Theory of everything

2007

All fields of the standard model and gravity are unified as an E8 principal bundle connection. A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs, and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E8 superconnection are described by the curvature and action over a four dimensional base manifold.

arXiv (Cornell University), 2010

The algebraic elements of gravitational and Standard Model gauge fields acting on a generation of fermions may be represented using real matrices. These elements match a subalgebra of spin(11,3) acting on a Majorana-Weyl spinor, consistent with GraviGUT unification. This entire structure embeds in the quaternionic real form of the largest exceptional Lie algebra, E8. These embeddings are presented explicitly and their implications discussed.

A candidate action for an Exceptional E8 gauge theory of gravity in 8D is constructed. It is obtained by recasting the E8 group as the semi-direct product of GL(8, R) with a deformed Weyl-Heisenberg group associated with canonical-conjugate pairs of vectorial and antisymmetric tensorial generators of rank two and three. Other actions are proposed, like the quartic E8 group-invariant action in 8D associated with the Chern-Simons E8 gauge theory defined on the 7-dim boundary of a 8D bulk. To finalize, it is shown how the E8 gauge theory of gravity can be embedded into a more general extended gravitational theory in Clifford spaces associated with the Cl(16) algebra and providing a solid geometrical program of a grand-unification of gravity with Yang-Mills theories. The key question remains if this novel gravitational model based on gauging the E8 group may still be renormalizable without spoiling unitarity at the quantum level.

General Relativity and Gravitation, 1986

Field Theories A grand superspace is proposed as the phase space for gauge field theories with a fixed structure group G over a fixed space-time manifold M. This superspace incorporates all principal fiber bundles with these data. This phase space is the space of isomorphism classes of all connections on all G-principal fiber bundles over M (fixed G and M). The justification for choosing this grand superspace for the phase space is that the space-time and the structure group are determinants of the physical theory, but the principal fiber bundle with the given G and M is not. Grand superspace is studied in terms of a natural universal principal fiber bundle over M, canonically associated with M alone, and with a natural universal connection on this bundle. This bundle and its connection are universal in the sense that all connections on all G-principal fiber bundles (any G) over M can be recovered from this universal bundle and its universal connection by a canonical construction. When G is Abelian, grand superspace is shown to be an Abelian group. Various subspaces of grand superspace consisting of the isomorphism classes of fiat connections and of Yang-Mills connections are also discussed.

International Journal of Geometric Methods in Modern Physics, 2009

We continue to study the Chern-Simons E 8 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 E 8 Generalized Yang-Mills (GYM) field theory, and is defined in the 15D boundary of a 16D bulk space. The Exceptional E 8 Geometry of the 256dim slice of the 256 × 256-dimensional flat Clifford space is explicitly constructed based on a spin connection Ω AB M , that gauges the generalized Lorentz transformations in the tangent space of the 256-dim curved slice, and the 256 × 256 components of the vielbein field E A M , that gauge the nonabelian translations. Thus, in one-scoop, the vielbein E A M encodes all of the 248 (nonabelian) E 8 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 E 8 ordinary Yang-Mills in 8D, after a sequence of symmetry breaking processes E 8 → E 7 → E 6 → SO(8, 2), and performing a Kaluza-Klein-Batakis compactification on CP 2 , 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) Clifford-algebra-valued gauge field theory. The Exceptional E 8 Geometry of Clifford (16) Superspace 387 Exceptional Groups, del Pezzo surfaces and the extra massless particles arising from rational double point singularities can be found in [10]. Clifford algebras and E 8 are key ingredients in Smith's

