Abstract: In physics, gauge theory is the study of gauge fields and their associated matter fields, such as the electromagnetic field and the associated electron field, which interact according to Maxwell’s equations of electromagnetism (such an interaction being called a “coupling” by physicists). Such gauge theories were developed in generality by Yang and Mills in the language of principal bundles, associated vector bundles, and connections, and this Yang–Mills theory now underpins the standard model of particle physics.
Mathematical gauge theory arose in the 1970s and 1980s as various mathematicians, most notably Michael Atiyah, demonstrated that interesting geometry constructions and invariants could be derived from physically meaningful gauge-theoretic equations, such as the Yang–Mills equations. Since the 1980s, mathematical gauge theory — the study of connections and curvature on vector bundles and principal bundles — has produced many interesting new geometric problems, techniques, structures, and solutions.
In this talk I will give a mathematicians account of the history of gauge theory, introducing the key players such as the connections, curvature, and the Yang–Mills equations, and go on to discuss the great successes of the theory. These include the deep relationships between gauge-theoretic structures and algebraic geometry through the Hitchin–Kobayashi correspondence, the novelty of moduli spaces of solutions to these equations, such as Higgs bundle moduli spaces, some of the first examples of compact non-symmetric hyper-Kähler manifolds, and the many powerful topological invariants that have arisen out of gauge-theoretic equations such as Donaldson invariants and Seiberg–Witten invariants.