Abstract
Motivated by reformulating Yangian invariants in planar $$ \mathcal{N} $$
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 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 $$ \mathcal{A} $$
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.
Twistor Space Structure Of One-Loop Amplitudes In Gauge Theory
