From Yang-Mills to Yang-Baxter: In Memory of Rodney Baxter and Chen--Ning Yang

TL;DR

The paper addresses the unifying question of how local consistency in two foundational theories—Yang-Mills gauge theory and the Yang-Baxter equation—produces global mathematical unity. It traces the development from intrinsic gauge symmetry and covariant geometry to topological invariants and from factorised scattering to quantum groups and Chern-Simons theory, highlighting the shared coherence principle across dimensions. Key contributions include geometric formulations of the Higgs mechanism, Donaldson-Floer theory, the Atiyah-Floer conjecture, Hitchin systems, and the emergence of YB structures via quantum groups, stable envelopes, and quantum cohomology. The work emphasizes the deep connections between geometry, topology, and quantum field theory, illustrating how coherence principles guide contemporary approaches to mathematical physics and suggesting a broad, ongoing synthesis of gauge theory and integrability across disciplines and dimensions.

Abstract

The year 2025 marked the passing of two towering figures of twentieth-century mathematical physics, Rodney Baxter and Chen-Ning Yang. Yang reshaped modern physics through the introduction of non-abelian gauge theory and, independently, through the consistency conditions underlying what is now called the Yang-Baxter equation. Baxter transformed those conditions into a systematic theory of exact solvability in statistical mechanics and quantum integrable systems. This article is written in memory of Baxter and Yang, whose work revealed how local consistency principles generate global mathematical structure. We review the Yang-Mills formulation of gauge theory, its mass obstruction and resolution via symmetry breaking, and the geometric framework it engendered, including instantons, Donaldson-Floer theory, magnetic monopoles, and Hitchin systems. In parallel, we trace the emergence of the Yang-Baxter equation from factorised scattering to solvable lattice models, quantum groups, and Chern-Simons theory. Rather than separate narratives, gauge theory and integrability are presented as complementary manifestations of a shared coherence principle, an ongoing journey from gauge symmetry toward mathematical unity.