  The major obstacle preventing Feynman diagrammatic expansions from accurately
solving many-fermion systems in strongly correlated regimes is the series slow
convergence or divergence problem. Several techniques have been proposed to
address this issue: series resummation by conformal mapping, changing the
nature of the starting point of the expansion by shifted action tools, and
applying the homotopy analysis method to the Dyson-Schwinger equation. They
emerge as dissimilar mathematical procedures aimed at different aspects of the
problem. The proposed homotopic action offers a universal and systematic
framework for unifying the existing -- and generating new -- methods and ideas
to formulate a physical system in terms of a convergent diagrammatic series. It
eliminates the need for resummation, allows one to introduce effective
interactions, enables a controlled ultraviolet regularization of
continuous-space theories, and reduces the intrinsic polynomial complexity of
the diagrammatic Monte Carlo method. We illustrate this approach by an
application to the Hubbard model.
