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

TL;DR

This paper commemorates the contributions of Baxter and Yang, highlighting how their work on gauge theory and the Yang-Baxter equation reveals a shared coherence principle underlying mathematical physics.

Contribution

It synthesizes the development of gauge theory and integrability, emphasizing their interconnectedness through a unified mathematical framework.

Findings

Gauge theory's geometric structures include instantons and monopoles.

Yang-Baxter equation underpins solvable models and quantum groups.

Unified view of gauge symmetry and integrability as coherence principles.

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.