A Note on Letters of Yangian Invariants

Abstract

Motivated by reformulating Yangian invariants in planar $\mathcal{N}=4$ SYM directly as $d\log$ forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the $d\log$'s, which we call "letters", for any Yangian invariant. These are functions of momentum twistors $Z$'s, given by the positive coordinates $\alpha$'s of parametrizations of the matrix $C(\alpha )$, evaluated on the support of polynomial equations $C(\alpha )\cdot Z=0$. We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian $G(4,n)$, which is relevant for the symbol alphabet of $n$-point scattering amplitudes. For $n=6,7$, the collection of letters for all Yangian invariants contains the cluster $\mathcal{A}$ coordinates of $G(4,n)$. We determine algebraic letters of Yangian invariant associated with any "four-mass" box, which for $n=8$ reproduce the $18$ multiplicative-independent, algebraic symbol letters discovered recently for two-loop amplitudes.

arXiv comment15 pages, 6 figures; v3: minor updates to match the published version in JHEP

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Cited by in corpus (15)

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