Yangian

Abstract: Preface 1. Introduction 2. The Hubbard Hamiltonian and its symmetries 3. The Bethe ansatz solution 4. String hypothesis 5. Thermodynamics in the Yang-Yang approach 6. Ground state properties in the thermodynamic limit 7. Excited states at zero temperature 8. Finite size corrections at zero temperature 9. Asymptotics of correlation functions 10. Scaling and continuum limits at half-filling 11. Universal correlations at low density 12. The algebraic approach to the Hubbard model 13. The path integral approach to thermodynamics 14. The Yangian symmetry of the Hubbard model 15. S-matrix and Yangian symmetry in the infinite interval limit 16. Hubbard model in the attractive case 17. Mathematical appendices References Index.

Abstract: Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices Bibliography

Abstract: Tree-level scattering amplitudes in = 4 super Yang-Mills theory have recently been shown to transform covariantly with respect to a `dual' superconformal symmetry algebra, thus extending the conventional superconformal symmetry algebra psu(2,2|4) of the theory. In this paper we derive the action of the dual superconformal generators in on-shell superspace and extend the dual generators suitably to leave scattering amplitudes invariant. We then study the algebra of standard and dual symmetry generators and show that the inclusion of the dual superconformal generators lifts the psu(2,2|4) symmetry algebra to a Yangian. The non-local Yangian generators acting on amplitudes turn out to be cyclically invariant due to special properties of psu(2,2|4). The representation of the Yangian generators takes the same form as in the case of local operators, suggesting that the Yangian symmetry is an intrinsic property of planar = 4 super Yang-Mills, at least at tree level.

Abstract: Recently, an explicit, recursive formula for the all-loop integrand of planar scattering amplitudes in $ \mathcal{N} = {4} $ SYM has been described, generalizing the BCFW formula for tree amplitudes, and making manifest the Yangian symmetry of the theory. This has made it possible to easily study the structure of multi-loop amplitudes in the theory. In this paper we describe a remarkable fact revealed by these investigations: the integrand can be expressed in an amazingly simple and manifestly local form when represented in momentum-twistor space using a set of chiral integrals with unit leading singularities. As examples, we present very-concise expressions for all 2- and 3-loop MHV integrands, as well as all 2-loop NMHV integrands. We also describe a natural set of manifestly IR-finite integrals that can be used to express IR-safe objects such as the ratio function. Along the way we give a pedagogical introduction to the foundations of the subject. The new local forms of the integrand are closely connected to leading singularities — matching only a small subset of all leading singularities remarkably suffices to determine the full integrand. These results strongly suggest the existence of a theory for the integrand directly yielding these local expressions, allowing for a more direct understanding of the emergence of local spacetime physics.