Four-dimensional Case Research Articles

SU(N) confining strings

String representations of the Wilson loop are constructed in the SU(N)-version of compact QED in three and four dimensions. This is done exactly in the case of the fundamental Wilson loop and in the large-N limit in the case of the adjoint Wilson loop. Using for concreteness the three-dimensional fundamental case, it is demonstrated how the resulting SU(N)-generalization of the so-called theory of confining strings can be obtained in various ways. In its weak-field limit (corresponding to the limit of low monopole densities), this theory enables one to fix the value of the string tension, which cannot be fixed when deduced from the mean magnetic field inside a flat contour (the derivation of this field is also presented). Moreover, the obtained theory enables one to find also the coupling constants of terms in the expansion of the nonlocal string effective action, which are higher in the derivatives than the Nambu-Goto term (some of these terms vanish at the flat surface). In the four-dimensional case with the theta-term, the critical values of the theta-parameter, at which the problem of crumpling of large world sheets might be solved, are found in both the fundamental case and in the large-N limit of the adjoint case. These values are only accessible provided the electric coupling constant is larger than a certain value, in accordance with the known fact that confinement in the four-dimensional compact QED holds in the strong-coupling regime.

Gravitational collapse in higher-dimensional charged-Vaidya space-time

We analyze here the gravitational collapse of higher-dimensional charged-Vaidya spacetime. We show that singularities arising in a charged null fluid in higher dimension are always naked violating at least strong cosmic censorship hypothesis (CCH), though not necessarily weak CCH. We show that earlier conclusions on the occurrence of naked singularities in four-dimensional case can be extended essentially in the same manner in 5D case also

Higher dimensional wormhole geometries with compact dimensions

This paper studies wormhole solutions to Einstein gravity with an arbitrary number of time dependent compact dimensions and a matter-vacuum boundary. A new gauge is utilized which is particularly suited for studies of the wormhole throat. The solutions possess arbitrary functions which allow for the description of infinitely many wormhole systems of this type and, at the stellar boundary, the matter field is smoothly joined to vacuum. It turns out that the classical vacuum structure differs considerably from the four-dimensional theory and is, therefore, studied in detail. The presence of the vacuum-matter boundary and extra dimensions places interesting restrictions on the wormhole. For example, in the static case, the radial size of a weak energy condition (WEC) respecting throat is restricted by the extra dimensions. There is a critical dimension, D=5, where this restriction is eliminated. In the time dependent case, one cannot respect the WEC at the throat as the time dependence actually tends the solution towards WEC violation. This differs considerably from the static case and the four-dimensional case.

Spin foam models of matter coupled to gravity

We construct a class of spin foam models describing matter coupled to gravity, such that the gravitational sector is described by the unitary irreducible representations of the appropriate symmetry group, while the matter sector is described by the finite-dimensional irreducible representations of that group. The corresponding spin foam amplitudes in the four-dimensional gravity case are expressed in terms of the spin network amplitudes for pentagrams with additional external and internal matter edges. We also give a quantum field theory formulation of the model, where the matter degrees of freedom are described by spin network fields carrying the indices from the appropriate group representation. In the non-topological Lorentzian gravity case, we argue that the matter representations should be appropriate SO(3) or SO(2) representations contained in a given Lorentz matter representation, depending on whether one wants to describe a massive or a massless matter field. The corresponding spin network amplitudes are given as multiple integrals of propagators which are matrix spherical functions.

Thermodynamics and kinetic theory of relativistic gases in 2D cosmological models

A kinetic theory of relativistic gases in a two-dimensional space is developed in order to obtain the equilibrium distribution function and the expressions for the fields of energy per particle, pressure, entropy per particle and heat capacities in equilibrium. Furthermore, by using the method of Chapman and Enskog for a kinetic model of the Boltzmann equation the non-equilibrium energy-momentum tensor and the entropy production rate are determined for a universe described by a two-dimensional Robertson-Walker metric. The solutions of the gravitational field equations that consider the non-equilibrium energy-momentum tensor - associated with the coefficient of bulk viscosity - show that opposed to the four-dimensional case, the cosmic scale factor attains a maximum value at a finite time decreasing to a "big crunch" and that there exists a solution of the gravitational field equations corresponding to a "false vacuum". The evolution of the fields of pressure, energy density and entropy production rate with the time is also discussed.

