Unified Field Theoretical Models from Generalized Affine Geometries III

In this work the underlying structure of new type of Unified Field Theoretical model introduced in by the authors is elucidated and analyzed from the geometrical and group theoretical point of view. Our goal is to take advantage of the geometrical and topological properties of this theory in order to determine the minimal group structure of the resultant spacetime manifold able to support a fermionic structure. From this fact, the relation between antisymmetric torsion and Dirac structure of the spacetime is determined and important physical consequences enumerated. In the case of spacetime with torsion the real meaning of the spin-frame alignment is find and the question of the minimal coupling is discussed based in the important cases of tratorial, totally antisymmetric and general torsion fields.     

On the axioms of topological electromagnetism

The axioms of topological electromagnetism that were given by Hehl, Obukhov, and Rubilar are refined by the use of geometrical and topological notions that are found on orientable manifolds. The central problem of defining the spacetime electromagnetic constitutive law in terms of the geometrical and topological structure of the spacetime manifold is elaborated upon in the linear and nonlinear cases. The manner by which the spacetime metric might follow from the electromagnetic constitutive law is examined in the linear case. The possibility that the intersection form of the spacetime manifold might play a role in defining a topological basis for a nonlinear electromagnetic constitutive law is explored. The manner by which electromagnetic wave motion relates to the geometric structure is also discussed.     

Superstrings with torsion

The conditions for spacetime supersymmetry of the heterotic superstring in backgrounds with arbitrary metric, torsion, Yang-Mills and dilaton expectation values are determined using the sigma model approach. The resulting equations are explicitly solved for the torsion and dilaton fields, and the remaining equations cast in a simple form. Previously unnoticed topological obstructions to solving these equations are found. The equations are shown to agree to leading order in perturbation theory with those derived in a field theory approach, provided one considers a more general ansatz than in previous analyses by allowing for a warp factor for the metric. Exact solutions with non-zero torsion are found, indicating a new class of finite sigma models. These solutions break the E_χ ⊗ E_χ or SO(32) gauge group down to a large variety of subgroups. Orbifolds with torsion are constructed. A perturbative analysis of the equations indicates a class of solutions whose existence has been recently argued for on other grounds. Brief comments are made on the implications for phenomenology.     

Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion

I explain the geometric basis for the recently-discovered nonholonomic mapping principle which permits deriving laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein's equivalence principle. As an important consequence, it yields a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force makes trajectories autoparallel rather than geodesic, as a manifestation of inertia. A generalization of the mapping principle transforms path integrals from flat spacetimes to those with curvature and torsion, thus playing the role of a quantum equivalence principle. This generalization yields consistent results only for completely antisymmetric or for gradient torsion.     

Spinning test particle in Kalb-Ramond background

In this work we explore the geodesic deviations of spinning test particles in a string inspired Einstein-Kalb-Ramond background. Such a background is known to be equivalent to a spacetime geometry with torsion. We have shown here that the antisymmetric Kalb-Ramond field has a significant effect on the geodesic deviation of a spinning test particle. A search for observational evidence of such an effect in astrophysical experiments may lead to a better understanding of the geometry of the background spacetime.Received: 5 April 2005, Revised: 19 May 2005, Published online: 8 July 2005     

Uniqueness of rotating charged black holes in five-dimensional minimal gauged supergravity

We study a five-dimensional spacetime admitting, in the presence of torsion, a non-degenerate conformal Killing–Yano 2-form which is closed with respect to both the usual exterior differentiation and the exterior differentiation with torsion. Furthermore, assuming that the torsion is closed and co-closed with respect to the exterior differentiation with torsion, we prove that such a spacetime is the only spacetime given by the Chong–Cvetič–Lü–Pope solution for stationary, rotating charged black holes with two independent angular momenta in five-dimensional minimal gauged supergravity.     

The geometrical form for the string space-time action

In the present article, we derive the space-time action of the bosonic string in terms of geometrical quantities. First, we study the space-time geometry felt by a probe bosonic string moving in antisymmetric and dilaton background fields. We show that the presence of the antisymmetric field leads to space-time torsion, and the presence of the dilaton field leads to space-time non-metricity. Using these results we obtain the integration measure for space-time with stringy non-metricity, requiring its preservation under parallel transport. We derive the Lagrangian depending on stringy curvature, torsion and non-metricity.     

