Unification of integrability in supersymmetric gauge theories
Kevin Costello, Junya Yagi
TL;DR
This work reveals that a six-dimensional topological--holomorphic theory, subjected to an $\Omega$-deformation, furnishes a common origin for four-dimensional Chern--Simons theory and a wide class of integrable lattice models. By embedding the construction in string theory and applying dualities, the authors unify appearances of the eight-vertex/XYZ models and their variants across supersymmetric gauge theories, via dynamical R-matrices, vertex–face correspondences and L-operators. The framework yields concrete realizations of spin chains (XXX/XXZ/XYZ) through brane tilings, class-S_k theories, and Nekrasov–Shatashvili-type correspondences, and explains the appearance of Q-operators as gauge-theoretic determinants. The results illuminate how topological and holomorphic twists, together with brane dynamics, generate a robust, duality-driven bridge between quantum field theory and integrable systems with rich mathematical structure. The approach provides a platform for exploring open spin chains and higher-genus spectral curves, highlighting the deep role of higher-dimensional gauge dynamics in integrable phenomena.
Abstract
A four-dimensional analog of Chern-Simons theory produces integrable lattice models from Wilson lines and surface operators. We show that this theory describes a quasi-topological sector of maximally supersymmetric Yang-Mills theory in six dimensions, topologically twisted and subjected to an Ω-deformation. By realizing the six-dimensional theory in string theory and applying dualities, we unify various phenomena in which the eight-vertex model and the XYZ spin chain, as well as variants thereof, emerge from supersymmetric gauge theories.