A novel Chern-Simons E 8 gauge theory of gravity in D = 15 based on an octic E 8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is 15 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) 17 of this Chern-Simons E 8 gauge theory. We review the construction showing why the ordinary 11D Chern-Simons gravity theory (based on the Anti de Sitter group) can 19 be embedded into a Clifford-algebra valued gauge theory and that an E 8 Yang-Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E 8 gauge 21 bundle formulation was instrumental in understanding the topological part of the 11dim M-theory partition function. The nature of this 11-dim E 8 gauge theory remains 23 unknown. We hope that the Chern-Simons E 8 gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional 25 reduction. 18:56 WSPC/IJGMMP-J043 00254 2 C. Castro symmetries for four-dimensional BPS black holes [6] also involved exceptional sym-1 metries associated with the exceptional magic Jordan algebras J 3 [R, C, H, O]. The discovery of the anomaly free 10-dim heterotic string for the algebra E 8 × E 8 was 3 another hallmark of the importance of exceptional Lie groups in Physics. The E 8 group was proposed long ago [24] as a candidate for a grand unification 5 model building in D = 4. An extensive review of the E 6 grand unified models may be found in [26]. The supersymmetric E 8 model has more recently been studied as a 7 fermion family and grand unification model [25] under the assumption that there is a vacuum gluino condensate but this condensate is not accompanied by a dynamical 9 generation of a mass gap in the pure E 8 gauge sector. A study of the interplay among exceptional groups, del Pezzo surfaces and the extra massless particles arising from 11 rational double point singularities can be found in [38]. Clifford algebras and E 8 are key ingredients in Smith's D 4 − D 5 − E 6 − E 7 − E 8 grand unified model in D = 8 [6]. 13 An E 8 gauge bundle was instrumental in the understanding the topological part of the M-theory partition function [27, 32]. A mysterious E 8 bundle which 15 restricts from 12-dim to the 11-dim bulk of M theory can be compatible with 11-dim supersymmetry. The nature of this 11-dim E 8 gauge theory remains unknown. We 17 hope that the Chern-Simons E 8 gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this question. 19 E 8 Yang-Mills theory can naturally be embedded into a Cl(16) algebra gauge theory [33] and the 11D Chern-Simons (super) gravity [4] is a very small sector of 21 a more fundamental polyvector-valued gauge theory in Clifford spaces. Polyvectorvalued supersymmetries [11] in Clifford-spaces [3] turned out to be more fundamen-23 tal than the supersymmetries associated with M, F theory superalgebras [7, 10]. For this reason, we believe that Clifford structures may shed some light into the origins 25 behind the hidden E 8 symmetry of 11D supergravity and reveal more important features underlying M, F theory.

1994

In this paper, we examine the coupling of matter fields to gravity within the framework of the Standard Model of particle physics. The coupling is described in terms of Weyl fermions of a definite chirality, and employs only (anti)self-dual or left-handed spin connection fields. It is known from the work of Ashtekar and others that such fields can furnish a complete description of gravity without matter. We show that conditions ensuring the cancellation of perturbative chiral gauge anomalies are not disturbed. We also explore a global anomaly associated with the theory, and argue that its removal requires that the number of fundamental fermions in the theory must be multiples of 16. In addition, we investigate the behavior of the theory under discrete transformations P, C and T; and discuss possible violations of these discrete symmetries, including CPT, in the presence of instantons and the Adler-Bell-Jackiw anomaly.

2006

A bstract. T he Standard M odel of the theory of elem entary particles is based on the U (1) SU (2) SU (3) sym m etry. In the presence of a gravitation eld, i.e. in a non-at space-tim e m anifold,this sym m etry is im plem ented through three special vector bundles. C onnections associated w ith these vector bundles are studied in this paper. In the Standard M odelthey are interpreted as gauge elds.

We discuss the physical implications of formulating the Standard Model (SM) in terms of the superconnection formalism involving the superalgebra su(2/1). In particular, we discuss the prediction of the Higgs mass according to the formalism and point out that it is ∼170 GeV, in clear disagreement with experiment. To remedy this problem, we extend the formalism to the superalgebra su(2/2), which extends the SM to the left-right symmetric model (LRSM) and accommodates a ∼126 GeV Higgs boson. Both the SM in the su(2/1) case and the LRSM in the su(2/2) case are argued to emerge at ∼4 TeV from an underlying theory in which the spacetime geometry is modified by the addition of a discrete extra dimension. The formulation of the exterior derivative in this model space suggests a deep connection between the modified geometry, which can be described in the language of noncommutative geometry, and the spontaneous breaking of the gauge symmetries. The implication is that spontaneous symmetry breaking could actually be geometric/quantum gravitational in nature. The nondecoupling phenomenon seen in the Higgs sector can then be reinterpreted in a new light as due to the mixing of low energy (SM) physics and high energy physics associated with quantum gravity, such as string theory. The phenomenology of a TeV scale LRSM is also discussed, and we argue that some exciting discoveries may await us at the LHC, and other near-future experiments.

Journal of Physics: Conference Series 222 (2010) 012017, 2010

Faced with the persisting problem of the unification of gravity with other fundamental interactions we investigate the possibility of a new paradigm, according to which the basic space of physics is a multidimensional space C associated with matter configurations. We consider general relativity in C. In spacetime, which is a 4-dimensional subspace of C, we have not only the 4-dimensional gravity, but also other interactions, just as in Kaluza-Klein theories. We then consider a finite dimensional description of extended objects in terms of the center of mass, area, and volume degrees of freedom, which altogether form a 16-dimensional manifold whose tangent space at any point is Clifford algebra Cl(1,3). The latter algebra is very promising for the unification, and it provides description of fermions.