The role of Killing and Killing-Yano tensors for studying the geodesic motion of the particle and the superparticle in a curved background is reviewed. Additionally, the Papadopoulos list [G. Papadopoulos, Class. Quantum Grav. 25, 105016 ( 2008)] for Killing-Yano tensors in G structures is reproduced by studying the torsion types these structures admit. The Papadopoulos list deals with groups G appearing in the Berger classification, and we enlarge the list by considering additional G structures which are not of the Berger type. Possible applications of these results in the study of supersymmetric particle actions and in the AdS/CFT correspondence are outlined.
We obtain the self-interaction for a point charge in the space-time of a cylindrical thin-shell wormhole connecting two identical locally flat geometries with a constant deficit angle. Although this wormhole geometry is locally indistinguishable from a cosmic string background, the corresponding selfforces are different even at the qualitative level. In fact in the cosmic string geometry the force is always repulsive while for the wormhole background we find that the force may point outwards or towards the wormhole throat depending on the parameters of the configuration. These results suggest that the study of the electromagnetic fields of charged particles is a useful tool for testing the global properties of a given background. *
A method for constructing explicit Calabi-Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are non linear in general, but become linear when the objects describing the metric depend on only one complex coordinate of the hyperkahler 4-dimensional space and its complex conjugated. This situation in particular gives a dual description of D6-branes wrapping a complex 1-cycle inside the hyperkahler space [9]. The present work generalize the construction given in that reference. But the explicit solutions we present correspond to the non linear problem. This is a non linear equation with respect to two variables which, with the help of some specific anzatz, is reduced to a non linear equation with a single variable solvable in terms of elliptic functions. In these terms we construct an infinite family of non compact Calabi-Yau metrics. Contents 1. Introduction 1 2. Calabi-Yau metrics with an isometry preserving the SU(3) structure 3 2.1 The general form of the SU(3) structure. .
As is well known, when D6 branes wrap a special lagrangian cycle on a non compact CY 3-fold in such a way that the internal string frame metric is Kahler there exists a dual description, which is given in terms of a purely geometrical eleven dimensional background with an internal metric of G 2 holonomy. It is also known that when D6 branes wrap a coassociative cycle of a non compact G 2 manifold in presence of a self-dual two form strength the internal part of the string frame metric is conformal to the G 2 metric and there exists a dual description, which is expressed in terms of a purely geometrical eleven dimensional background with an internal non compact metric of Spin (7) holonomy. In the present work it is shown that any G 2 metric participating in the first of these dualities necessarily participates in one of the second type. Additionally, several explicit Spin(7) holonomy metrics admitting a G 2 holonomy reduction along one isometry are constructed. These metrics can be described as R-fibrations over a 6-dimensional Kahler metric, thus realizing the pattern Spin(7) → G 2 → (Kahler) mentioned above. Several of these examples are further described as fibrations over the Eguchi-Hanson gravitational instanton and, to the best of our knowledge, have not been previously considered in the literature. *
Non-compact G 2 holonomy metrics that arise from a T 2 bundle over a hyper-Kähler space are constructed. These are one parameter deformations of certain metrics studied by Gibbons, Lü, Pope and Stelle in [1]. Seven-dimensional spaces with G 2 holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational instantons are found, together with other examples. By using the Apostolov-Salamon theorem [2], we construct a new example that, still being a T 2 bundle over hyper-Kähler, represents a non trivial two parameter deformation of the metrics studied in [1]. We then review the Spin(7) metrics arising from a T 3 bundle over a hyper-Kähler and we find two parameter deformation of such spaces as well. We show that if the hyper-Kähler base satisfies certain properties, a non trivial three parameter deformations is also possible. The relation between these spaces with half-flat and almost G 2 holonomy structures is briefly discussed.The problem of classifying spaces of G 2 holonomy possessing one isometry whose Killing vector orbits form a Kähler six-dimensional space was analyzed both by physicist and mathematicians. In Ref. [23], it was concluded that such geometries are described by a sort of holomorphic monopole equation together with a condition related to the integrability of the complex structure. Such condition turns out to be stronger than the one required by supersymmetry. On the other hand, Apostolov and Salamon have proven in [2] that the Kähler condition yields the existence of a new Killing vector that commutes with the first, so that these metrics are toric. Besides, it was shown that such a G 2 metric yields a four-dimensional manifold equipped with a complex symplectic structure and a one-parameter family of functions and 2forms linked by second order equations (henceforth called Apostolov-Salamon equation). The inverse problem, i.e. the one of constructing a torsion-free G 2 structure starting from such a four-dimensional space was also discussed in [2]. Then, a natural question arises as to whether both description of this classification problem are equivalent. In Ref [24], it was argued that this is indeed the case. Moreover, in Ref. [1,2,24] such a construction was employed to generate new G 2 -metrics. In the present work, the solution generating technique will be analyzed in detail and a wider family of G 2 -metrics will be written down. In particular, the Eguchi-Hanson and the Taub-Nut metrics will be dimensionally extended to new examples of G 2 holonomy following the construction proposed in [1]. In section 2 we discuss the Apostolov-Salamon theorem [2], which formalizes a method for systematically constructing spaces with special holonomy G 2 by starting with a hyper-Kähler space in four-dimensions. This construction is actually the one previously employed by Gibbons, Lü, Pope and Stelle in Ref.[1] and here we discuss it within the framework of [2]. Then, we describe some explicit examples in order to illustrate the procedure. In particular, we show how some of the G 2 metrics obtained i...
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