Toric G 2 and Spin(7) Holonomy Spaces from Gravitational Instantons and Other Examples

Abstract

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...

Spin(7)-manifolds with three-torus symmetry

“…In the final section of the paper we present two examples illustrating our reduction and reconstruction procedures. One of the examples complements previous ones that have appeared in the context of domain wall problems in supergravity theories [12,13].…”

New spin(7) holonomy metrics admittingG2holonomy reductions and M-theory/type-IIA dualities

“…The local expression (2.27) describe infinite family of Spin(7) metrics admitting G 2 reductions. In this section we found the metrics corresponding to known G 2 holonomy cases [21]. The lift in this case present no difficulties, but serves as warmup for the next section, in which less trivial examples will be worked out.…”

“…This statement is not difficult to see. The calibration 3-form corresponding to the metrics (2.9) is 16) and the dual form * Φ corresponding to (2.16) is given by [21]…”

“…Thus, we have found the Spin(7) metrics we were looking for. The reason for which the family is infinite is because the G 2 family over which are fibered is also infinite [20], [21].…”

M-theory moduli spaces and torsion-free structures

“…For instance, in reference [24] it was shown that a configuration of D6-branes wrapping a special Lagrangian cycle of a non-compact Calabi-Yau manifold such that the internal string frame metric is Kähler, is dual to a purely geometrical background in eleven dimensions with an internal metric of G 2 -holonomy. Other examples of torsion-free structures dual to configuration with fluxes, in other words, non-zero torsion, can be found in references [25][26][27][28], this time from six-dimensions to a G 2 -holonomy. For further examples and results in the Spin(7) and G 2 cases see reference [29,30].…”