  Cold three flavor quark matter at large (but not asymptotically large)
densities may exist in a crystalline color superconducting phase. These phases
are characterized by a gap parameter $\Delta$ that varies periodically in
space, forming a crystal structure. A Ginzburg-Landau expansion in $\Delta$
shows that two crystal structures based on cubic symmetry are particularly
favorable, and may be the ground state of matter at densities present in
neutron star cores. We derive the effective action for the phonon fields that
describe space- and time-dependent fluctuations of the crystal structure formed
by $\Delta$, and obtain the shear modulus from the coefficients of the spatial
derivative terms. Within a Ginzburg-Landau approximation, we find shear moduli
which are 20 to 1000 times larger than those of neutron star crusts. This phase
of matter is thus more rigid than any known material in the universe, but at
the same time the crystalline color superconducting phase is also superfluid.
These properties raise the possibility that the presence of this phase within
neutron stars may have distinct implications for their phenomenology. For
example, (some) pulsar glitches may originate in crystalline superconducting
neutron star cores.