REFLEXIVE POLYHEDRA, WEIGHTS AND TORIC CALABI-YAU FIBRATIONS

During the last years we have generated a large number of data related to Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive polyhedra. We classified all reflexive polyhedra in three dimensions leading to K3 hypersurfaces and have also completed the four-dimensional case relevant to Calabi-Yau threefolds. In addition, we have analysed for many of the resulting spaces whether they allow fibration structures of the types that are relevant in the context of superstring dualities. In this survey we want to give background information both on how we obtained these data, which can be found at our web site, and on how they may be used. We give a complete exposition of our classification algorithm at a mathematical (rather than algorithmic) level. We also describe how fibration structures manifest themselves in terms of toric diagrams and how we managed to find the respective data. Both for our classification scheme and for simple descriptions of fibration structures the concept of weight systems plays an important role.

Nonperturbative continuity in graviton mass versus perturbative discontinuity

We address the question whether a graviton could have a small nonzero mass. The issue is subtle for two reasons: there is a discontinuity in the mass in the lowest tree-level approximation, and, moreover, the nonlinear four-dimensional theory of a massive graviton is not defined unambiguously. First, we reiterate the old argument that for the vanishing graviton mass the lowest tree-level approximation breaks down since the higher order corrections are singular in the graviton mass. However, there exist nonperturbative solutions which correspond to the summation of the singular terms and these solutions are continuous in the graviton mass. Furthermore, we study a completely nonlinear and generally covariant five-dimensional model which mimics the properties of the four-dimensional theory of massive gravity. We show that the exact solutions of the model are continuous in the mass, yet the perturbative expansion exhibits the discontinuity in the leading order and the singularities in higher orders as in the four-dimensional case. Based on exact cosmological solutions of the model we argue that the helicity-zero graviton state which is responsible for the perturbative discontinuity decouples from the matter in the limit of vanishing graviton mass in the full classical theory.

Yang–Mills Algebra

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincare series of this algebra $$\mathcal{A}$$ and for the dimension in degree n of the graded Lie algebra of which $$\mathcal{A}$$ is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.

Spatially anisotropic four-dimensional gauge interactions, planar fermions, and magnetic catalysis

We consider magnetic catalysis in a field-theoretic system of (3+1)-dimensional Dirac fermions with anisotropic kinetic term. By placing the system in a strong external magnetic field, we examine magnetically-induced fermion mass generation. When the coupling anisotropy is strong, in which case the fermions effectively localize on the plane, we find a significant enhancement of the induced mass gap compared to the isotropic four-dimensional case of quantum electrodynamics. As expected on purely dimensional grounds, the mass and critical temperature scale with the square root of the magnetic field. This phenomenon might be related to recent experimental findings on magnetically-induced gaps at the nodes of d-wave superconducting gaps in high-temperature cuprates.

Absolute Points of Continuous and Smooth Polarities

This paper deals with sets of absolute points of continuous or smooth polarities in compact, connected or smooth projective planes, called topological polar unitals or smooth polar unitals, respectively. We will show that topological polar unitals are Z2-homology spheres. In the four-dimensional case, a topological polar unital U is either a topological oval, or any line which intersects U in more than one point intersects in a set homeomorphic to S1. Smooth polar unitals turn out to be smoothly embedded submanifolds of the point space. Moreover, secants intersect such unitals transversally. For these unitals, we will obtain full information on the existence of secants, tangents and exterior lines through given points according to their position with respect to the unital. The main result of this paper states that the possible dimensions of smooth polar unitals coincide with those of sets of absolute points of continuous polarities in the classical projective planes P2F, Fϵ{R,C,H,O}. Finally, we will prove that smooth polar unitals in four-dimensional smooth projective planes are topological ovals or are homeomorphic to S3.

