The geometry of the Yang-Mills theory

This master's thesis focuses on the mathematical formalization of gauge theories, with a particular focus on the Yang-Mills theory. The underlying framework of gauge theories is differential geometry, including smooth manifolds, Lie groups and algebras, fiber bundles, and connections. Special attention is given to principal bundles and their associated bundles, as these provide the language for formulating gauge fields and curvature, interpreted respectively as gauge bosons and field strength tensors in physics. The thesis studies the necessary concepts to construct gauge theories on flat and curved spacetimes and generalizes the Maxwell Lagrangian to non-abelian SU(n) gauge groups. This approach highlights the central role of geometry in modern theoretical physics and the deep connections between mathematics and the fundamental interactions described by the Standard Model.