SU(5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling

摘要

We associate the lepton-quark families with the vertices of the 4D polytopes 5-cell (0001) (A4) and the rectified 5-cell (0100) (A4) derived from the SU(5) Coxeter-Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope (1001) A(4) whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the SU(5) charge conservation. The Dynkin diagram symmetry of the SU(5) diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes (1000) (A4) + (0100) (A4) + (0010) (A4) + (0001) (A4) whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to SO(10) and even to SO(11) by noting the Coxeter-Dynkin diagram embedding A(4) subset of D-5 subset of B-5. Another embedding can be made through the relation A(4) subset of D-5 subset of E-6 for more popular GUT's. Appendix A includes the quaternionic representations of the Coxeter-Weyl groups W(A(4)) subset of W(H-4) which can be obtained directly from W(E-8) by projection. This leads to relations of the SU(5) polytopes with the quasicrystallography in 4D and E-8 polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter-Weyl group W(A(4)) approximate to S-5.