Dimensional Regularization and the n-Wave Procedure for Scalar Fields in Many-Dimensional Quasi-Euclidean Spaces

We obtain the vacuum expectation values of the energy–momentum tensor for a scalar field arbitrarily coupled to a curvature in the case of an N-dimensional quasi-Euclidean space–time; the vacuum is defined in accordance with the Hamiltonian diagonalization method. We extend the n-wave procedure to the many-dimensional case. We find all the counterterms in the case N=5 and the counterterms for the conformal scalar field in the cases N=6,7. We determine the geometric structure of the first three counterterms in the N-dimensional case. We show that all the subtractions in the four-dimensional case and the first three subtractions in the many-dimensional case correspond to the renormalization of the parameters in the bare gravitational Lagrangian. We discuss the geometric structure of the other counterterms in the many-dimensional case and the problem of eliminating the conformal anomaly in the four-dimensional case.

Conformal Einstein Spaces in N-Dimensions

This paper presents a set of necessary and sufficient conditions for ann-dimensional semi-Riemannian manifold to be conformal to an Einstein space. We extend results due to C. N. Kozameh, E. T. Newman and K. P. Tod who solved the problem in the four-dimensional Lorentz case formanifolds with nondegenerate Weyl tensor, i.e. for spacetimes withJ ≠ 0. In particular, in n-dimension we will find tensorialconditions if the Weyl tensor operates injectively on the alternatingtwo-forms. Moreover, in the four-dimensional Riemannian case we canalways decide whether a manifold is locally conformal to an Einsteinspace.

The Pomeron intercept in λφ3 theory in 4 Minkowski +1 compact dimensions

We calculate the total cross section for two scalar particles scattering at high energies in λφ 3 theory in five dimensions, four of which are usual Minkowski ones and the fifth is compact. It is shown that the cross section is dominated by exchange of Pomeron whose intercept is larger than in usual four-dimensional case.

Thelarge cosmological constant approximation to classical andquantum gravity: model examples

We have recently introduced an approach for studying perturbativelyclassical and quantum canonical general relativity. The perturbativetechnique appears to preserve many of the attractive features of thenon-perturbative quantization approach based on Ashtekar's newvariables and spin networks. With this approach one can findperturbatively classical observables (quantities that have vanishingPoisson brackets with the constraints) and quantum states (states thatare annihilated by the quantum constraints). The relative ease withwhich the technique appears to deal with these traditionally hardproblems opensup several questions concerning how relevant the resultsproduced can possibly be. Among the questions is the issue of howuseful are results for large values of the cosmological constant andhow the approach can deal with several pathologies that are expectedto be present in the canonical approach to quantum gravity. With theaim of clarifying these points, and to make our construction asexplicit as possible, we study its application in several simplemodels. We consider Bianchi cosmologies, the asymmetric top,coupled harmonic oscillators with constant energy density and a simplequantum mechanical system with two Hamiltonian constraints. We findthat the technique satisfactorily deals with the pathologies of thesemodels and offers promise for finding (at least some) results even forsmall values of the cosmological constant. Finally, we briefly sketchhow the method would operate in the full four-dimensional quantumgeneral relativity case.

VISION GUIDED CIRCUMNAVIGATING AUTONOMOUS ROBOTS

We present a system for vision guided autonomous circumnavigation, allowing a mobile robot to navigate safely around objects of arbitrary pose, and avoid obstacles. The system performs model-based object recognition from an intensity image. By enabling robots to recognize and navigate with respect to particular objects, this system empowers robots to perform deterministic actions on specific objects, rather than general exploration and navigation as emphasized in much of the current literature. This paper describes a fully integrated system, and, in particular, introduces canonical-views. Further, we derive a direct algebraic method for finding object pose and position for the four-dimensional case of a ground-based robot with uncalibrated vertical movement of its camera. Vision for mobile robots can be treated as a very different problem to traditional computer vision, as mobile robots have a characteristic perspective, and there is a causal relation between robot actions and view changes. Canonical-views are a novel, active object representation designed specifically to take advantage of the constraints of the robot navigation problem to allow efficient recognition and navigation.

