Mathematical Uses of Gauge Theory

1.1. This article surveys some developments in pure mathematics which have, to varying degrees, grown out of the ideas of gauge theory in Mathematical Physics. The realisation that the gauge fields of particle physics and the connections of differential geometry are one and the same has had wide-ranging consequences, at different levels. Most directly, it has lead mathematicians to work on new kinds of questions, often shedding light later on well-established problems. Less directly, various fundamental ideas and techniques, notably the need to work with the infinite-dimensional gauge symmetry group, have found a place in the general world-view of many mathematicians, influencing developments in other fields. Still less direct, the work in this area—between geometry and mathematical physics—has been a prime example of the interaction between these fields which has been so fruitful over the past thirty years.