Slave-variable techniques are very successful semi-analytic methods to treat the quantum many-body problem. Still simplified but numerically much cheaper than the more accurate Dynamical Mean-Field Theory (DMFT), these are ideally complementary techniques. In particular the Slave-Spin method is particularly versatile in approaching multi-orbital Hubbard models, which are essential in interpreting and describing the electronic properties of several strongly-correlated materials, like e.g. the Fe-based high-Tc superconductors. Still, the method has to be generalized to “rotationally invariance”, which implies the possibility of treating off-diagonal interactions and spin-orbit coupling, for instance, and off-diagonal symmetry breaking orders like superconductivity. This generalization is the main subject of this internship. The new method will have a vast number of applications both in and out of equilibrium. First problems to be tackled will be the effect of spin-orbit coupling on the correlations of Hund metals (see below) and multi-orbital photo-doped Mott insulators.