Experimental evidence for the occurrence of E 8 in nature and the radii of the Gosset circles

Abstract

A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a (1989) Zamolodchikov model involving the exceptional Lie group E 8. The model predicts 8 particles and predicts the ratio of their masses. The conjectures have now been validated experimentally, at least for the first five masses. The Zamolodchikov model was extended in 1990 to a Kateev–Zamolodchikov model involving E 6 and E 7 as well. In a seemingly unrelated matter, the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of E 8. Under the famous two-dimensional (Peter McMullen) projection of the polytope, the images of the vertices are arranged in eight concentric circles, hereafter referred to as the Gosset circles. The McMullen projection generalizes to any complex simple Lie algebra (in particular not restricted to A-D-E types) whose rank is greater than 1. The Gosset circles generalize as well, using orbits of the Coxeter element on roots. Applying results in Kostant (Am J Math 81:973–1032, 1959), I found some time ago a very easily defined operator A on a Cartan subalgebra, the ratio of whose eigenvalues is exactly the ratio of squares of the radii r i of the generalized Gosset circles. The two matters considered above relate to one another in that the ratio of the masses in the E 6, E 7, E 8 Kateev–Zamolodchikov models are exactly equal to the ratios of the radii of the corresponding generalized Gosset circles.