  We present a formalism for strongly correlated systems with fermions coupled
to bosonic modes. We construct the three-particle irreducible functional
$\mathcal{K}$ by successive Legendre transformations of the free energy of the
system. We derive a closed set of equations for the fermionic and bosonic
self-energies for a given $\mathcal{K}$. We then introduce a local
approximation for $\mathcal{K}$, which extends the idea of dynamical mean field
theory (DMFT) approaches from two- to three-particle irreducibility. This
approximation entails the locality of the three-leg electron-boson vertex
$\Lambda(i\omega,i\Omega)$, which is self-consistently computed using a quantum
impurity model with dynamical charge and spin interactions. This local vertex
is used to construct frequency- and momentum-dependent electronic self-energies
and polarizations. By construction, the method interpolates between the
spin-fluctuation or GW approximations at weak coupling and the atomic limit at
strong coupling. We apply it to the Hubbard model on two-dimensional square and
triangular lattices. We complement the results of Phys.Rev. B 92, 115109 by (i)
showing that, at half-filling, as DMFT, the method describes the Fermi-liquid
metallic state and the Mott insulator, separated by a first-order
interacting-driven Mott transition at low temperatures, (ii) investigating the
influence of frustration and (iii) discussing the influence of the bosonic
decoupling channel.
