
TL;DR
This paper explores the algebraic structure of Yangian invariants in planar ${ m ext{N}=4}$ SYM, linking their 'letters' to cluster algebras and identifying algebraic letters relevant for two-loop amplitudes.
Contribution
It introduces an algebraic approach to determine the 'letters' of Yangian invariants and connects them to cluster algebras of Grassmannian $G(4,n)$, including algebraic letters for specific amplitude configurations.
Findings
Letters of Yangian invariants relate to cluster ${ m ext{A}}$ coordinates of $G(4,n)$.
For $n=6,7$, all Yangian invariants' letters include cluster ${ m ext{A}}$ coordinates.
Algebraic letters for four-mass boxes at $n=8$ match recent two-loop amplitude symbol letters.
Abstract
Motivated by reformulating Yangian invariants in planar SYM directly as forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the 's, which we call "letters", for any Yangian invariant. These are functions of momentum twistors 's, given by the positive coordinates 's of parametrizations of the matrix , evaluated on the support of polynomial equations . We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian , which is relevant for the symbol alphabet of -point scattering amplitudes. For , the collection of letters for all Yangian invariants contains the cluster coordinates of . We determine algebraic letters of Yangian invariant associated with any "four-mass" box, which for…
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