Gauge Theory and Integrability, I

TL;DR

This paper develops a detailed gauge theory approach to deriving solutions of the Yang-Baxter equation, including rational, trigonometric, elliptic, and dynamical types, with explicit Feynman diagram computations.

Contribution

It extends previous gauge theory methods to systematically derive and compute various solutions of the Yang-Baxter equation, clarifying their origins and properties.

Findings

Explicit computation of R-matrix contributions via Feynman diagrams

Derivation of rational, trigonometric, and elliptic solutions within gauge theory

Explanation of the emergence of the dynamical Yang-Baxter equation

Abstract

Several years ago, it was proposed that the usual solutions of the Yang-Baxter equation associated to Lie groups can be deduced in a systematic way from four-dimensional gauge theory. In the present paper, we extend this picture, fill in many details, and present the arguments in a concrete and down-to-earth way. Many interesting effects, including the leading nontrivial contributions to the $R$-matrix, the operator product expansion of line operators, the framing anomaly, and the quantum deformation that leads from $\mathfrak{g}[[z]]$ to the Yangian, are computed explicitly via Feynman diagrams. We explain how rational, trigonometric, and elliptic solutions of the Yang-Baxter equation arise in this framework, along with a generalization that is known as the dynamical Yang-Baxter equation.