Analytical and Geometric approches of non-perturbative Quantum Field Theories

TL;DR

This paper explores non-perturbative quantum field theories using analytical and geometric methods, focusing on Hopf algebras, Schwinger-Dyson equations, and BV formalism, providing exact solutions and analyzing singularities.

Contribution

It introduces a geometric framework for non-perturbative QFT, studies exact solutions of Schwinger-Dyson equations, and links BV formalism with geometric integration methods.

Findings

Exact solutions to linear Schwinger-Dyson equations

Analysis of asymptotics and singularities in the Wess-Zumino model

Presentation of BV formalism as a theory of integration

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.