  Calculation of conductivity in the Hubbard model is a challenging task.
Recent years have seen much progress in this respect and numerically exact
solutions are now possible in certain regimes. In this paper we discuss the
calculation of conductivity for the square lattice Hubbard model in the
presence of a perpendicular magnetic field, focusing on orbital effects. We
present the relevant formalism in all detail and in full generality, and then
discuss the simplifications that arise at the level of the dynamical mean field
theory (DMFT). We prove that the Kubo bubble preserves gauge and translational
invariance, and that in the DMFT the vertex corrections cancel regardless of
the magnetic field. We present the DMFT results for the spectral function and
both the longitudinal and Hall conductivity in several regimes of parameters.
We analyze thoroughly the quantum oscillations of the longitudinal conductivity
and identify a high-frequency oscillation component, arising as a combined
effect of scattering and temperature, in line with recent experimental
observations in moir\'e systems.
