571 research outputs found
The Role of Spin(9) in Octonionic Geometry
Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some
interactions between Spin(9) and geometries related to octonions. Several
topics are discussed in this respect: explicit descriptions of the Spin(9)
canonical 8-form and its analogies with quaternionic geometry as well as the
role of Spin(9) both in the classical problems of vector fields on spheres and
in the geometry of the octonionic Hopf fibration. Next, we deal with locally
conformally parallel Spin(9) manifolds in the framework of intrinsic torsion.
Finally, we discuss applications of Clifford systems and Clifford structures to
Cayley-Rosenfeld planes and to three series of Grassmannians.Comment: 25 page
Spin(9) geometry of the octonionic Hopf fibration
We deal with Riemannian properties of the octonionic Hopf fibration
S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In
particular, we show that any vertical vector field has at least one zero, thus
reproving the non-existence of S^1 subfibrations. We then discuss
Spin(9)-structures from a conformal viewpoint and determine the structure of
compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a
list of examples of locally conformally parallel Spin(9)-manifolds.Comment: Proofs and Examples revised, some references adde
The Octonions
The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction
Octonionic Yang-Mills Instanton on Quaternionic Line Bundle of Spin(7) Holonomy
The total space of the spinor bundle on the four dimensional sphere S^4 is a
quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider
octonionic Yang-Mills instanton on this eight dimensional gravitational
instanton. This is a higher dimensional generalization of (anti-)self-dual
instanton on the Eguchi-Hanson space.
We propose an ansatz for Spin(7) Yang-Mills field and derive a system of
non-linear ordinary differential equations. The solutions are classified
according to the asymptotic behavior at infinity. We give a complete solution,
when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7).
The existence of more general Spin(7) valued solutions can be seen by making an
asymptotic expansion.Comment: A reference added; 22 pages,a final version to appear J.Geom.Phy
Unified Octonionic Representation of the 10-13 Dimensional Clifford Algebra
We give a one dimensional octonionic representation of the different Clifford
algebra Cliff(5,5)\sim Cliff(9,1), Cliff(6,6)\sim Cliff(10,2) and lastly
Cliff(7,6)\sim Cliff(10,3) which can be given by (8x8) real matrices taking
into account some suitable manipulation rules.Comment: RevTex file, 19 pages, to be published in Int. J. of Mod. Phys.
Exceptional quantum geometry and particle physics II
We continue the study undertaken in [13] of the relevance of the exceptional
Jordan algebra of hermitian octonionic matrices for the
description of the internal space of the fundamental fermions of the Standard
Model with 3 generations. By using the suggestion of [30] (properly justified
here) that the Jordan algebra of hermitian octonionic
matrices is relevant for the description of the internal space of the
fundamental fermions of one generation, we show that, based on the same
principles and the same framework as in [13], there is a way to describe the
internal space of the 3 generations which avoids the introduction of new
fundamental fermions and where there is no problem with respect to the
electroweak symmetry.Comment: 18 page
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