Notices of the International Congress of Chinese Mathematicians

Volume 6 (2018)

Number 1

Gauge Theory And Integrability, I

Pages: 46 – 119

DOI: http://dx.doi.org/10.4310/ICCM.2018.v6.n1.a6

Authors

Kevin Costello (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)

Edward Witten (School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A)

Masahito Yamazaki (Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Japan)

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.

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