Minuscule ABCDE Lax Operators from 4D Chern-Simons Theory

TL;DR

This paper constructs explicit minuscule Lax operators for integrable spin chains using 4D Chern-Simons theory, covering classical and exceptional gauge groups, and explores their symmetries and interrelations.

Contribution

It provides a comprehensive derivation of oscillator realizations of minuscule L-operators for various gauge groups, including previously missing non-simply laced cases, and analyzes their symmetry links.

Findings

Derived explicit Lax operators for classical and exceptional groups.

Completed missing results for non-simply laced gauge symmetries.

Identified outer-automorphism symmetries linking different Lax operators.

Abstract

Using 4D Chern-Simons (CS) theory with gauge symmetry $G$ having minuscule coweights, we develop a suitable operator basis to deal with the explicit calculation of the Lax operator of integrable spin chain satisfying the RLL equation. Using this basis, we derive the oscillator realisations of the full list of the minuscule L-operators which are classified by the gauge symmetries A$_{N}$, B$_{N}$, C$_{N}$, D$_{N}$, E$_{6}$, E$_{7}$. We also complete missing results regarding the non simply laced $SO_{2N+1}$ and $SP_{2N}$ gauge symmetries and comment on their intrinsic features. Moreover, we investigate the properties of links reported in Yangian spin chain studies between the (A-,D-) Lax operators and the (C-,B-) homologue. We show that these links are due to discrete outer-automorphism symmetries that are explicitly worked out.