Cluster Polylogarithms for Scattering Amplitudes

Yangian invariants and cluster adjacency in $$ \mathcal{N} $$ = 4 Yang-Mills

“…It is however interesting to note that if we uplift (3.2) by introducing trivial dependence on additional particles, this simple argument no longer applies. For example, [1,3,5,7,9] still passes the bracket test even though it does not involve any frozen coordinates. The fact that the five-bracket [i, j, k, l, m] passes the bracket test for any choice of indices can be understood in terms of the weak separation criterion [30] for determining when two Plücker coordinates are cluster adjacent.…”

“…In recent years cluster algebras have shed interesting light on the mathematical properties of scattering amplitudes in planar N = 4 supersymmetric Yang-Mills (SYM) theory [1]. Cluster algebraic structure manifests itself in several distinct ways, notably including the appearance of certain Gr(4, n) cluster coordinates in the symbol alphabets [1][2][3][4], cobrackets [1,[5][6][7][8], and integrands [9] of n-particle amplitudes.…”

Descent Equation for superloop and cyclicity of OPE

“…It would be important to extend it to higher orders, especially in a fully analytic manner relying on the methods of Ref. [31], using the heptagon bootstrap program [32,33,34,35] that generalizes earlier results on the hexagon [36,37] .…”

The Two-Loop Remainder Function for Eight and Nine Particles

“…This has been especially true in the planar limit of N = 4 super-Yang-Mills (sYM) theory [1,2], where nontrivial amplitudes have been computed in six-and seven-particle kinematics through (respectively) seven and four loops [3][4][5][6][7][8][9][10][11][12], three-particle form factors have been computed through five loops [13,14], and methods exist for calculating the symbols of several classes of two-loop amplitudes to all particle multiplicity [15][16][17]. With the benefit of this concrete data, a number of unexpected analytic [7,18], cluster-algebraic [19][20][21][22][23][24][25][26][27][28], number-theoretic [18], and positivity [29,30] properties have been discovered, which have in turn stimulated the development of increasingly advanced computational techniques. The success of these explorations clearly motivates the further study of amplitudes involving eight and more particles, especially as new algebraic and analytic features are known to arise in these higher-point amplitudes.…”