  Cavity control of quantum matter may offer new ways to study and manipulate
many-body systems. A particularly appealing idea is to use cavities to enhance
superconductivity, especially in unconventional or high-$T_c$ systems.
Motivated by this, we propose a scheme for coupling Terahertz resonators to the
antiferromagnetic fluctuations in a cuprate parent compound, which are believed
to provide the glue for Cooper pairs in the superconducting phase. First, we
derive the interaction between magnon excitations of the Ne\'el-order and polar
phonons associated with the planar oxygens. This mode also couples to the
cavity electric field, and in the presence of spin-orbit interactions mediates
a linear coupling between the cavity and magnons, forming hybridized
magnon-polaritons. This hybridization vanishes linearly with photon momentum,
implying the need for near-field optical methods, which we analyze within a
simple model. We then derive a higher-order coupling between the cavity and
magnons which is only present in bilayer systems, but does not rely on
spin-orbit coupling. This interaction is found to be large, but only couples to
the bimagnon operator. As a result we find a strong, but heavily damped,
bimagnon-cavity interaction which produces highly asymmetric cavity line-shapes
in the strong-coupling regime. To conclude, we outline several interesting
extensions of our theory, including applications to carrier-doped cuprates and
other strongly-correlated systems with Terahertz-scale magnetic excitations.
