The Geometry of Self-dual 2-forms

We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$ and the dimension of the maximal linear subspaces of ${\cal S}_{2n}$ is equal to the (Radon-Hurwitz) number of linearly independent vector fields on the sphere $S^{2n-1}$. We provide a direct proof that for $n$ odd ${\cal S}_{2n}$ has only one-dimensional linear submanifolds. We exhibit $2^c-1$ dimensional subspaces in dimensions which are multiples of $2^c$, for $c=1,2,3$. In particular, we demonstrate that the seven dimensional linear subspaces of ${\cal S}_{8}$ also include among many other interesting classes of self-dual 2-forms, the self-dual 2-forms of Corrigan, Devchand, Fairlie and Nuyts and a representation of ${\cal C}l_7$ given by octonionic multiplication. We discuss the relation of the linear subspaces with the representations of Clifford algebras.Comment: Latex, 15 page

Monopole equations on 8-manifolds with Spin(7) holonomy

We construct a consistent set of monopole equations on eight-manifolds with Spin(7) holonomy. These equations are elliptic and admit non-trivial solutions including all the 4-dimensional Seiberg-Witten solutions as a special case.Comment: 14 pages, LATEX (No figures

Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields F_{\mu \nu}. We derive a topological bound on {\bf R}^8, \int_{M} ( F,F )^2 \geq k \int_{M} p_1^2 where p_1 is the first Pontrjagin class of the SO(n) Yang-Mills bundle and k is a constant. Strongly self-dual Yang-Mills fields realise the lower bound

Statistical properties of the deviations of f 0 F 2 from monthly medians

The deviations of hourly f 0 F 2 from monthly medians for 20 stations in Europe during the period 1958-1998 are studied. Spectral analysis is used to show that, both for original data (for each hour) and for the deviations from monthly medians, the deterministic components are the harmonics of 11 years (solar cycle), 1 year and its harmonics, 27 days and 12 h 50.49 m (2nd harmonic of lunar rotation period L 2 ) periodicities. Using histograms for one year samples, it is shown that the deviations from monthly medians are nearly zero mean (mean < 0.5) and approximately Gaussian (relative difference range between %10 to %20) and their standard deviations are larger for daylight hours (in the range 5-7). It is shown that the amplitude distribution of the positive and negative deviations is nearly symmetrical at night hours, but asymmetrical for day hours. The positive and negative deviations are then studied separately and it is observed that the positive deviations are nearly independent of R12 except for high latitudes, but negative deviations are modulated by R12 . The 90% confidence interval for negative deviations for each station and each hour is computed as a linear model in terms of R12. After correction for local time, it is shown that for all hours the confidence intervals increase with latitude but decrease above 60N. Long-term trend analysis showed that there is an increase in the amplitude of positive deviations from monthly means irrespective of the solar conditions. Using spectral analysis it is also shown that the seasonal dependency of negative deviations is more accentuated than the seasonal dependency of positive deviations especially at low latitudes. In certain stations, it is also observed that the 4th harmonic of 1 year corresponding to a periodicity of 3 months, which is missing in f 0 F 2 data, appears in the spectra of negative variations

The collision and snapping of cosmic strings generating spherical impulsive gravitational waves

The Penrose method for constructing spherical impulsive gravitational waves is investigated in detail, including alternative spatial sections and an arbitrary cosmological constant. The resulting waves include those that are generated by a snapping cosmic string. The method is used to construct an explicit exact solution of Einstein's equations describing the collision of two nonaligned cosmic strings in a Minkowski background which snap at their point of collision.Comment: 10 pages, 6 figures, To appear in Class. Quantum Gra