Kinematic dynamo action in a sphere. II. Symmetry selection

The magnetic fields of the planets are generated by dynamo action in their electrically conducting interiors. The Earth possesses an axial dipole magnetic field but other planets have other configurations: Uranus has an equatorial dipole for example. In a previous paper we explored a two-parameter class of flows, comprising convection rolls, differential rotation (D) and meridional circulation (M), for dynamo generation of steady fields with axial dipole symmetry by solving the kinematic dynamo equations. In this paper we explore generation of the remaining three allowed symmetries: axial quadrupole, equatorial dipole and equatorial quadrupole. The results have implications for the fully nonlinear dynamical dynamo because the flows qualitatively resemble those driven by thermal convection in a rotating sphere, and the symmetries define separable solutions of the nonlinear equations. Axial dipole solutions are generally preferred (they have lower critical magnetic Reynolds number) for D > 0, corresponding to westward surface drift. Axial quadrupoles are preferred for D 0), axial dipoles are preferred. The equatorial dipole must change sign between east and west hemispheres, and is not favoured by any elongation of the flux in longitude (caused by D) or polar concentrations (caused by M): they are preferred for small D and M. Polar and equatorial concentrations can be related to dynamo waves and the sign of Parker's dynamo number. For the three-dimensional flow considered here, the sign of the dynamo number is related to the sense of spiralling of the convection rolls, which must be the same as the surface drif

A String and M-theory Origin for the Salam-Sezgin Model

An M/string-theory origin for the six-dimensional Salam-Sezgin chiral gauged supergravity is obtained, by embedding it as a consistent Pauli-type reduction of type I or heterotic supergravity on the non-compact hyperboloid ${\cal H}^{2,2}$ times $S^1$. We can also obtain embeddings of larger, non-chiral, gauged supergravities in six dimensions, whose consistent truncation yields the Salam-Sezgin theory. The lift of the Salam-Sezgin (Minkowski)$_4\times S^2$ ground state to ten dimensions is asymptotic at large distances to the near-horizon geometry of the NS5-brane.Comment: Latex, 18 pages; minor correction

3-Branes and Uniqueness of the Salam-Sezgin Vacuum

We prove the uniqueness of the supersymmetric Salam-Sezgin (Minkowski)_4\times S^2 ground state among all nonsingular solutions with a four-dimensional Poincare, de Sitter or anti-de Sitter symmetry. We construct the most general solutions with an axial symmetry in the two-dimensional internal space, and show that included amongst these is a family that is non-singular away from a conical defect at one pole of a distorted 2-sphere. These solutions admit the interpretation of 3-branes with negative tension.Comment: Latex, 12 pages; typos corrected, discussion of brane tensions amende

Kerr-de Sitter Black Holes with NUT Charges

The four-dimensional Kerr-de Sitter and Kerr-AdS black hole metrics have cohomogeneity 2, and they admit a generalisation in which an additional parameter characterising a NUT charge is included. In this paper, we study the higher-dimensional Kerr-AdS metrics, specialised to cohomogeneity 2 by appropriate restrictions on their rotation parameters, and we show how they too admit a generalisation in which an additional NUT-type parameter is introduced. We discuss also the supersymmetric limits of the new metrics. If one performs a Wick rotation to Euclidean spacetime signature, these yield new Einstein-Sasaki metrics in odd dimensions, and Ricci-flat metrics in even dimensions. We also study the five-dimensional Kerr-AdS black holes in detail. Although in this particular case the NUT parameter is trivial, our investigation reveals the remarkable feature that a five-dimensional Kerr-AdS ``over-rotating'' metric is equivalent, after performing a coordinate transformation, to an under-rotating Kerr-AdS metric.Comment: Latex, 21 page

New Black Holes in Five Dimensions

We construct new stationary Ricci-flat metrics of cohomogeneity 2 in five dimensions, which generalise the Myers-Perry rotating black hole metrics by adding a further non-trivial parameter. We obtain them via a construction that is analogous to the construction by Plebanski and Demianski in four dimensions of the most general type D metrics. Limiting cases of the new metrics contain not only the general Myers-Perry black hole with independent angular momenta, but also the single rotation black ring of Emparan and Reall. In another limit, we obtain new static metrics that describe black holes whose horizons are distorted lens spaces L(n;m)= S^3/\Gamma(n;m), where m\ge n+2\ge 3. They are asymptotic to Minkowski spacetime factored by \Gamma(m;n). In the general stationary case, by contrast, the new metrics describe spacetimes with an horizon and with a periodicity condition on the time coordinate; these examples can be thought of as five-dimensional analogues of the four-dimensional Taub-NUT metrics.Comment: 25 page