Gravity theories with asymptotically flat instantons

We construct explicitly a gravitational instanton which is asymptotically flat and globally euclidean. The instanton is a solution of euclidean gravity coupled to a scalar field in first-order formulation. The presence of a zero in the vierbeins and the consequent non-vanishing torsion allow us to obtain non-trivial topological excitations of the metric field without changing the topologically trivial structure of the underlying (euclidean) spacetime.     

Simplest classical models of topological transitions

It is shown that the simplest classical models of topological transitions have scalar singularity of curvature with a point carrier that is a source of spacetime incompleteness. It is also shown that, close to topological transition, the condition of energy dominance is violated, while the asymptotic behavior of the curvature tensor (increase in curvature on approaching topological transition) and the energy-momentum tensor (violation of the energy-dominance condition) is a common property of the given models and is determined overall by the type of topological transition.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 42–46, December, 1983.     

Unified Field Theoretical Models from Generalized Affine Geometries

New model of a non-dualistic Unified Theory is analyzed. This model is based in a manifold equipped with an underlying hypercomplex structure and zero non-metricity, that makes it geometricaly and physically consistent. Wormhole solution from this new model is presented and is explicitly compared with our previous one coming from the Einstein-Non Abelian Born-Infeld theory (in Class. Quantum Gravity 22:4987–5004, 2005). We find that the torsion plays in this unified theory similar role that Yang Mills type strength field coming from the non-Abelian Born-Infeld energy momentum tensor. The meaning of the Yang-Mills ansatz based in the alignment of the isospin with the frame geometry of the spacetime is discussed.     

Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors

A new form of superselection sectors of topological origin is developed. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C*–algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts’ cohomological analysis to the case where 1–cocycles bear non-trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in the case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1–cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1–cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part much resembles what in literature is known as geometric phases. Indeed, by the very geometrical origin of the 1–cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations. Dedicated to Klaus Fredenhagen on the occasion of his sixtieth birthday     

On unified field theories,dynamical torsion and geometrical models

A new model of a nondualistic unified theory is proposed. This model is absolutely consistent from the mathematical and geometrical points of view and is based on a manifold equipped with an underlying hypercomplex structure and zero nonmetricity. Also we showed that interesting wormhole solutions, similar to the non-Abelian Born-Infeld theory of our previous work [14] can be obtained. The solution of this model is explicitly compared with our previous one and we find that the torsion plays in this unified theory a role similar to that of Yang-Mills type strength field coming from the non-Abelian Born-Infeld energy momentum tensor. The meaning of the Hosoya-Ogura ansatz (namely, the alignment of the isospin with the frame geometry of the space-time) is completely elucidated.     

A minimal time and time-temperature uncertainty principle

We show that introducing torsion in general relativity, that is, physically, considering the effect of the spin and linking the torsion to defects in spacetime topology, we can have a minimal unit of time. Also an uncertainty relation between time and temperature is suggested. The interesting thing is that with this minimal time we can eliminate the divergence of the self-energy integral without introducing any ad hoc cut-off, and it is also possible to understand black-hole evaporation as a process of quantum diffusion which leads directly to the Hawking formula. A minimal operationally definable temperature in a cosmological context is discussed.     

New curvature-torsion relations through decomposition of the Bianchi Identities

The Bianchi Identities relating asymmetric curvature to torsion are obtained as a new set of equations governing second-order curvature tensors. The usual contribution of symmetric curvature to the gravitational field is found to be a subset of these identities though with an added contribution due to torsion gradients. The antisymmetric curvature two-tensor is shown to be related to the divergence of the torsion. Using a model of particle-antiparticle pair production, identification of certain torsion components with electroweak fields is proposed. These components obey equations, similar to Maxwell's, that are subsets of these linear Bianchi identities. These results are shown to be consistent with gauge and other previous analyses.     

Unification,KK-thresholds and the top Yukawa coupling in F-theory GUTs

In a class of F-theory SU(5) GUTs the low energy chiral mass spectrum is obtained from rank one fermion mass textures with a hierarchical structure organized by U(1) symmetries embedded in the exceptional E ₈ group. In these theories chiral fields reside on matter ‘curves’ and the tree-level masses are computed from integrals of overlapping wave functions of the particles at the triple intersection points. This calculation requires knowledge of the exact form of the wave functions. In this work we propose a way to obtain a reliable estimate of the various quantities which determine the strength of the Yukawa couplings. We use previous analysis of KK-threshold effects to determine the (ratios of) heavy mass scales of the theory which are involved in the normalization of the wave functions. We consider similar effects from the chiral spectrum of these models and discuss possible constraints on the emerging matter content. In this approach, we find that the Yukawa couplings can be determined solely from the U(1) charges of the states in the ‘intersection’ and the torsion which is a topological invariant quantity. We apply the results to a viable SU(5) model with minimal spectrum which satisfies all the constraints imposed by our analysis. We use renormalization group analysis to estimate the top and bottom masses and find that they are in agreement with the experimental values.     