Environmental contour analysis in earthquake engineering

In many engineering problems the uncertainty in the design and analysis process is dominated by uncertainty concerning the environmentally induced loading on the system. One such type of system is a structure located in high seismic zones, where there is a high level of uncertainty associated with the ground acceleration. In this study the demand caused by the environmental loads is statistically characterized in terms of magnitude, site-to-source distance and attenuation error at a specified structural period. Through the use of a reliability-based procedure known as the Inverse-First Order Reliability Method all of the combinations of the random loading variables that produce a response spectrum with the specified return period may be identified. These infinite number of combinations produce an Environmental Contour that may be derived for the two, three, or four-dimensional case. Because these contours represent an infinite number of combinations of environmental loading random variables, and in turn, a family of response spectra with the same return period, one need only search the Environmental Contour for the response spectrum producing the peak spectral acceleration at the structural period of interest. This study presents a general multi-variate framework focusing on the derivation of these two, three, and four-dimensional Environmental Contours. Initially, the magnitude and site-to-source distance are assumed to be statistically independent. Two and three-dimensional Environmental Contours are derived for this case and critical response spectra for three different, commonly used return periods are examined. Then a hypothetical example in which the site-to-source distance is assumed LogNormal and dependent on magnitude, is used as a basis for discussion for a generic California site. A qualitative discussion between the assumption of independence and dependence between magnitude and site-to-source distance is addressed. Finally, the peak spectral accelerations for the dependent two, three, and four-dimensional cases are compared to one another and the possible consequences associated with increasing the complexity of the environmental contour model analyzed and discussed.

(Anti-)Instantons and the Atiyah-Hitchin manifold

The Atiyah-Hitchin manifold arises in many different contexts, ranging from its original occurrence as the moduli space of two SU(2) 't Hooft-Polyakov monopoles in 3+1 dimensions, to supersymmetric backgrounds of string theory. In all these settings, (super)symmetries require the metric to be hyperk\"ahler and have an SO(3) transitive isometry, which in the four-dimensional case essentially selects out the Atiyah-Hitchin manifold as the only such smooth manifold with the correct topology at infinity. In this paper, we analyze the exponentially small corrections to the asymptotic limit, and interpret them as infinite series of instanton corrections in these various settings. Unexpectedly, the relevant configurations turn out to be bound states of $n$ instantons and $\bar n$ anti-instantons, with $|n-\bar n|=0,1$ as required by charge conservation. We propose that the semi-classical configurations relevant for the higher monopole moduli space are Euclidean open branes stretched between the monopoles.

A study of holographic renormalization group flows in d = 6 and d = 3

We present an explicit study of the holographic renormalization group (RG) in six dimensions using minimal gauged supergravity. By perturbing the theory with the addition of a relevant operator of dimension four one flows to a non-supersymmetric conformal fixed point. There are also solutions describing non-conformal vacua of the same theory obtained by giving an expectation value to the operator. One such vacuum is supersymmetric and is obtained by using the true superpotential of the theory. We discuss the physical acceptability of these vacua by applying the criteria recently given by Gubser for the four dimensional case and find that those criteria give a clear physical picture in the six dimensional case as well. We use this example to comment on the role of the Hamilton-Jacobi equations in implementing the RG. We conclude with some remarks on AdS_4 and the status of three dimensional superconformal theories from squashed solutions of M-theory.

Space-time uncertainty relation and Lorentz invariance

We discuss a Lorentz covariant space-time uncertainty relation, which agrees with that of Karolyhazy-Ng-van Dam when an observational time period delta t is larger than the Planck time lp. At delta t < lp, this uncertainty relation takes roughly the form delta t delta x > lp^2, which can be derived from the condition prohibiting the multi-production of probes to a geometry. We show that there exists a minimal area rather than a minimal length in the four-dimensional case. We study also a three-dimensional free field theory on a non-commutative space-time realizing the uncertainty relation. We derive the algebra among the coordinate and momentum operators and define a positive-definite norm of the representation space. In four-dimensional space-time, the Jacobi identity should be violated in the algebraic representation of the uncertainty relation.

Geometric description of a relativistic string

Classical three-dimensional relativistic string theory is considered in terms of world sheet quadratic forms. Taking the second quadratic form, not only the first one, into account is essential. A system of nonlinear evolution equations describing the string dynamics at the surface of primary constraints in a conformally invariant manner is derived. The results are generalized to the four-dimensional case.