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From Amplituhedron to Associahedron

Matti Pitkänen


Nima Arkani-Hamed et al have published an article generalizing the notion of amplituhedron to associahedron and shown that it emerges also in string theory. Associahedron characterizes the non-associativity for a product of n algebraic objects. Each face of associahedron defined in n-2-D space corresponds to one particular association for the product (particular bracketing). Also the proposal is made that color corresponds to something less trivial than Chan-Paton factors. In TGD non-associativity at the level of arguments of scattering amplitude is induced from that for octonions: one can assign to space-time surfaces octonionic polynomials and induce arithmetic operations for space-time surface from those for polynomials (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 (space-time 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 space-time geometry. In TGD framework the description of color and four-momentum is unified at the level and the notion of twistor generalizes: one has twistors in 8-D space-time instead of twistors in 4-D space-time so Chan-Paton factors are replaced with something non-trivial.

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