Space Of Bundle Research Articles

The Ooguri-Vafa Space as a Moduli Space of Framed Wild Harmonic Bundles

The Ooguri-Vafa space is a 4-dimensional incomplete hyperkähler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri-Vafa hyperkähler metric should be part of the local model of the hyperkähler metric of the Hitchin moduli spaces, near the most generic kind of singular locus of the Hitchin fibration. In order to relate the Ooguri-Vafa space with the Hitchin moduli spaces, we show that the Ooguri-Vafa space can be interpreted as a set of rank 2, framed wild harmonic bundles over $\mathbb{C}P^1$, with one irregular singularity. Along the way we show that a certain twistor family of holomorphic Darboux coordinates, which describes the hyperkähler geometry of the Ooguri-Vafa space, has an interpretation in terms of Stokes data associated to our framed wild harmonic bundles.

Stratification of the moduli space of vector bundles

Stratification of the moduli space of vector bundles

Rational curves on moduli spaces of vector bundles

W describe the unobstructed components of the Hilbert Scheme of rational curves of fixed degree [Formula: see text] in the moduli space [Formula: see text] of stable vector bundles of rank [Formula: see text] and determinant [Formula: see text] on a curve [Formula: see text]. We show that for every [Formula: see text], there are [Formula: see text] such components. We construct obstructed components of the Hilbert Scheme. We also obtain an upper bound on the degree of rational connectedness of [Formula: see text] which is linear in the dimension.

Parabolic Hitchin connection

In this paper, we present an algebro-geometric construction of the Hitchin connection in the parabolic setting for a fixed determinant line bundle. Our strategy is based on Hecke modifications, where we provide a decomposition formula for the parabolic determinant line bundle and the canonical line bundle of the moduli space of parabolic bundles. As a special case, we construct a Hitchin connection on the moduli space of vector bundles with fixed, not necessarily trivial, determinant.

A functorial approach to the stability of vector bundles

On a normal projective variety the locus of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable bundles that remain μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles. As an application we obtain a very different behaviour of the étale fundamental group in positive vs. characteristic 0.

Heavenly metrics, hyper‐Lagrangians and Joyce structures

AbstractIn [Proc. Sympos. Pure Math., American Mathematical Society, Providence, RI, 2021, pp. 1–66], Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space of stability conditions of a triangulated category. Given a non‐degeneracy assumption, a feature of this structure is a complex hyper‐Kähler metric with homothetic symmetry on the total space of the holomorphic tangent bundle. Generalising the isomonodromy calculation which leads to the Joyce structure in [Math. Ann. 385 (2023), 193–258], we obtain an explicit expression for a hyper‐Kähler metric with homothetic symmetry via construction of the isomonodromic flows of a Schrödinger equation with deformed polynomial oscillator potential of odd‐degree . The metric is defined on a total space of complex dimension and fibres over a ‐dimensional manifold which can be identified with the unfolding of the ‐singularity. The hyper‐Kähler structure is shown to be compatible with the natural symplectic structure on in the sense of admitting an affine symplectic fibration as defined in [Lett. Math. Phys. 111 (2021), 54]. Separately, using the additional conditions imposed by a Joyce structure, we consider reductions of Plebański's heavenly equations that govern the hyper‐Kähler condition. We introduce the notion of a projectable hyper‐Lagrangian foliation and show that in dimension four such a foliation of leads to a linearisation of the heavenly equation. The hyper‐Kähler metrics constructed here are shown to admit such a foliation.

Preparation and formation mechanism of (W) WC/Fe bundle reinforced iron matrix composites

To address the issue of strength and toughness inversion in traditional metal matrix composites, which are reinforced by uniformly distributed ceramic particles, this research developed a novel space bundle configuration of reinforced iron matrix composite using tungsten wire (W) and tungsten carbide (WC). This composite was prepared using a lost foam casting process combined with in-situ reaction. Microstructural analysis revealed that WC particles are distributed along the tungsten wire, with larger particles at the center and smaller ones at the edges. The formation of the reinforcement occurs in two stages: first, a solid-liquid reaction between the solid tungsten wire and molten iron during the lost foam casting process, promoting WC formation at high temperatures and carbon potential; second, an in situ solid-solid reaction where Fe diffuses and Fe3W3C decomposes, forming a composite structure of WC and α-Fe. The composite obtained at 1100 °C for 9 h exhibited a compressive strength of 836.59 MPa and a fracture strain of 18.69 %, representing increases of 48.46 % and 48.69 % respectively compared to gray cast iron (GCI). The (W) WC/Fe bundle reinforcement effectively enhances the strength and toughness of the composite through the high volume fraction and gradient distribution of WC particles, combined with the synergistic effect of α-Fe. This research provides a new strategy for improving the mechanical properties of composite materials and resolving the issue of strength-toughness inversion.

Cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures on the following classes of torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds: the Euclidean R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} with its standard G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

Rational homotopy type of projectivization of the tangent bundle of certain spaces

PurposeThe paper aims to determine the rational homotopy type of the total space of projectivized bundles over complex projective spaces using Sullivan minimal models, providing insights into the algebraic structure of these spaces.Design/methodology/approachThe paper utilises techniques from Sullivan’s theory of minimal models to analyse the differential graded algebraic structure of projectivized bundles. It employs algebraic methods to compute the Sullivan minimal model of P(E) and establish relationships with the base space.FindingsThe paper determines the rational homotopy type of projectivized bundles over complex projective spaces. Of great interest is how the Chern classes of the fibre space and base space, play a critical role in determining the Sullivan model of P(E). We also provide the homogeneous space of P(E) when n = 2. Finally, we prove the formality of P(E) over a homogeneous space of equal rank.Research limitations/implicationsLimitations may include the complexity of computing minimal models for higher-dimensional bundles.Practical implicationsUnderstanding the rational homotopy type of projectivized bundles facilitates computations in algebraic topology and differential geometry, potentially aiding in applications such as topological data analysis and geometric modelling.Social implicationsWhile the direct social impact may be indirect, advancements in algebraic topology contribute to broader mathematical knowledge, which can underpin developments in science, engineering, and technology with societal benefits.Originality/valueThe paper’s originality lies in its application of Sullivan minimal models to determine the rational homotopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles.

Hodge-to-singular correspondence for reduced curves

We study the summands of the decomposition theorem for the Hitchin system for \operatorname{GL}_{n} , in arbitrary degree, over the locus of reduced spectral curves. A key ingredient is a new correspondence between these summands and the topology of hypertoric quiver varieties. In contrast to the case of meromorphic Higgs fields, the intersection cohomology groups of moduli spaces of regular Higgs bundles depend on the degree. We describe this dependence.

Cohomological $\chi$-independence for Higgs bundles and Gopakumar–Vafa invariants

The aim of this paper is two-fold. Firstly, we prove Toda’s \chi -independence conjecture for Gopakumar–Vafa invariants of arbitrary local curves. Secondly, following Davison’s work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank r and Euler characteristic \chi which are not necessary coprime, and show that it does not depend on \chi . This result extends the Hausel–Thaddeus conjecture on the \chi -independence of E-polynomials proved by Mellit, Groechenig–Wyss–Ziegler and Yu in two ways: We obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda.

Connes fusion of spinors on loop space

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

Hofer–Zehnder capacity of disc tangent bundles of projective spaces

AbstractWe compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.

Multiplicative Higgs bundles and involutions

In this paper we generalize the theory of multiplicative G-Higgs bundles over a curve to pairs (G,θ), where G is a reductive algebraic group and θ is an involution of G. This generalization involves the notion of a multiplicative Higgs bundle taking values in a symmetric variety associated to θ, or in an equivariant embedding of it. We also study how these objects appear as fixed points of involutions of the moduli space of multiplicative G-Higgs bundles, induced by the involution θ.

Simplicity of Tangent bundles on the moduli spaces of symplectic and orthogonal bundles over a curve

Simplicity of Tangent bundles on the moduli spaces of symplectic and orthogonal bundles over a curve

Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected. In contrast, moduli spaces of sheaves on minimal rational surfaces and certain del Pezzo surfaces are irreducible and smooth along the locus of stable bundles. We find examples of moduli spaces of vector bundles on more general blowups of the projective plane that are disconnected and have components of different dimensions. In fact, assuming the SHGH Conjecture, we can find moduli spaces with arbitrarily many components of arbitrarily large dimension.

Automorphisms of moduli spaces of principal bundles over a smooth curve

For any almost-simple group [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero, we describe the automorphism group of the moduli space of semistable [Formula: see text]-bundles over a connected smooth projective curve [Formula: see text] of genus at least [Formula: see text]. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers.

On the motives of moduli of parabolic chains and parabolic Higgs bundles

Analogous to the Higgs bundles on a Riemann surface which played an important role in the study of representations of the fundamental group of the surface, the parabolic Higgs bundles play also their importance in the study of the fundamental group but of the punctured surface. In this paper, we give an algorithm to calculate the (virtual) motive (i.e. in a suitable Grothendieck group) of the moduli spaces of parabolic Higgs bundles of fixed rank and fixed parabolic structure, using localization with respect to the circle action.

Chen–Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

We consider the moduli space of stable parabolic Higgs bundles of rank r and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth connected complex projective curve X of genus g, with g≥2. The group Γ of r-torsion points of the Jacobian of X acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from Γ. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen–Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen–Ruan product structure in special cases.

Multi-scale collaborative prediction of optimal configuration for carbon fiber woven composites based on deep learning neural networks

Since carbon fiber woven composite panels contain complex fiber bundle structures, the aim of this paper is to investigate the factors influencing the macro-equivalent properties and topological flexibility of composite panels resulting from variations in fiber bundle configuration. Firstly, the structural topology optimization design of carbon fiber composite plates was achieved using a multi-scale modeling approach with the variable density method. Nonlinear relationships between carbon fiber bundle configuration parameters and macroscopic equivalent properties, as well as macroscale topological flexibility, were determined using deep learning methods, respectively. The maximum error in the predicted values of the deep learning network model does not exceed 0.5% when compared with the results of the finite element calculations. In addition, utilizing deep learning networks to predict macro-equivalent properties of composites can significantly decrease computational time. The impact of varying the number of filaments in fiber bundles and the spacing of the fiber bundles on the flexibility of the topology was determined using a deep learning network. The optimal topological flexibility, along with its corresponding fiber bundle spacing and number of fiber bundle filaments, are globally searched within the design domain using an adaptive genetic algorithm. The study in this paper can provide a reference for designing fiber bundle configurations of composites for practical applications.