  We study cavity-mediated interactions that are generated in a two-dimensional
two-band Hubbard model coupled to an optical cavity, when it is driven in-gap
by a strong laser. Starting from a Floquet description of the driven system, we
derive effective low-energy Hamiltonians by projecting out the high-energy
degrees of freedom and treating intrinsic interactions on a mean field level.
We then investigate how the emergence of high-energy Frenkel excitons from the
electronic interband coupling, which form near the upper electronic band,
affects the interactions as well as the laser-induced Floquet renormalization
of the electronic band structure. Cavity-mediated interactions are enhanced
strongly when the light couples to an excitonic transition. Additionally, the
interaction as well as the Floquet renormalization are strongly broadened in
reciprocal space, which could further boost the impact of cavity-mediated
interactions on the driven-dissipative steady state.
