Lectures on Yangian Symmetry

TL;DR

This paper provides an introductory overview of Yangian symmetry, exploring its classical and quantum aspects, mathematical structure, and applications in integrable models, spin chains, and super Yang-Mills theory.

Contribution

It offers a comprehensive introduction to Yangian symmetry, connecting classical nonlocal charges, algebraic structures, and physical applications across various models.

Findings

Yangian symmetry extends Noether symmetries in 2D field theories.

Implementation of Yangian algebra in quantum theories and spin chains.

Implications of Yangian symmetry on scattering matrices and super Yang-Mills theory.

Abstract

In these introductory lectures we discuss the topic of Yangian symmetry from various perspectives. Forming the classical counterpart of the Yangian and an extension of ordinary Noether symmetries, first the concept of nonlocal charges in classical, two-dimensional field theory is reviewed. We then define the Yangian algebra following Drinfeld's original motivation to construct solutions to the quantum Yang-Baxter equation. Different realizations of the Yangian and its mathematical role as a Hopf algebra and quantum group are discussed. We demonstrate how the Yangian algebra is implemented in quantum, two-dimensional field theories and how its generators are renormalized. Implications of Yangian symmetry on the two-dimensional scattering matrix are investigated. We furthermore consider the important case of discrete Yangian symmetry realized on integrable spin chains. Finally we give a brief introduction to Yangian symmetry in planar, four-dimensional super Yang-Mills theory and indicate its impact on the dilatation operator and tree-level scattering amplitudes. These lectures are illustrated by several examples, in particular the two-dimensional chiral Gross-Neveu model, the Heisenberg spin chain and N=4 superconformal Yang-Mills theory in four dimensions. This review arose from lectures given at the Young Researchers Integrability School at Durham University (UK).