Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes

Abstract

We propose that Baxter's Z-invariant six-vertex model at the rational gl(2) point on a planar but in general not rectangular lattice provides a way to study Yangian invariants. These are identified with eigenfunctions of certain monodromies of an auxiliary inhomogeneous spin chain. As a consequence they are special solutions to the eigenvalue problem of the associated transfer matrix. Excitingly, this allows to construct them using Bethe ansatz techniques. Conceptually, our construction generalizes to general (super) Lie algebras and general representations. Here we present the explicit form of sample invariants for totally symmetric, finite-dimensional representations of gl(n) in terms of oscillator algebras. In particular, we discuss invariants of three- and four-site monodromies that can be understood respectively as intertwiners of the bootstrap and Yang-Baxter equation. We state a set of functional relations significant for these representations of the Yangian and discuss their solutions in terms of Bethe roots. They arrange themselves into exact strings in the complex plane. In addition, it is shown that the sample invariants can be expressed analogously to Grassmannian integrals. This aspect is closely related to a recent on-shell formulation of scattering amplitudes in planar N=4 super Yang-Mills theory.

56 pages, 14 figures; v2: appendix B added, references updated and typos fixed; v3: published version

References in corpus (38)

• Perturbative Gauge Theory As A String Theory In Twistor Space
• Review of AdS/CFT Integrability: An Overview
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• How Algebraic Bethe Ansatz works for integrable model
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• A Duality For The S Matrix
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• Stringing Spins and Spinning Strings
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• The S-Matrix in Twistor Space
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data
• Algebraic Aspects of Bethe-Ansatz
• Introduction to Yangian symmetry in integrable field theory
• Exacting N=4 Superconformal Symmetry
• An Introduction to Yangian Symmetries
• Baxter Q-Operators and Representations of Yangians
• The Yangian origin of the Grassmannian integral
• Yangians, Grassmannians and T-duality
• Superconformal invariants for scattering amplitudes in N=4 SYM theory
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• Yang-Baxter operators and scattering amplitudes in $\mathcal{N}=4$ super-Yang-Mills theory
• Oscillator Construction of su(n|m) Q-Operators
• Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory
• Hexagon Wilson Loop OPE and Harmonic Polylogarithms
• Algebraic Bethe ansatz for singular solutions
• On Yangian Symmetry in Planar N=4 SYM
• Descent equations for superamplitudes
• Exact Superconformal and Yangian Symmetry of Scattering Amplitudes
• Yangian symmetric correlators
• Self Duality and Quantization
• Yang--Baxter symmetry in integrable models: new light from the Bethe Ansatz solution
• Tree-level amplitudes and dual superconformal symmetry
• Introduction to the Bethe ansatz I
• Yangian Symmetry in N=4 super Yang-Mills
• Coordinate space wave function from the Algebraic Bethe Ansatz for the inhomogeneous six-vertex model
• From Baxter Q-Operators to Local Charges
• A conjugate for the Bargmann representation

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• On-shell Diagrams, Graßmannians and Integrability for Form Factors
• Amplitudes, Form Factors and the Dilatation Operator in $\mathcal{N}=4$ SYM Theory
• Towards the Amplituhedron Volume
• N=4 Scattering Amplitudes and the Deformed Grassmannian
• A dictionary between R-operators, on-shell graphs and Yangian algebras
• Integrable Amplitude Deformations for N=4 Super Yang-Mills and ABJM Theory
• Novel construction and the monodromy relation for three-point functions at weak coupling
• String Bits and the Spin Vertex
• On Soft Theorems And Form Factors In N=4 SYM Theory
• Grassmannians and form factors with $q^2=0$ in N=4 SYM theory
• Wilson lines, Grassmannians and Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Four dimensional ambitwistor strings and form factors of local and Wilson line operators
• Yangian Symmetry for the Tree Amplituhedron
• On the singlet projector and the monodromy relation for psu(2, 2|4) spin chains and reduction to subsectors
• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Grassmannian Integrals as Matrix Models for Non-Compact Yangian Invariants
• Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes
• Ambitwistor strings and reggeon amplitudes in N=4 SYM
• Light-cone limits of large rectangular fishnets
• Boundary Perimeter Bethe Ansatz
• Grassmannian Integrals in Minkowski Signature, Amplitudes, and Integrability
• Bethe ansatz inside Calogero-Sutherland models
• Coherent-State Overcompleteness, Path Integrals, and Weak Values
• Divergences in maximal supersymmetric Yang-Mills theories in diverse dimensions
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes