" gauge theories, that capture some of the
phenomenology of the Fracton order. It is shown that these theories loose 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 \emph{geometric} in nature and is interpreted as a theory of
\emph{fractional topological elasticity}. This theory exhibits 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 the Fracton
phenomenology, suggesting that the Fracton order itself is \emph{geometric} in
nature.