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Marcos Jardim

Publications and source records attributed to Marcos Jardim.

At least 73 records · Page 4Linked to original sources
arXiv2009

Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons

We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 singularity removal theorem of Sibner--Sibner and Rade we prove that given a smooth, finite energy, self-dual SU(2) connection over a complete ALF space, its energy is congruent to a Chern--Simons invariant of the boundary three-manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly. Then we introduce framed moduli spaces of self-dual connections over Ricci flat ALF spaces. We prove that the moduli space of smooth, irreducible, rapidly decaying self-dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold. We calculate its dimension by a variant of the Gromov--Lawson relative index theorem. As an application, we study Yang--Mills instantons over the flat R^3 x S^1, the multi-Taub--NUT family, and the Riemannian Schwarzschild space.

math.DG↗
arXiv2008

Nahm Transform for Higgs bundles

We construct the Nahm transform for Higgs bundles over a Riemann surface of genus at least 2 as hyperholomorphic connections on the total space of the tangent bundle of its dual Jacobian.

math.DG↗
arXiv2007

On Moduli Spaces for Abelian Categories

We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.

math.AG↗
arXiv2007

Nonsingular solutions of Hitchin's equations for noncompact gauge groups

We consider a general ansatz for solving the 2-dimensional Hitchin's equations, which arise as dimensional reduction of the 4-dimensional self-dual Yang-Mills equations, with remarkable integrability properties. We focus on the case when the gauge group G is given by a real form of SL(2,C). For G=SO(2,1), the resulting field equations are shown to reduce to either the Liouville, elliptic sinh-Gordon or elliptic sine-Gordon equations. As opposed to the compact case, given by G=SU(2), the field equations associated with the noncompact group SO(2,1) are shown to have smooth real solutions with nonsingular action densities, which are furthermore localized in some sense. We conclude by discussing some particular solutions, defined on R^2, S^2 and T^2, that come out of this ansatz.

math-ph↗
arXiv2007

Linear and Steiner bundles on projective varieties

We use a generalization of Horrocks monads for arithmetic Cohen-Macaulay (ACM) varieties to establish a cohomological characterization of linear and Steiner bundles over projective spaces and quadric hypersurfaces. We also study resolutions of bundles on ACM varieties by line bundles, and characterize linear homological dimension in the case of quadric hypersurfaces.

math.AG↗
arXiv2006

Stable bundles on 3-fold hypersurfaces

Using monads, we construct a large class of stable bundles of rank 2 and 3 on 3-fold hypersurfaces, and study the set of all possible Chern classes of stable vector bundles.

math.AG↗
arXiv2006

On the semistability of instanton sheaves over certain projective varieties

We show that instanton bundles of rank $r\le 2n-1$, defined as the cohomology of certain linear monads, on an $n$-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford-Takemoto. Furthermore, we show that rank $r\le n$ linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.

math.AG↗
arXiv2005

Instanton sheaves on complex projective spaces

We study a class of torsion-free sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable if its rank is not too large, while semistable torsion-free sheaves satisfying certain cohomological conditions are instanton. We also study a few examples of moduli spaces of instanton sheaves.

math.AG↗
arXiv2004

Admissible sheaves on P^3

Admissible locally-free sheaves on P^3, also known in the literature as mathematical instanton bundles, arise in twistor theory, and are in 1-1 correspondence with instantons on R^4. In this paper, we study admissible sheaves on P^3 from the algebraic geometric point of view. We discuss examples and compare the admissibility condition with semistability and splitting type.

math.AG↗
arXiv2004

Complex ADHM equations, sheaves on P^3 and quantum instantons

We use a complex version of the celebrated Atiyah-Hitchin-Drinfeld-Manin matrix equations to construct admissible torsion-free sheaves on $\p^3$ and complex quantum instantons over our quantum Minkowski space-time. We identify the moduli spaces of various subclasses of sheaves on $\p^3$, and prove their smoothness. We also define the Laplace equation in the quantum Minkowski space-time, study its solutions and relate them to the admissibility condition for sheaves on $\p^3$.

math.RT↗
arXiv2004

A Fourier-Mukai transform for real torus bundles

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $π:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$ with a flat partial unitary connection, that is families or deformations of flat vector bundles (or unitary local systems) on the torus $T.$ This leads to a correspondence between such objects on $M$ and relative skyscraper sheaves $\cS$ supported on a spectral covering $Σ\hra \what M,$ where $\hatπ:\what{M} \to B$ is the flat dual fiber bundle. Additional structures on $(E,\nabla)$ (flatness, anti-self-duality) will be reflected by corresponding data on the transform $(\cS, Σ).$ Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture

math.DG↗
arXiv2004

A survey on Nahm transform

We review the construction known as the Nahm transform in a generalized context, which includes all the examples of this construction already described in the literature. The Nahm transform for translation invariant instantons on $\real^4$ is presented in an uniform manner. We also analyze two new examples, the first of which being the first example involving a four-manifold that is not complex.

math.DG↗
arXiv2003

Hyperkähler Nahm transform

Given two hyperkähler manifolds $M$ and $N$ and a quaternionic instanton on their product, a hyperkähler Nahm transform can be defined, which maps quaternionic instantons on $M$ to quaternionic instantons on $N$. This construction includes the case of Nahm transform for periodic instantons on $\bR^4$, the Fourier-Mukai transform for instantons on K3 surfaces, as well as the Nahm transform for ALE instantons.

math.DG↗
arXiv2003

Quantum instantons with classical moduli spaces

We introduce a quantum Minkowski space-time based on the quantum group SU(2)q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective twistor space.

math.QA↗