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.
Highlights
Introduction and overviewSome time ago, a remarkable observation has been made in the field of scattering amplitudes of planar N = 4 super Yang-Mills theory, namely their Yangian structure [1]
It was obtained by combining superconformal symmetry and a hidden dual superconformal symmetry [2]
What is the nature of Yangian symmetry, as it appears in the scattering amplitudes, from the view point of integrability and the quantum inverse scattering method (QISM)? In order to answer this question, we focus on Yangian invariants |Ψ, which are defined in the following way, Mab(u)|Ψ = δab|Ψ, (1.1)
Summary
This should play the role of a toy model of the N = 4 scattering amplitudes, where suitable non-compact representations of gl(4|4) are needed instead The latter are built from continuous generalizations of the oscillators mentioned above, which are essentially the spinor-helicity variables and their derivatives. Just like in the toy model, it is imperative that the Yangian invariants and the tree-level amplitudes do not depend on the spectral parameter u The latter merely serves as a suitable device for applying the QISM and for employing (a adequate generalization of) the Bethe ansatz to the problem. This opens the way to derive a perimeter Bethe ansatz for the latter.
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