  In the study of strongly interacting systems, correlations on the
two-particle level are receiving more and more attention. In this work, we
study a particular two-particle correlation function: the fermion-boson vertex.
It describes the response of the Green's function when an external field is
applied and is an important ingredient of diagrammatic extensions of Dynamical
Mean-Field Theory. We provide several perspectives on this object, using Ward
identities, sum rules, perturbative analysis and asymptotic relations. We then
use these tools to study the vertex across the doping-driven metal-insulator
transition and find a divergence in the imaginary part.
