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The noncommutative and nonassociative geometry of octonionic spacetime, modified dispersion relations and grand unification
 J. Math. Phys
, 2007
"... The Octonionic Geometry (Gravity) developed long ago by Oliveira and Marques is extended to Noncommutative and Nonassociative Spacetime coordinates associated with octonionicvalued coordinates and momenta. The octonionic metric Gµν already encompasses the ordinary spacetime metric gµν, in addition ..."
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The Octonionic Geometry (Gravity) developed long ago by Oliveira and Marques is extended to Noncommutative and Nonassociative Spacetime coordinates associated with octonionicvalued coordinates and momenta. The octonionic metric Gµν already encompasses the ordinary spacetime metric gµν, in addition to the Maxwell U(1) and SU(2) YangMills fields such that implements the KaluzaKlein Grand Unification program without introducing extra spacetime dimensions. The color group SU(3) is a subgroup of the exceptional G2 group which is the automorphism group of the octonion algebra. It is shown that the flux of the SU(2) YangMills field strength ⃗ Fµν through the areamomentum ⃗ Σ µν in the internal isospin space yields corrections O(1/M 2 P lanck) to the energymomentum dispersion relations without violating Lorentz invariance as it occurs with Hopf algebraic deformations of the Poincare algebra. The known Octonionic realizations of the Clifford Cl(8), Cl(4) algebras should permit the construction of octonionic string actions that should have a correspondence with ordinary string actions for strings moving in a curved Cliffordspace target background associated with a Cl(3, 1) algebra.
A Chern–Simons E8 gauge theory of gravity in D = 15, grandunification and generalized gravity in clifford spaces
, 2007
"... A novel Chern–Simons E8 gauge theory of gravity in D = 15 based on an octic E8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extensio ..."
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A novel Chern–Simons E8 gauge theory of gravity in D = 15 based on an octic E8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern–Simons E8 gauge theory. We review the construction showing why the ordinary 11D Chern–Simons gravity theory (based on the Anti de Sitter group) can be embedded into a Cliffordalgebra valued gauge theory and that an E8 Yang–Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E8 gauge bundle formulation was instrumental in understanding the topological part of the 11dim Mtheory partition function. The nature of this 11dim E8 gauge theory remains unknown. We hope that the Chern–Simons E8 gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction.
Nonassociative Octonionic Ternary Gauge Field Theories
, 2010
"... A novel (to our knowledge) nonassociative and noncommutative octonionic ternary gauge field theory is explicitly constructed that it is based on a ternarybracket structure involving the octonion algebra. The ternary bracket was defined earlier by Yamazaki. The field strengths Fµν are given in term ..."
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A novel (to our knowledge) nonassociative and noncommutative octonionic ternary gauge field theory is explicitly constructed that it is based on a ternarybracket structure involving the octonion algebra. The ternary bracket was defined earlier by Yamazaki. The field strengths Fµν are given in terms of the 3bracket [Bµ, Bν,Φ] involving an auxiliary octonionicvalued scalar field Φ = Φaea which plays the role of a ”coupling ” function. In the concluding remarks a list of relevant future investigations are briefly outlined.
Exceptional Jordan Strings/Membranes and Octonionic Gravity/pbranes
, 2011
"... Nonassociative Octonionic Ternary Gauge Field Theories are revisited paving the path to an analysis of the many physical applications of Exceptional Jordan Strings/Membranes and Octonionic Gravity. The old octonionic gravity constructions based on the split octonion algebra Os (which strictly speak ..."
