The Coupled Seiberg-Witten Equations, vortices, and Moduli spaces of stable pairs

We introduce coupled Seiberg-Witten equations, and we prove, using a generalized vortex equation, that, for Kaehler surfaces, the moduli space of solutions of these equations can be identified with a moduli space of holomorphic stable pairs. In the rank 1 case, one recovers Witten's result identifying the space of irreducible monopoles with a moduli space of divisors. As application, we give a short proof of the fact that a rational surface cannot be diffeomorphic to a minimal surface of general type.Comment: late

Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.Comment: LaTeX, 44 pages, to appear in Comm. Math. Phy

Moduli of vortices and Grassmann manifolds

We use the framework of Quot schemes to give a novel description of the moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface with target Mat(r x n, C), where n >= r. We then show that these moduli spaces embed canonically into certain Grassmann manifolds, and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are smooth at least in the local case r=n. For abelian local vortices we prove that, if a certain "quantization" condition is satisfied, the embedding can be chosen in such a way that the induced Fubini-Study structure realizes the Kaehler class of the usual L^2 metric of gauged vortices.Comment: 22 pages, LaTeX. Final version: last section removed, typos corrected, two references added; to appear in Commun. Math. Phy

Hyperholomorpic connections on coherent sheaves and stability

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for instance, when one deals with higher direct images of holomorphic bundles. We show that such sheaves are stable.Comment: 37 pages, version 11, reference updated, corrected many minor errors and typos found by the refere

Toward a Comprehensive Approach to the Collection and Analysis of Pica Substances, with Emphasis on Geophagic Materials

Pica, the craving and subsequent consumption of non-food substances such as earth, charcoal, and raw starch, has been an enigma for more than 2000 years. Currently, there are little available data for testing major hypotheses about pica because of methodological limitations and lack of attention to the problem.In this paper we critically review procedures and guidelines for interviews and sample collection that are appropriate for a wide variety of pica substances. In addition, we outline methodologies for the physical, mineralogical, and chemical characterization of these substances, with particular focus on geophagic soils and clays. Many of these methods are standard procedures in anthropological, soil, or nutritional sciences, but have rarely or never been applied to the study of pica.Physical properties of geophagic materials including color, particle size distribution, consistency and dispersion/flocculation (coagulation) should be assessed by appropriate methods. Quantitative mineralogical analyses by X-ray diffraction should be made on bulk material as well as on separated clay fractions, and the various clay minerals should be characterized by a variety of supplementary tests. Concentrations of minerals should be determined using X-ray fluorescence for non-food substances and inductively coupled plasma-atomic emission spectroscopy for food-like substances. pH, salt content, cation exchange capacity, organic carbon content and labile forms of iron oxide should also be determined. Finally, analyses relating to biological interactions are recommended, including determination of the bioavailability of nutrients and other bioactive components from pica substances, as well as their detoxification capacities and parasitological profiles.This is the first review of appropriate methodologies for the study of human pica. The comprehensive and multi-disciplinary approach to the collection and analysis of pica substances detailed here is a necessary preliminary step to understanding the nutritional enigma of non-food consumption

Symmetric theta divisors of Klein surfaces

This is a slightly expanded version of the talk given by the first author at the conference Instantons in complex geometry, at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces. In this paper we compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry