A Comprehensive Approach to the Yang-Mills Existence and Mass Gap Problem Using Differential Geometry and Dynamical Systems

The Yang-Mills existence and mass gap problem is a fundamental challenge in mathematical physics, posing significant implications for our understanding of quantum field theories. This paper presents a comprehensive theoretical framework integrating differential geometry and dynamical systems to address this problem. By introducing curvature concepts and Lie derivatives, we explore the topological invariants associated with Yang-Mills fields, particularly focusing on winding numbers and their relation to the mass gap. The approach incorporates various methodologies, including Poincaré/Lorentz covariance, effective field theories with cut-offs, and the calculation of Wilson loops, to establish the existence of a mass gap. Detailed mathematical derivations and the construction of relevant theorems are provided to substantiate the proposed framework.