Analytical and Geometric approches of non-perturbative Quantum Field Theories

Abstract: We present the Hopf algebra of renormalization and introduce the renormalization group equation in this framework. Some linear Schwinger--Dyson equations are studied, and exact solutions are presented. Then we study the Schwinger--Dyson equation of the massless Wess--Zumino model in the physical plane and in the Borel plane. In the former the asymptotics of the solution of the Schwinger--Dyson equation is found and its perturbations are computed. In the later we study the singularities of the solution and their transcendental contents. The last chapter is a presentation of the BV formalism seen as theory of integration for the polyvector fields. The appendices contain a presentation of the geometric approach of the BRST formalism, an alternative description of the BV formalism that underlines the link between BRST and BV and some computations of Feynman integrals.