A note on letters of Yangian invariants

Song He,Zhenjie Li

doi:10.1007/jhep02(2021)155

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 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} 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.

Highlights

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
We make a simple observation that an alternative construction, based on solving polynomial equations associated with plabic graphs for Yangian invariants, may provide another route to symbol alphabet including algebraic letters
We will argue that the alphabet of Yangian invariants, which will be defined shortly, contains the symbol letters of n = 6 and n = 7 amplitudes and the algebraic letters for two-loop n = 8 NMHV amplitude mentioned above

Summary

Letters of Yangian invariants

We will illustrate our algorithm by finding (subsets of) alphabets of Yangian invariants for certain n and k. The procedure goes as follows: for any given n and k, we first scan through all possible 4k-dimensional cell of G+(k, n) and list the resulting. The representation of any Yangian invariant can be obtained using the procedure given in [1], in terms of the matrix C({α}) with canonical coordinates {α}1≤i≤4k for the cell; equivalently these coordinates can be identified with non-trivial edge variables of a representation plabic graph. It is straightforward to find all equivalence moves and obtain the complete alphabet of the Yangian invariant; for more involved cases, we will content with ourselves in finding a subset of the alphabet by applying such move once, and the results turn out to be already illuminating

Letters of NMHV and MHV invariants

Algebraic letters of N2MHV invariants

Discussions