From amplituhedron to associahedronLubos has a nice article (see this) explaining the proposal represented in the newest article by Nima ArkaniHamed, Yuntao Bai, Song He, Gongwang Yan (see this). Amplituhedron is generalized to a purely combinatorial notion of associahedron and shown to make sense also in string theory context (particular bracketing). The hope is that the generalization of amplituhedron to associahedron allows to compute also the contributions of nonplanar diagrams to the scattering amplitudes  at least in N=4 SYM. Also the proposal is made that color corresponds to something less trivial than ChanPaton factors. The remaining problem is that 4D conformal invariance requires massless particles and TGD allows to overcome this problem by using a generalization of the notion of twistor: masslessness is realized in 8D sense and particles massless in 8D sense can be massive in 4D sense. In TGD nonassociativity at the level of arguments of scattering amplitude corresponds to that for octonions: one can assign to spacetime surfaces octonionic polynomials and induce arithmetic operations for spacetime surface from those for polymials (or even rational or analytic functions). I have already earlier demonstrated that associahedron and construction of scattering amplitudes by summing over different permutations and associations of external particles (spacetime surfaces). Therefore the notion of associahedron makes sense also in TGD framework and summation reduces to "integration" over the faces of associahedron. TGD thus provides a concrete interpretation for the associations and permutations at the level of spacetime geometry. In TGD framework the description of color and fourmomentum is unified at the level and the notion of twistor generalizes: one has twistors in 8D spacetime instead of twistors in 4D spacetime so ChanPaton factors are replaced with something nontrivial. 1. Associahedrons and scattering amplitudes The following describes briefly the basic idea between associahedrons. 1.1 Permutations and associations One starts from a noncommutative and nonassociative algebra with product (in TGD framework this algebra is formed by octonionic polynomials with real coefficients defining spacetime surfaces as the zero loci of their real or imaginary parts in quaternionic sense. One can indeed multiply spacetime surface by multiplying corresponding polynomials! Also sum is possible. If one allows rational functions also division becomes possible. All permutations of the product of n elements are in principle different. This is due to noncommutativity. All associations for a given ordering obtained by scattering bracket pairs in the product are also different in general. In the simplest case one has either a(bc) or (ab)c and these 2 give different outcomes. These primitive associations are building bricks of general associations: for instance, abc does not have welldefined meaning in nonassociative case. If the product contains n factors, one can proceed recursively to build all associations allowed by it. Decompose the n factors to groups of m and nm factors. Continue by decomposing these two groups to two groups and repeat until you have have groups consisting of 1 or two elements. You get a large number of associations and you can write a computer code computing recursively the number N(n) of associations for n letters. Two examples help to understand. For n=3 letters one obviously has N(3)= 2. For n=4 one has N(4)=5: decompose abcd to (abc)d and a(bcd) and (ab)(cd) and then the 3 letter groups to two groups: this gives 2+2+1 =5 associations and associahedron in 3D space has therefore 5 faces. 1.2 Geometric representation of association as face of associahedron Associations of n letters can be represented geometrically as so called Stasheff polytope (see this). The idea is that each association of n letters corresponds to a face of polytope in n2dimensional space with faces represented by the associations. Associahedron is constructed by using the condition that adjacent faces (now 2D polygons) intersecting along common face (now 1D edges). The number of edges of the face codes for the structure particular association. Neighboring faces are obtained by doing minimal change which means replacement of some (ab)c with a(bc) appearing in the association as a building bricks or vice versa. This means that the changes are carried out at the root level. 1.3 How does this relate to particle physics? In scattering amplitude letters correspond to external particles. Scattering amplitude must be invariant under permutations and associations of the external particles. In particular, this means that one sums over all associations by assigning an amplitude to each association. Geometrically this means that one "integrates" over the boundary of associahedron by assigning to each face an amplitude. This leads to the notion of associahedron generalizing that of amplituhedron. Personally I find it difficult to believe that the mere combinatorial structure leading to associahedron would fix the theory completely. It is however clear that it poses very strong conditions on the structure of scattering amplitudes. Especially so if the scattering amplitudes are defined in terms of "volumes" of the polyhedrons involved so that the scattering amplitude has singularities at the faces of associahedron. An important constraint on the scattering amplitudes is the realization of the Yangian generalization of conformal symmetries of Minkowski space. The representation of the scattering amplitudes utilizing moduli spaces (projective spaces of various dimensions) and associahedron indeed allows Yangian symmetries as diffeomorphisms of associahedron respecting the positivity constraint. The hope is that the generalization of amplituhedron to associahedron allows to generalize the construction of scattering amplitudes to include also the contribution of nonplanar diagrams of at N=4 SYM in QFT framework. 2. Associations and permutations in TGD framework Also in the number theoretical vision about quantum TGD one encounters associativity constraints leading to the notion of associahedron. This is closely related to the generalization of twistor approach to TGD forcing to introduce 8D analogs of twistors (see this).
Nima et al talk also about color structure of the scattering amplitudes usually regarded as trivial. It is claimed that this is actually not the case and that there is nontrivial dynamics involved. This is indeed the case in TGD framework. Also color quantum numbers are twistorialized in terms of the twistor space of CP_{2}, and one performs a twistorialization at the level of M^{8} and M^{4}× CP_{2}. At the level of M^{8} momenta and color quantum numbers correspond to associative 8momenta. Massless particles are now massless in 8D sense but can be massive in 4D sense. This solves one of the basic difficulty of the ordinary twistor approach. A further bonus is that the choice of the imbedding space H becomes unique: only the twistor spaces of S^{4} (and generalized twistor space of M^{4} and CP_{2} have Kähler structure playing a crucial role in the twistorialization of TGD. To sum up, all roads lead to Rome. Everyone is wellcome to Rome! See the articles Does M 8 − H duality reduce classical TGD to octonionic algebraic geometry? and From amplituhedron to associahedron. Addition: Marni Lee Sheppard wrote a thesis in which the notion of associahedron appeared. I remember discussions in some net group. Her motivations came from category theory. Marni had bad luck. Big boys rarely remember who proposed the idea first if she/he is not a name. See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article From amplituhedron to associahedron.
