A genus six cyclic tetragonal reduction of the Benney equations

A reduction of Benney’s equations is constructed corresponding to Schwartz–Christoffel maps associated with a family of genus six cyclic tetragonal curves. The mapping function, a second kind Abelian integral on the associated Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian σ-function of the curve

Gravitational Instantons, Confocal Quadrics and Separability of the Schr\"odinger and Hamilton-Jacobi equations

A hyperk\"ahler 4-metric with a triholomorphic SU(2) action gives rise to a family of confocal quadrics in Euclidean 3-space when cast in the canonical form of a hyperk\"ahler 4-metric metric with a triholomorphic circle action. Moreover, at least in the case of geodesics orthogonal to the U(1) fibres, both the covariant Schr\"odinger and the Hamilton-Jacobi equation is separable and the system integrable.Comment: 10 pages Late

Moduli, Scalar Charges, and the First Law of Black Hole Thermodynamics

We show that under variation of moduli fields $\phi$ the first law of black hole thermodynamics becomes $dM = {\kappa dA\over 8\pi} + \Omega dJ + \psi dq + \chi dp - \Sigma d\phi$, where $\Sigma$ are the scalar charges. We also show that the ADM mass is extremized at fixed $A$, $J$, $(p,q)$ when the moduli fields take the fixed value $\phi_{\rm fix}(p,q)$ which depend only on electric and magnetic charges. It follows that the least mass of any black hole with fixed conserved electric and magnetic charges is given by the mass of the double-extreme black hole with these charges. Our work allows us to interpret the previously established result that for all extreme black holes the moduli fields at the horizon take a value $\phi= \phi_{\rm fix}(p,q)$ depending only on the electric and magnetic conserved charges: $\phi_{\rm fix}(p,q)$ is such that the scalar charges $\Sigma ( \phi_{\rm fix}, (p,q))=0$.Comment: 3 pages, no figures, more detailed versio

Bulk/Boundary Thermodynamic Equivalence, and the Bekenstein and Cosmic-Censorship Bounds for Rotating Charged AdS Black Holes

We show that one may pass from bulk to boundary thermodynamic quantities for rotating AdS black holes in arbitrary dimensions so that if the bulk quantities satisfy the first law of thermodynamics then so do the boundary CFT quantities. This corrects recent claims that boundary CFT quantities satisfying the first law may only be obtained using bulk quantities measured with respect to a certain frame rotating at infinity, and which therefore do not satisfy the first law. We show that the bulk black hole thermodynamic variables, or equivalently therefore the boundary CFT variables, do not always satisfy a Cardy-Verlinde type formula, but they do always satisfy an AdS-Bekenstein bound. The universal validity of the Bekenstein bound is a consequence of the more fundamental cosmic censorship bound, which we find to hold in all cases examined. We also find that at fixed entropy, the temperature of a rotating black hole is bounded above by that of a non-rotating black hole, in four and five dimensions, but not in six or more dimensions. We find evidence for universal upper bounds for the area of cosmological event horizons and black-hole horizons in rotating black-hole spacetimes with a positive cosmological constant.Comment: Latex, 42 page

Branes as BIons

A BIon may be defined as a finite energy solution of a non-linear field theory with distributional sources. By contrast a soliton is usually defined to have no sources. I show how harmonic coordinates map the exteriors of the topologically and causally non-trivial spacetimes of extreme p-branes to BIonic solutions of the Einstein equations in a topologically trivial spacetime in which the combined gravitational and matter energy momentum is located on distributional sources. As a consequence the tension of BPS p-branes is classically unrenormalized. The result holds equally for spacetimes with singularities and for those, like the M-5-brane, which are everywhere singularity free.Comment: Latex, 9 pages, no figure

Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)

The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, SIM(n-2). Ricci-flat metrics with SIM(2) holonomy were constructed by Kerr and Goldberg, and a single four-dimensional example with a non-zero cosmological constant was exhibited by Ghanam and Thompson. Here we reduce the problem of finding the general $n$-dimensional Einstein metric of SIM(n-2) holonomy, with and without a cosmological constant, to solving a set linear generalised Laplace and Poisson equations on an (n-2)-dimensional Einstein base manifold. Explicit examples may be constructed in terms of generalised harmonic functions. A dimensional reduction of these multi-centre solutions gives new time-dependent Kaluza-Klein black holes and monopoles, including time-dependent black holes in a cosmological background whose spatial sections have non-vanishing curvature.Comment: Typos corrected; 29 page

Studies on the meiofauna of rocky shores

Bibliography: leaves 88-97.Annual macrofaunal and meiofaunal standing stocks were estimated on an exposed rocky shore along the west coast of False Bay, South Africa, using comparable area based sampling techniques. While meiofaunal densities exceeded those of macrofauna in all zones, by an overall ratio of approximately 400:1, macrofaunal biomass exceeded that of meiofauna by an overall ratio of 10:1. The numbers of meiofauna were not evenly distributed across the shore but varied with the algal standing stocks in each zone and their sediment load. By incorporating turnover ratios from the literature, mean annual productivity ratios were calculated which suggested that meiofauna were responsible for 25 of total (excluding bacterial) secondary production. To follow this up, the impact of wave exposure on the meiofauna of one species of alga (viz. Gelidium pristoides) was examined on five shores around False Bay. Meiofaunal densities (dominated by animals between 63um-280um) were significantly greater on sheltered than exposed shores. As the minimum width of Gelidium fronds exceeds that of these permanent meiofauna, and tufts offer little resistance to wave action, only those individuals living in the dense, holdfast region of plants could escape the impact of waves on exposed shores. Total meiofaunal biomass per plant remained constant irrespective of shore type, due to the greater numbers of juvenile bivalves and amphipods on exposed shores. Algal and herbivore biomass were not significantly different between shore types around False Bay and therefore, the proportional contribution by meiofauna to total secondary production on sheltered shores was predicted to be greater than on exposed shores, where the biomass of macrofaunal filter feeders was very high. It has previously been argued that differences in meiofaunal communities between plant species are a result of differential surface area, number of habitats and refugia from predation. The possible fate of meiofaunal productivity as food for higher trophic levels (fish) and the mediating role played by algal complexity was investigated in a series of carefully designed laboratory and field experiments

Kleinian Geometry and the N=2 Superstring

This paper is devoted to the exploration of some of the geometrical issues raised by the $N=2$ superstring. We begin by reviewing the reasons that $\beta$-functions for the $N=2$ superstring require it to live in a four-dimensional self-dual spacetime of signature $(--++)$, together with some of the arguments as to why the only degree of freedom in the theory is that described by the gravitational field. We then move on to describe at length the geometry of flat space, and how a real version of twistor theory is relevant to it. We then describe some of the more complicated spacetimes that satisfy the $\beta$-function equations. Finally we speculate on the deeper significance of some of these spacetimes.Comment: 30 pages, AMS-Te