Lectures on Yangian Symmetry

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).

arXiv comment75 pages, v2: references added, typos corrected

References in corpus (70)

• Large N Field Theories, String Theory and Gravity
• The Bethe-Ansatz for N=4 Super Yang-Mills
• Perturbative Gauge Theory As A String Theory In Twistor Space
• Review of AdS/CFT Integrability: An Overview
• Hidden Symmetries of the AdS_5 x S^5 Superstring
• The N=4 SYM Integrable Super Spin Chain
• The Dilatation Operator of Conformal N=4 Super Yang-Mills Theory
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• Fermi gases in one dimension: From Bethe Ansatz to experiments
• How Algebraic Bethe Ansatz works for integrable model
• The Complete One-Loop Dilatation Operator of N=4 Super Yang-Mills Theory
• The Dilatation Operator of N=4 Super Yang-Mills Theory and Integrability
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Remarks on the notion of quantum integrability
• Planar N=4 gauge theory and the Inozemtsev long range spin chain
• A Relation Between Approaches to Integrability in Superconformal Yang-Mills Theory
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Quantum Inverse Scattering Method. Selected Topics
• Introduction to Yangian symmetry in integrable field theory
• Exacting N=4 Superconformal Symmetry
• An Introduction to Yangian Symmetries
• Symmetries of Tree-level Scattering Amplitudes in N=6 Superconformal Chern-Simons Theory
• Quantum group symmetry in sine-Gordon and affine Toda field theories on the half-line
• One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills
• Long-Range Deformations for Integrable Spin Chains
• Bethe Ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions
• Local conserved charges in principal chiral models
• Review of AdS/CFT Integrability, Chapter VI.1: Superconformal Symmetry
• Boundary remnant of Yangian symmetry and the structure of rational reflection matrices
• Lectures on the Bethe Ansatz
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• The Bound State S-Matrix for AdS5 x S5 Superstring
• Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory
• Yangian Symmetry of smooth Wilson Loops in N=4 super Yang-Mills Theory
• Boosting Nearest-Neighbour to Long-Range Integrable Spin Chains
• A Spin Chain Primer
• Amplitudes, Form Factors and the Dilatation Operator in $\mathcal{N}=4$ SYM Theory
• Introduction to the thermodynamic Bethe ansatz
• A Secret Symmetry of the AdS/CFT S-matrix
• On Yangian Symmetry in Planar N=4 SYM
• Infinite Quantum Group Symmetry of Fields in Massive 2D Quantum Field Theory
• Yangians, S-matrices and AdS/CFT
• Fixing the Quantum Three-Point Function
• Bonus Yangian Symmetry for the Planar S-Matrix of N=4 Super Yang-Mills
• From Scattering Amplitudes to the Dilatation Generator in N=4 SYM
• Integrable Amplitude Deformations for N=4 Super Yang-Mills and ABJM Theory
• The S-Matrix of AdS/CFT and Yangian Symmetry
• Lectures on S-matrices and Integrability
• Yangian symmetry of the Y=0 maximal giant graviton
• Novel construction and the monodromy relation for three-point functions at weak coupling
• Spin Models and Superconformal Yang-Mills Theory
• Exact Superconformal and Yangian Symmetry of Scattering Amplitudes
• String Bits and the Spin Vertex
• Supersymmetry and the Nonlocal Yangian Deformation Symmetry
• Lectures on Integrable Structures in Quantum Field Theory and Massive ODE/IM Correspondence
• Yangian Symmetry at Two Loops for the su(2|1) Sector of N=4 SYM
• Yangian Symmetry of Long-Range gl(N) Integrable Spin Chains
• Secret Symmetries in AdS/CFT
• Integrability of Smooth Wilson Loops in N=4 Superspace
• Integrable Deformations of the XXZ Spin Chain
• Classically integrable boundary conditions for symmetric-space sigma models
• Bonus Symmetry for Super Wilson Loops
• Recursion Relations for Long-Range Integrable Spin Chains with Open Boundary Conditions
• Lectures on Hopf Algebras, Quantum Groups and Twists
• Yangians in Integrable Field Theories, Spin Chains and Gauge-String Dualities
• Minimal Surfaces of the $Ad{S}_5\times S^5$ Superstring and the Symmetries of Super Wilson Loops at Strong Coupling
• On Yangian symmetry of scattering amplitudes and the dilatation operator in N=4 super Yang-Mills
• On the bootstrap structure of Yangian-invariant factorized S-matrices
• Massive current algebra in the many-flavor chiral Gross-Neveu model
• Q-operators, Yangian invariance and the quantum inverse scattering method

Cited by in corpus (47)

• Constructing Integrable Lindblad Superoperators
• Integrable Deformations of Sigma Models
• Lectures on the Bethe Ansatz
• Yangian Bootstrap for Conformal Feynman Integrals
• An Integrability Primer for the Gauge-Gravity Correspondence: an Introduction
• Tensorial Gross-Neveu models
• Yang-Baxter and the Boost: splitting the difference
• Classifying nearest-neighbour interactions and deformations of AdS
• Classifying integrable spin-1/2 chains with nearest neighbour interactions
• Equivalences between three presentations of orthogonal and symplectic Yangians
• Lectures on S-matrices and Integrability
• Flag manifold sigma models: spin chains and integrable theories
• Introduction to classical and quantum integrability
• Lectures on Integrable Structures in Quantum Field Theory and Massive ODE/IM Correspondence
• Yangian Bootstrap for Massive Feynman Integrals
• Yangian Symmetry for the Action of Planar N=4 Super Yang-Mills and N=6 Super Chern-Simons Theories
• Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case
• Massive Fishnets
• Amplituhedra, and Beyond
• Ultralocal Lax connection for para-complex ${\mathbb{Z}}_T$-cosets
• Integrability for Feynman Integrals
• Nonlocal Symmetries, Spectral Parameter and Minimal Surfaces in AdS/CFT
• Minkowski Box from Yangian Bootstrap
• The $\mathsf{C}{\mathsf{P}}^{\mathsf{n}-1}$-model with fermions: a new look
• Yangian Symmetry in Holographic Correlators
• Yangian Algebra and Correlation Functions in Planar Gauge Theories
• Dynamical Spin Chains in 4D $\mathcal{N}=2$ SCFTs
• Nonlocal Symmetries and Factorized Scattering
• di-Langlands correspondence and extended observables
• A study of integrable form factors in massless relativistic AdS_3
• Properties of the Conformal Yangian in Scalar and Gauge Field Theories
• On the classification of rational K-matrices
• Non-hermitian integrable systems from constant non-invertible solutions of the Yang-Baxter equation
• Lectures on Classical Integrability
• The Feyn-Structure of Yangian Symmetry
• Non-Local Symmetries of Planar Feynman Integrals
• The Yang-Baxter integrability of the critical Ising chain
• Grassmannian Integrals in Minkowski Signature, Amplitudes, and Integrability
• A Drinfeld type presentation of twisted Yangians
• On Exact Solvability of $\mathcal{N}$=4 super Yang-Mills
• Nonlocal charges from marginal deformations of 2D CFTs: Holographic $T\bar{T}$, $T\bar{J}$ and Yang-Baxter deformations
• An integrable deformed Landau-Lifshitz model with particle production?
• Jordanian spin chains for twisted strings in $Ad{S}_5\times S^5$
• Almost local integrable models from supersymmetry algebras
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions
• Boundary bound states and integrable Wilson loops in ABJM
• The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space