Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions

It is well known that in four or more dimensions, there exist exotic manifolds; manifolds that are homeomorphic but not diffeomorphic to each other. More precisely, exotic manifolds are the same topological manifold but have inequivalent differentiable structures. This situation is in contrast to the uniqueness of the differentiable structure on topological manifolds in one, two and three dimensions. As exotic manifolds are not diffeomorphic, one can argue that quantum amplitudes for gravity formulated as functional integrals should include a sum over not only physically distinct geometries and topologies but also inequivalent differentiable structures. But can the inclusion of exotic manifolds in such sums make a significant contribution to these quantum amplitudes? This paper will demonstrate that it will. Simply connected exotic Einstein manifolds with positive curvature exist in seven dimensions. Their metrics are found numerically; they are shown to have volumes of the same order of magnitude. Their contribution to the semiclassical evaluation of the partition function for Euclidean quantum gravity in seven dimensions is evaluated and found to be nontrivial. Consequently, inequivalent differentiable structures should be included in the formulation of sums over histories for quantum gravity.Comment: AmsTex, 23 pages 5 eps figures; replaced figures with ones which are hopefully viewable in pdf forma

The index of symmetry of compact naturally reductive spaces

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.submittedVersionFil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Reggiani, Silvio Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Tamuru, Hiroshi. Universidad de Hiroshima. Escuela de Ciencias. Departamento de Matemática; Japón.Matemática Pur

M-Theory on Spin(7) Manifolds

We study M-theory on two classes of manifolds of Spin(7) holonomy that are developing an isolated conical singularity. We construct explicitly a new class of Spin(7) manifolds and analyse in detail the topology of the corresponding classical spacetimes. We discover also an intricate interplay between various anomalies in M-theory, string theory, and gauge theory within these models, and in particular find a connection between half-integral G-fluxes in M-theory and Chern-Simons terms of the N=1, D=3 effective theory.Comment: 81 pages; minor corrections/clarifications, references adde

Cohomogeneity One Manifolds of Spin(7) and G(2) Holonomy

In this paper, we look for metrics of cohomogeneity one in D=8 and D=7 dimensions with Spin(7) and G_2 holonomy respectively. In D=8, we first consider the case of principal orbits that are S^7, viewed as an S^3 bundle over S^4 with triaxial squashing of the S^3 fibres. This gives a more general system of first-order equations for Spin(7) holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over \CP^3, with a non-trivial parameter that characterises the homogeneous squashing of CP^3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k,\ell)=SU(3)/U(1), where the integers k and \ell characterise the embedding of U(1). We find new ALC and AC metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N(k,\ell), and numerical and perturbative results for ALC families with AC limits. We then study D=7 metrics of $G_2$ holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU(3)/(U(1)\times U(1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S^3\times S^3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N(k,\ell), and an explicit parameterisation of SU(3).Comment: Latex, 60 pages, references added, formulae corrected and additional discussion on the asymptotic flow of N(k,l) cases adde

Supercoset CFT's for String Theories on Non-compact Special Holonomy Manifolds

We study aspects of superstring vacua of non-compact special holonomy manifolds with conical singularities constructed systematically using soluble N = 1 superconformal field theories (SCFT's). It is known that Einstein homogeneous spaces G/H generate Ricci flat manifolds with special holonomies on their cones R_+ x G/H, when they are endowed with appropriate geometrical structures, namely, the Sasaki-Einstein, tri-Sasakian, nearly Kahler, and weak G_2 structures for SU(n), Sp(n), G_2, and Spin(7) holonomies, respectively. Motivated by this fact, we consider the string vacua of the type: R^{d-1,1} x (N = 1 Liouville) x (N=1 supercoset CFT on G/H) where we use the affine Lie algebras of G and H in order to capture the geometry associated to an Einstein homogeneous space G/H. Remarkably, we find the same number of spacetime and worldsheet SUSY's in our ``CFT cone'' construction as expected from the analysis of geometrical cones over G/H in many examples. We also present an analysis on the possible Liouville potential terms (cosmological constant type operators) which provide the marginal deformations resolving the conical singularities.Comment: 60 pages, no figure, 2 tables; v2 typos correcte

استخدام الانظمة المتسقة في تقسيم الترددات المتعامدة في انظمة الاتصالات الضوئية

Orthogonal frequency division multiplexing (OFDM) has quickly gained its attraction in optical communications that are evolving towards software-enhanced optical transmissions. Coherent optical OFDM (CO-OFDM) takes advantage of software capabilities of electronic digital signal processing (DSP) to perform sophisticated operations and has demonstrated its easiness of realizing high spectral efficiency and combating various distortions at the same time. Coherent optical OFDM (CO-OFDM) has attracted lots of interest due to its high spectral efficiency (SE) and robustness to fiber dispersion and is considered as a promising candidate for long haul optical fiber transmission systems. In the beginning of this study, we will focus in (OFDM) theoretically with basic initial concepts, then a theoretical study of (CO-OFDM) and Direct detection (DD-OFDM) systems in deep with comparing between them, and identifying the advantages and disadvantages for both systems. Next, a practical study for the previous systems, by using (Optisystem), a special simulation program to simulate and analyze the system. This simulation we will simulate (CO-OFDM) for long-haul transmissions with its analysis such as optical signal to noise ratio (OSNR), RF spectrum, and constellation diagrams .In addition, we will simulate (DD-OFDM) with the same previous parameters and compare between the two systems .Then, we will compare (CO-OFDM) with dispersion compensating fiber (DCF) as a treatment to increase the transmission distance. Finally, we will integrate the wavelength division multiplexing system (WDM) with (CO-OFDM) system, to increase the system performance and achieve high data rates