The dual curvature tensors and dynamics of gravitomagnetic matter

Gravitomagnetic charge that can also be referred to as the dual mass or magnetic mass is the topological charge in gravity theory. A gravitomagnetic monopole at rest can produce a stationary gravitomagnetic field. Due to the topological nature of gravitomagnetic charge, the metric of spacetime where the gravitomagnetic matter is present will be nonanalytic. In this paper both the dual curvature tensors (which can characterize the dynamics of gravitational charge/monopoles) and the antisymmetric gravitational field equation of gravitomagnetic matter are presented. We consider and discuss the mathematical formulation and physical properties of the dual curvature tensors and scalar, antisymmetric source tensors, dual spin connection (including the low‐motion weak‐field approximation), dual vierbein field as well as dual current densities of gravitomagnetic charge. It is shown that the dynamics of gravitomagnetic charge can be founded within the framework of the above dual quantities. In addition, the duality relationship in the dynamical theories between the gravitomagnetic charge (dual mass) and the gravitoelectric charge (mass) is also taken into account in the present paper.     

Exact vacuum solutions of 4-dimensional metric-affine gauge theories of gravitation

We presenttwo exact spherically symmetric vacuum solutions of gauge theories of gravity on a spacetime with non metric-compatible connection. One of them is defined on a Weyl-Cartan spacetime and the other on a general metric-affine space. We consider Lagrangians which include terms quadratic in the irreducible parts of the curvature, the torsion, and the nonmetricity. The metric part of both solutions is of the Reissner-Nordström type and includes a contribution of an effectivedilatation charge. A nontrivial Weyl 1-form is also common to both solutions. It resembles a Coulomb potential originating from thedilatation charge. The torsion is closely related to the nonmetricity.Supported by the Consejo Superior de Investigaciones Científicas, Serrano 123, E-28006 Madrid, Spain     

Parity violation in four and higher dimensional spacetime with torsion

The possibility of parity violation in a gravitational theory with torsion is extensively explored in four and higher dimensions. In the former case, we have listed our conclusions on when and whether parity ceases to be conserved, with both two- and three-index antisymmetry of the torsion field. In the latter, the bulk spacetime is assumed to have torsion, and the survival of parity violating terms in the four dimensional effective action is studied, using the compactification schemes proposed by Arkani-Hamed-Dimopoulos-Dvali and Randall-Sundrum. An interesting conclusion is that the torsion-axion duality arising in a stringy scenario via the second rank antisymmetric Kalb-Ramond field leads to conservation of parity in the gravity sector in any dimension. However, parity violating interactions do appear for spin-1/2 fermions in such theories, which can have crucial phenomenological implications.Received: 17 July 2003, Revised: 12 February 2004, Published online: 23 April 2004     

The cornerstone role of the torsion in Finslerian physical worlds

It is shown that there are no metric-compatible connections with zero torsion onproperly Finslerian, i.e. post-Riemannian, metrics. Since Finslerian connections exist on Riemannian metrics, the torsion rather than the metric becomes the object which determines whether the geometry is properly Finslerian or not. On the other hand, the solder forms and connection are determined by the torsion if the affine curvature is zero, the torsion then containing all the information about the geometric reality of spacetime. Since the metric curvature may still be Riemannian, the question arises of whether its present central role in spacetime physics is but a consequence of requiring that all the geometric content of spacetime be contained in the metric.     

Charge Without Charge,Regular Spherically Symmetric Solutions and the Einstein-Born-Infeld Theory

The aim of this paper is to continue the research (J. Math. Phys. 46:042501, 2005) of regular static spherically symmetric spacetimes in Einstein-Born-Infeld theories from the point of view of the spacetime geometry and the electromagnetic structure. The energy conditions, geodesic completeness and the main features of the horizons of this spacetime are explicitly shown. A new static spherically symmetric dyonic solution in Einstein-Born-Infeld theory with similar good properties as in the regular pure electric and magnetic cases of our previous work, is presented and analyzed. Also, the circumvention of a version of “no go” theorem claiming the non existence of regular electric black holes and other electromagnetic static spherically configurations with regular center is explained by dealing with a more general statement of the problem.