Notices of the International Consortium of Chinese Mathematicians

Volume 6 (2018)

Number 1

Gauge Theory And Integrability, II

Pages: 120 – 146



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)


Starting with a four-dimensional gauge theory approach to rational, elliptic, and trigonometric solutions of the Yang–Baxter equation, we determine the corresponding quantum group deformations to all orders in $\hbar$ by deducing their RTT presentations. The arguments we give are a mix of familiar ones with reasoning that is more transparent from the four-dimensional gauge theory point of view. The arguments apply most directly for $\mathfrak{gl}_N$ and can be extended to all simple Lie algebras other than $\mathfrak{e}_8$ by taking into account the self-duality of some representations, the framing anomaly for Wilson operators, and the existence of quantum vertices at which several Wilson operators can end.

Published 14 September 2018