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Yangians, Grassmannians and T-duality

J. M. Drummond, L. Ferro

TL;DR

The paper elucidates Yangian symmetry in planar N=4 SYM amplitudes, revealing a T-duality between twistor and momentum-twistor formulations. It shows the full Yangian can be viewed as the Yangian of the dual superconformal algebra after factoring the MHV piece, and provides a direct invariance proof for Grassmannian formulas using level-one generators. By comparing two Grassmannian constructions, it demonstrates that both encode Yangian invariants and are related by a change of variables, reinforcing the role of Grassmannians as a universal invariant framework for leading singularities. These results deepen the understanding of symmetry in N=4 SYM and suggest avenues for extending Yangian invariance to broader contexts and loop-level analyses.

Abstract

We investigate the Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory and show that its formulations in twistor and momentum twistor space can be interchanged. In particular we show that the full symmetry can be thought of as the Yangian of the dual superconformal algebra, annihilating the amplitude with the MHV part factored out. The equivalence of this picture with the one where the ordinary superconformal symmetry is thought of as fundamental is an algebraic expression of T-duality. Motivated by this, we analyse some recently proposed formulas, which reproduce different contributions to amplitudes through a Grassmannian integral. We prove their Yangian invariance by directly applying the generators.

Yangians, Grassmannians and T-duality

TL;DR

The paper elucidates Yangian symmetry in planar N=4 SYM amplitudes, revealing a T-duality between twistor and momentum-twistor formulations. It shows the full Yangian can be viewed as the Yangian of the dual superconformal algebra after factoring the MHV piece, and provides a direct invariance proof for Grassmannian formulas using level-one generators. By comparing two Grassmannian constructions, it demonstrates that both encode Yangian invariants and are related by a change of variables, reinforcing the role of Grassmannians as a universal invariant framework for leading singularities. These results deepen the understanding of symmetry in N=4 SYM and suggest avenues for extending Yangian invariance to broader contexts and loop-level analyses.

Abstract

We investigate the Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory and show that its formulations in twistor and momentum twistor space can be interchanged. In particular we show that the full symmetry can be thought of as the Yangian of the dual superconformal algebra, annihilating the amplitude with the MHV part factored out. The equivalence of this picture with the one where the ordinary superconformal symmetry is thought of as fundamental is an algebraic expression of T-duality. Motivated by this, we analyse some recently proposed formulas, which reproduce different contributions to amplitudes through a Grassmannian integral. We prove their Yangian invariance by directly applying the generators.

Paper Structure

This paper contains 11 sections, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On-shell scattering amplitudes
  3. Symmetries
  4. T-dual representation of the symmetries
  5. Grassmannian formulas
  6. Yangian invariance of the Grassmannian formulas
  7. Conclusions
  8. Formulae for both superconformal algebras
  9. Some generalities on $gl(n|n)$ and its Yangian
  10. Induced transformation of the minors
  11. Details of invariance of the gauge-fixed integral

Figures (1)

  • Figure 1: The tower of symmetries acting on scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. The original superconformal charges are denoted by $j$ and the dual ones by $J$. Each can be thought of as the level-zero part of the Yangian $Y(psu(2,2|4))$. The dual superconformal charges $K$ and $S$ form part of the level-one $j^{(1)}$ while the original superconformal charges $k$ and $s$ form part of the level one charges $J^{(1)}$. In each representation the 'negative' level ($P$ and $Q$ or $p$ and $q$) is trivialised. T-duality maps $j$ to $J$ and $j^{(1)}$ to $J^{(1)}$.