Gauge Theory and Integrability: An Overview

Abstract

While general quantum field theories (QFTs) have yet to be rigorously defined in mathematics, they have generated new mathematics and have served as a unifying principle connecting different branches of the subject. In 1989, Witten made a profound impact on the mathematical community by systematically constructing knot invariants via the three-dimensional Chern-Simons theory. One of the historical roots of knot invariants was integrable models, whose explanation in terms of QFT remained unsolved for decades. Recently, this problem was solved by a perturbative analysis of the four-dimensional Chern-Simons theory, which provides a novel framework for understanding and unifying many different aspects of integrable models. In this article, we summarize the basic aspects of these developments for non-experts in both physics and mathematics.