Geometric and Extensor Algebras and the Differential Geometry of Arbitrary Manifolds

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U of M, based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field gamma defining a parallelism structure on U, which represents in a well defined way the action on U of the restriction there of some given connection del defined on M. Also we give a novel and intrinsic presentation (i.e., one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's connection operator fields are identified, and both appear in the intrinsic Cartan's structure equations satisfied by the Cartan's torsion and curvature extensor fields. We introduce moreover a metrical extensor g in U corresponding to the restriction there of given metric tensor \slg defined on M and also introduce the concept a geometric structure (U,gamma,g) for U and study metric compatibility of covariant derivatives induced by the connection extensor gamma. This permits the presentation of the concept of gauge (deformed) derivatives which satisfy noticeable properties useful in differential geometry and geometrical theories of the gravitational field. Several derivatives operators in metric and geometrical structures, like ordinary and covariant Hodge coderivatives and some duality identities are exhibit.Comment: This paper is an improved version of material contained in math.DG/0501560, math.DG/0501561, math.DG/050200

The central parsecs of active galactic nuclei: challenges to the torus

Type 2 AGN are by definition nuclei in which the broad-line region and continuum light are hidden at optical/UV wavelengths by dust. Via accurate registration of infrared (IR) Very Large Telescope adaptive optics images with optical \textit{Hubble Space Telescope} images we unambiguously identify the precise location of the nucleus of a sample of nearby, type 2 AGN. Dust extinction maps of the central few kpc of these galaxies are constructed from optical-IR colour images, which allow tracing the dust morphology at scales of few pc. In almost all cases, the IR nucleus is shifted by several tens of pc from the optical peak and its location is behind a dust filament, prompting to this being a major, if not the only, cause of the nucleus obscuration. These nuclear dust lanes have extinctions $A_V \geq 3-6$ mag, sufficient to at least hide the low-luminosity AGN class, and in some cases are observed to connect with kpc-scale dust structures, suggesting that these are the nuclear fueling channels. A precise location of the ionised gas H$\alpha$ and [\textsc{Si\,vii}] 2.48 $\mu$m coronal emission lines relative to those of the IR nucleus and dust is determined. The H$\alpha$ peak emission is often shifted from the nucleus location and its sometimes conical morphology appears not to be caused by a nuclear --torus-- collimation but to be strictly defined by the morphology of the nuclear dust lanes. Conversely, [\textsc{Si\,vii}] 2.48 $\mu$m emission, less subjected to dust extinction, reflects the truly, rather isotropic, distribution of the ionised gas. All together, the precise location of the dust, ionised gas and nucleus is found compelling enough to cast doubts on the universality of the pc-scale torus and supports its vanishing in low-luminosity AGN. Finally, we provide the most accurate position of the NGC 1068 nucleus, located at the South vertex of cloud B.Comment: 23 pages, 10 figures, accepted for publication in MNRA

Geometric Algebras and Extensors

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to metric geometric algebras Cl(V,G) and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in Cl(V,G) with easier ones in Cl(V,G_{E}) (e.g., a noticeable relation between the Hodge star operators associated to G and G_{E}). Several useful examples are worked in details fo the purpose of transmitting the "tricks of the trade".Comment: This paper (to appear in Int. J. Geom. Meth. Mod. Phys. 4 (6) 2007) is an improved version of material appearing in math.DG/0501556, math.DG/0501557, math.DG/050155

Valence fluctuations in a lattice of magnetic molecules: application to iron(II) phtalocyanine molecules on Au(111)

We study theoretically a square lattice of the organometallic Kondo adsorbate iron(II) phtalocyanine (FePc) deposited on top of Au(111), motivated by recent scanning tunneling microscopy experiments. We describe the system by means of an effective Hubbard-Anderson model, where each molecule has degenerate effective $d-$orbitals with $xz$ and $yz$ symmetry, which we solve for arbitrary occupation and arbitrary on-site repulsion $U$. To that end, we introduce a generalized slave-boson mean-field approximation (SBMFA) which correctly describes both the non-interacting limit (NIL) $U=0$ and the strongly-interacting limit $U \rightarrow \infty$, where our formalism reproduces the correct value of the Kondo temperature for an isolated FePc molecule. Our results indicate that while the isolated molecule can be described by an SU(4) Anderson model in the Kondo regime, the case of the square lattice corresponds to the intermediate-valence regime, with a total occupation of nearly 1.65 holes in the FePc molecular orbitals. Our results have important implications for the physical interpretation of the experiment.Comment: 7 pages, 2 figure

A family of complex potentials with real spectrum

We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other PT symmetric models, which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.Comment: Six figures and four table