  We discuss the higher-order topological field theory and response of
topological crystalline insulators with no other symmetries. We show how the
topology and geometry of the system is organised in terms of the elasticity
tetrads which are ground state degrees of freedom labelling lattice topological
charges, higher-form conservation laws and responses on sub-dimensional
manifolds of the bulk system. In a crystalline insulator, they classify
higher-order global symmetries in a transparent fashion. This coincides with
the dimensional hierarchy of topological terms, the multipole expansion, and
anomaly inflow, related to a mixed number of elasticity tetrads and
electromagnetic gauge fields. In the continuum limit of the elasticity tetrads,
the semi-classical expansion can be used to derive the higher-order or embedded
topological responses to global U(1) symmetries, such as electromagnetic gauge
fields with explicit formulas for the higher-order quasi-topological invariants
in terms of the elasticity tetrads and Green's functions. The topological
responses and readily generalized in parameter space to allow for e.g.
multipole pumping. Our simple results further bridge the recently appreciated
connections between topological field theory, higher form symmetries and gauge
fields, fractonic excitations and topological defects with restricted mobility
elasticity in crystalline insulators.