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Nonassociative Octonionic Ternary Gauge Field Theories are revisited paving the path to an analysis of the many physical applications of Exceptional Jordan Strings/Membranes and Octonionic Gravity. The old octonionic gravity constructions based on the split octonion algebra Os (which strictly speaking is not a division algebra) is extended to the full fledged octonion division algebra O. A realvalued analog of the EinsteinHilbert Lagrangian L = R involving sums of all the possible contractions of the Ricci tensors plus their octonioniccomplex conjugates is presented. A discussion follows of how to extract the Standard Model group (the gauge fields) from the internal part of the octonionic gravitational connection. The role of Exceptional Jordan algebras, their automorphism and reduced structure groups which play the roles of the rotation and Lorentz groups is also reexamined. Finally, we construct (to our knowledge) generalized novel octonionic string and pbrane actions and raise the possibility that our generalized 3brane action (based on a quartic product) in octonionic flat backgrounds of 7, 8 octonionic dimensions may display an underlying E7, E8 symmetry, respectively. We conclude with some final remarks pertaining to the developments related to Jordan exceptional algebras, octonions, blackholes in string theory and quantum information theory.
Jordan Algebras and Extremal Black Holes
, 2008
"... We review various properties of the exceptional Euclidean Jordan algebra of degree three. Euclidean Jordan algebras of degree three and their corresponding Freudenthal triple systems were recently shown to be intimately related to extremal black holes in N = 2, d = 4 homogeneous supergravities. Usin ..."
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We review various properties of the exceptional Euclidean Jordan algebra of degree three. Euclidean Jordan algebras of degree three and their corresponding Freudenthal triple systems were recently shown to be intimately related to extremal black holes in N = 2, d = 4 homogeneous supergravities. Using a novel type of eigenvalue problem with eigenmatrix solutions, we elucidate the rich matrix geometry underlying the exceptional N = 2, d = 4 homogeneous supergravity and explore the relations to extremal black holes.
THE EXCEPTIONAL E8 GEOMETRY OF CLIFFORD (16) SUPERSPACE AND CONFORMAL GRAVITY YANG–MILLS GRAND UNIFICATION
, 2008
"... We continue to study the Chern–Simons E8 Gauge theory of Gravity developed by the author which is a unified field theory (at the Planck scale) of a Lanczos–Lovelock Gravitational theory with a E8 Generalized Yang–Mills (GYM) field theory, and is defined in the 15D boundary of a 16D bulk space. The E ..."
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We continue to study the Chern–Simons E8 Gauge theory of Gravity developed by the author which is a unified field theory (at the Planck scale) of a Lanczos–Lovelock Gravitational theory with a E8 Generalized Yang–Mills (GYM) field theory, and is defined in the 15D boundary of a 16D bulk space. The Exceptional E8 Geometry of the 256dim slice of the 256 × 256dimensional flat Clifford (16) space is explicitly constructed based on a spin connection ΩAB M, that gauges the generalized Lorentz transformations in the tangent space of the 256dim curved slice, and the 256 × 256 components of the vielbein field EA M, that gauge the nonabelian translations. Thus, in onescoop, the vielbein EA M encodes all of the 248 (nonabelian) E8 generators and 8 additional (abelian) translations associated with the vectorial parts of the generators of the diagonal subalgebra [Cl(8) ⊗ Cl(8)]diag ⊂ Cl(16). The generalized curvature, Ricci tensor, Ricci scalar, torsion, torsion vector and the Einstein–Hilbert–Cartan action is constructed. A preliminary analysis of how to construct a Clifford Superspace (that is far richer than ordinary superspace) based on orthogonal and symplectic Clifford algebras is presented. Finally, it is shown how an E8 ordinary Yang–Mills in 8D, after a sequence of symmetry breaking processes E8 → E7 → E6 → SO(8, 2), and performing a Kaluza–Klein–Batakis compactification on CP2, involving a nontrivial torsion, leads to a (Conformal) Gravity and Yang–Mills theory based on the Standard Model in 4D. The conclusion is devoted to explaining how Conformal (super) Gravity and (super) Yang–Mills theory in any dimension can be embedded into a (super) Cliffordalgebravalued gauge field theory. Keywords: Cspace gravity; Clifford algebras; grand unification; exceptional algebras; string theory.