Orientifolds and Slumps in G_2 and Spin(7) Metrics

We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by C_8, which are complete on a complex line bundle over CP^3. The principal orbits are S^7, described as a triaxially squashed S^3 bundle over S^4. The behaviour in the S^3 directions is similar to that in the Atiyah-Hitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S^4. We then consider new G_2 metrics which we denote by C_7, which are complete on an R^2 bundle over T^{1,1}, with principal orbits that are S^3\times S^3. We study the C_7 metrics using numerical methods, and we find that they have the remarkable property of admitting a U(1) Killing vector whose length is nowhere zero or infinite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S^2 cycles, and both carry magnetic charge with respect to the R-R vector field. We also discuss some four-dimensional hyper-Kahler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(\infty) Toda equation, which can provide a way of studying their interior structure.Comment: Latex, 45 pages; minor correction

Separability and Killing Tensors in Kerr-Taub-NUT-de Sitter Metrics in Higher Dimensions

A generalisation of the four-dimensional Kerr-de Sitter metrics to include a NUT charge is well known, and is included within a class of metrics obtained by Plebanski. In this paper, we study a related class of Kerr-Taub-NUT-de Sitter metrics in arbitrary dimensions D \ge 6, which contain three non-trivial continuous parameters, namely the mass, the NUT charge, and a (single) angular momentum. We demonstrate the separability of the Hamilton-Jacobi and wave equations, we construct a closely-related rank-2 Staeckel-Killing tensor, and we show how the metrics can be written in a double Kerr-Schild form. Our results encompass the case of the Kerr-de Sitter metrics in arbitrary dimension, with all but one rotation parameter vanishing. Finally, we consider the real Euclidean-signature continuations of the metrics, and show how in a limit they give rise to certain recently-obtained complete non-singular compact Einstein manifolds.Comment: Author added, title changed, references added, focus of paper changed to Killing tensors and separability. Latex, 13 page

New Complete Non-compact Spin(7) Manifolds

We construct new explicit metrics on complete non-compact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A_8, is topologically R^8 and another, which we denote by B_8, is the bundle of chiral spinors over $S^4$. Unlike the previously-known complete non-compact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S^4, our new metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP^3. We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L^2-normalisable harmonic 4-form for the A_8 manifold, and two such 4-forms (of opposite dualities) for the B_8 manifold. We use the metrics to construct new supersymmetric brane solutions in M-theory and string theory. In particular, we construct resolved fractional M2-branes involving the use of the L^2 harmonic 4-forms, and show that for each manifold there is a supersymmetric example. An intriguing feature of the new A_8 and B_8 Spin(7) metrics is that they are actually the same local solution, with the two different complete manifolds corresponding to taking the radial coordinate to be either positive or negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which by contrast do not have special holonomy. In an appendix we construct the general solution of our first-order equations for Spin(7) holonomy, and obtain further regular metrics that are complete on manifolds B^+_8 and B^-_8 similar to B_8.Comment: Latex, 29 pages. Appendix obtaining general solution of first-order equations and additional complete Spin(7) manifolds adde

Horava-Witten Stability: eppur si muove

We construct exact time-dependent solutions of the supergravity equations of motion in which two initially non-singular branes, one with positive and the other with negative tension, move together and annihilate each other in an all-enveloping spacetime singularity. Among our solutions are the Horava-Witten solution of heterotic M-theory and a Randall-Sundrum I type solution, both of which are supersymmetric, i.e. BPS, in the time-independent case. In the absence of branes our solutions are of Kasner type, and the source of instability may ascribed to a failure to stabilise some of the modulus fields of the compactification. It also raises questions about the viability of models based on some sorts of negative tension brane.Comment: Latex, 22 pages, extended discussion of the global spacetime structure, and reference adde

A G_2 Unification of the Deformed and Resolved Conifolds

We find general first-order equations for G_2 metrics of cohomogeneity one with S^3\times S^3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics \bB_7 and \bD_7, which have weak-coupling limits that are S^1 times the deformed conifold and the resolved conifold respectively. Our more general first-order equations provide a supersymmetric unification of the two Calabi-Yau manifolds, since the metrics \bB_7 and \bD_7 arise as solutions of the {\it same} system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G_2 solutions to these first-order equations, which we denote by \wtd\bC_7, whose topology is an \R^2 bundle over T^{1,1}. There are two non-trivial parameters characterising the homogeneous squashing of the T^{1,1} bolt. Like the previous examples of the \bB_7 and \bD_7 ALC metrics, here too there is a U(1) isometry for which the circle has everywhere finite and non-zero length. The weak-coupling limit of the \wtd\bC_7 metrics gives S^1 times a family of Calabi-Yau metrics on a complex line bundle over S^2\times S^2, with an adjustable parameter characterising the relative sizes of the two S^2 factors.Comment: Latex, 14 pages, Major simplification of first-order equations; references amende