A note on letters of Yangian invariants
Song He, Zhenjie Li
Abstract
A bstract Motivated by reformulating Yangian invariants in planar $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 α ’s of parametrizations of the matrix C ( α ), evaluated on the support of polynomial equations C ( α ) · 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 $$ \mathcal{A} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> 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.