Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their variation with this parameter.Comment: 78 pages. Extended version. Added a section with the fixed determinant case. To appear in Memoirs of the AM

Coupled equations for Kähler metrics and Yang-Mills connections

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kahler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kahler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K-energy and geodesic stability. We finish by giving some examples of solutions.Comment: 61 pages; v2: introduction partially rewritten; minor corrections and improvements in presentation, especially in Section 4; added references; v3: To appear in Geom. Topol. Minor corrections and improvements, following comments by referee

A Prym-Narasimhan-Ramanan construction of principal bundle fixed points

Let $X$ be a compact Riemann surface and $G$ be a connected reductive complex Lie group with centre $Z$. Consider the moduli space $M(X,G)$ of polystable principal holomorphic $G$-bundles on $X$. There is an action of the group $H^1(X,Z)$ of isomorphism classes of $Z$-bundles over $X$ on $M(X,G)$ induced by the multiplication $Z\times G\to G.$ Let $\Gamma$ be a finite subgroup of $H^1(X,Z)$. Our goal is to find a Prym--Narasimhan--Ramanan-type construction to describe the fixed points of $M(X,G)$ under the action of $\Gamma$. A main ingredient in this construction is the theory of twisted equivariant bundles on an \'etale cover of $X$ developed in arXiv:2208.0902(2).Comment: 52 pages. In this version we have substantially restructured the content of the pape

Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

We consider U(n+1) Yang-Mills instantons on the space \Sigma\times S^2, where \Sigma is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n+1) instanton equations on \Sigma\times S^2 are equivalent to non-Abelian vortex equations on \Sigma. Solutions to these equations are given by pairs (A,\phi), where A is a gauge potential of the group U(n) and \phi is a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g>1, when \Sigma\times S^2 becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions.Comment: 8 pages; v2: typos fixe

Two more disk galaxies with global gas counterrotation

We report a discovery of extended counterrotating gaseous disks in early-type disk galaxies NGC 2551 and NGC 5631. To find them, we have undertaken complex spectral observations including integral-field spectroscopy for the central parts of the galaxies and long-slit deep spectroscopy to probe the external parts. The line-of-sight velocity fields have been constructed and compared to the photometric structure of the galaxies. As a result, we have revealed full-size counterrotating gaseous disks, the one coplanar to the stellar disk in NGC 2551 and the other inclined to the main stellar disk in NGC 5631. We suggest that we observe the early stages of minor-merger events which may be two different stages of the process of lenticular galaxy formation in rather sparse environments.Comment: 8 pages, 8 EPS figures, accepted for publication in Ap

Non-Abelian Vortices, Super-Yang-Mills Theory and Spin(7)-Instantons

We consider a complex vector bundle E endowed with a connection A over the eight-dimensional manifold R^2 x G/H, where G/H = SU(3)/U(1)xU(1) is a homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A of the Spin(7)-instanton equations on R^2 x G/H and general solutions of non-Abelian coupled vortex equations on R^2. These vortices are BPS solitons in a d=4 gauge theory obtained from N=1 supersymmetric Yang-Mills theory in ten dimensions compactified on the coset space G/H with an SU(3)-structure. The novelty of the obtained vortex equations lies in the fact that Higgs fields, defining morphisms of vector bundles over R^2, are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in N=4 super Yang-Mills theory and show that they have the same feature.Comment: 14 pages; v2: typos fixed, published versio

On the curvature of vortex moduli spaces

We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric powers of the surface), we prove that, for genus g>1, the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kaehler metrics on certain symmetric powers. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.Comment: 25 pages; final version, to appear in Math.

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

Rank two quadratic pairs and surface group representations

Let $X$ be a compact Riemann surface. A quadratic pair on $X$ consists of a holomorphic vector bundle with a quadratic form which takes values in fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the $\mathrm{SO}_0(2,3)$-character variety of $X$.Comment: 37 pages, 1 figur

Skin gene therapy for acquired and inherited disorders

The rapid advances associated with the Human Genome Project combined with the development of proteomics technology set the bases to face the challenge of human gene therapy. Different strategies must be evaluated based on the genetic defect to be corrected. Therefore, the re-expression of the normal counterpart should be sufficient to reverse phenotype in single-gene inherited disorders. A growing number of candidate diseases are being evaluated since the ADA deficiency was selected for the first approved human gene therapy trial (Blaese et al., 1995). To cite some of them: sickle cell anemia, hemophilia, inherited immune deficiencies, hyper-cholesterolemia and cystic fibrosis. The approach does not seem to be so straightforward when a polygenic disorder is going to be treated. Many human traits like diabetes, hypertension, inflammatory diseases and cancer, appear to be due to the combined action of several genes and environment. For instance, several wizard gene therapy strategies have recently been proposed for cancer treatment, including the stimulation of the immune system of the patient (Xue et al., 2005), the targeting of particular signalling pathways to selectively kill cancer cells (Westphal and Melchner, 2002) and the modulation of the interactions with the stroma and the vasculature (Liotta, 2001; Liotta and Kohn, 2001).Our work is supported by grants SAF-2004-07717 from Ministerio de Ciencia y Tecnología (Spain) and LSHG-512073 from UE to M. Del Rio, LSHG-503447 from UE to J.L. Jorcano and LSHG-512102 from UE to F. Larcher. We express our gratitude to Dr. Y. Gache, Dr. F. Spirito and Dr. G. Meneguzzi for providing EM pictures to illustrate this work