TITLE:
On Discrete Hopf Fibrations, Grand Unification Groups, the Barnes-Wall, Leech Lattices, and Quasicrystals
AUTHORS:
Carlos Castro Perelman
KEYWORDS:
Division Algebras, Hopf Fibrations, Barnes-Wall Lattice, Leech Lattice, Exceptional Lie Algebras, Grand Unification, Quasicrystals
JOURNAL NAME:
Journal of High Energy Physics, Gravitation and Cosmology,
Vol.10 No.4,
October
16,
2024
ABSTRACT: A discrete Hopf fibration of S15 over S8 with S7 (unit octonions) as fibers leads to a 16D Polytope P16 with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ16. It is argued (conjectured) how a subsequent 2-1 mapping (projection) of P16 onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 241 polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes
2
41
,
2
31
,
2
21
,
2
11
in
D=8,7,6,5
dimensions, leading, respectively, to the sequence of Coxeter groups
E
8
,
E
7
,
E
6
,SO(
10
)
which are putative GUT group candidates. An embedding of the
E
8
⊕
E
8
and
E
8
⊕
E
8
⊕
E
8
lattice into the Barnes-Wall Λ16 and Leech Λ24 lattices, respectively, is explicitly shown. From the 16D lattice
E
8
⊕
E
8
one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H4 (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s
Q×Q
spanning an 8D space. Similarly, from the 24D lattice
E
8
⊕
E
8
⊕
E
8
one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s)
Q×Q×Q
with H4 symmetry and spanning a 12D space.