I detail three major mathematical developments that led to the emergence of Yang-Mills theories as the foundation for the standard model of particle physics. In less than ten years, work on renormalizability, the renormalization group, and lattice quantum field theory highlighted the utility of Yang-Mills type models of quantum field theory by connecting poorly understood candidate dynamical models to emerging experimental results.The model was the result of developments in mathematical physics from 1960-1975. The paper describes these pretty well.
As the paper explains, renormalizing gauge theories was the crucial development.
The story ends in about 1975, with a satisfactory theory of all four fundamental forces, and agreement with experiment for the foreseeable future. So did physicists celebrate solving all their big problems? No. They embarked on string theory and billion-dollar searches for SUSY particles. None of this has amounted to anything.
Historians will record a golden age of theoretical physics that ran from Maxwell in about 1860 to the standard model in 1975.