Space Of Bundle Research Articles

On Fiber Bundles and Quaternionic Slice Regular Functions

The papers (González-Cervantes in Adv Appl Clifford Algebras 31:55, 2021; González-Cervantes Complex Variables and Elliptic Equations 2021) are the first works to apply the theory of fiber bundles in the study of the quaternionic slice regular functions. The main goal of the present work is to extend the results given in González-Cervantes (2021), where the quaternionic right linear space of quaternionic slice regular functions was presented as the base space of a fiber bundle. When the quaternionic right linear space of quaternionic slice regular functions is associated to certain domains then this paper shows that the elements of total space, given in González-Cervantes (2021), are defined from a pair of harmonic functions and a pair of orthogonal vectors. Simplifying the computations presented in González-Cervantes (2021), where each element of the total space is formed by two pair of conjugate harmonic functions and a pair of orthogonal unit vectors. This work also gives some interpretations of the behavior of the zero sets of some quaternionic slice regular polynomials in terms of the theory of fiber bundles.

An analytic approach to the quasi-projectivity of the moduli space of Higgs bundles

The moduli space of Higgs bundles can be defined as a quotient of an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with the structure of a normal complex space. In this paper, we will use analytic methods to show that the moduli space is quasi-projective. In fact, following Hausel's method, we will use the symplectic cut to construct a normal and projective compactification of the moduli space, and hence prove the quasi-projectivity. The main difference between this paper and Hausel's is that the smoothness of the moduli space is not assumed.

On the rationality of moduli spaces of vector bundles over chain-like curves

Let C be a chain-like curve over C. In this paper, we investigate the rationality of moduli spaces of w-semistable vector bundles on C of arbitrary rank and fixed determinant by putting some restrictions on the Euler characteristics.

Simpson–Mochizuki correspondence for λ-flat bundles

The notion of flat λ-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi–Hitchin-type theorem for λ-flat bundles (λ≠0), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable λ-flat bundles (λ≠0). On the other hand, we study two applications of the Simpson–Mochizuki correspondence to moduli spaces. More concretely, we show this correspondence provides a homeomorphism between the moduli space of (semi)stable λ-flat bundles over a complex projective manifold and the Dolbeault moduli space, and also provides dynamical systems with two parameters on the latter moduli space. We investigate such dynamical systems, in particular, we calculate the first variation, the fixed points and discuss the asymptotic behavior.

P=W conjectures for character varieties with symplectic resolution

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-etale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, or the intersection theory of some singular Lagrangian cycles. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.

Chen–Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface

Let \((X,\,D)\) be an m-pointed compact Riemann surface of genus at least 2. For each \(x \,\in \, D\), fix full flag and concentrated weight system \(\alpha \). Let \(P \mathcal {M}_{\xi }\) denote the moduli space of semi-stable parabolic vector bundles of rank r and determinant \(\xi \) over X with weight system \(\alpha \), where r is a prime number and \(\xi \) is a holomorphic line bundle over X of degree d which is not a multiple of r. We compute the Chen–Ruan cohomology of the orbifold for the action on \(P \mathcal {M}_{\xi }\) of the group of r-torsion points in \(\mathrm{Pic}^0(X)\).

Associations of Lumber Disc Degeneration With Paraspinal Muscles Myosteatosis in Discogenic Low Back Pain.

Accompanied with intervertebral disc (IVD) degeneration, increasing fat infiltration of paraspinal muscles may be related to discogenic low back pain (DLBP), but their relationship is still unclear and the classical animal models are not completely applicable. The purpose of this study was to assess the paraspinal muscle fat infiltration in patients with DLBP by quantitative MRI, and to develop a novel DLBP rat model to explore the potential relationship between DLBP paraspinal muscle fat infiltration and TNF-α levels. We measured the proton density fat fraction (PDFF) of the multifidus and erector spinae muscles of 70 DLBP patients and 36 healthy volunteers by using quantitative MRI IDEAL-IQ. In addition, we developed a DLBP experimental rat model by puncturing the L4/5 and L5/6 IVDs under the guidance of X-ray fluoroscopy. Then various behavioral experiments, MRI and pathological examination of IVDs were used to evaluate the performance of the DLBP animal model. The gait analysis, hot plate test, acetone test, grasping test and tail suspension test were used to evaluate the pain and muscle dysfunction in rats. Through quantitative MRI and histological examination, the degeneration of IVDs and fat infiltration in the muscles were observed in vivo and ex vivo. Enzyme linked immunosorbent assay detects the level of TNF-α in rat IVDs and paraspinal muscles. In the human study, compared with healthy volunteers, the PDFF of multifidus and erector muscles of DLBP patients increased significantly at L4/5 and L5/S1 levels (p<0.05). In the rat experiment, compared with control group and sham group, DLBP group had reduced gait score, shortened response time to cold and heat stimuli, prolonged bending time, and shortened struggling time. Rat lumbar MRI T2WI showed that the signal intensity of L4/5 and L5/6 IVDs were progressively decreased. Histological examination revealed that IVDs had increased collagen fibers, reduced nucleus pulposus, thickened annulus fibrosus, and distorted shape. The PDFF of multifidus muscle at L4/5 and L5/6 level in the DLBP group were more than that in other groups (p<0.05), and HE staining and oil red O staining of paraspinal muscles showed that the muscle bundle space of the DLBP group muscles increased, and the muscle tissues Increased lipid droplets. Finally, the expression of TNF-α in IVDs and paraspinal muscles in the DLBP group were significantly higher than that in the control group (p<0.05). It is reliable and feasible to establish a DLBP rat model by puncturing the lumbar IVDs under the guidance of X-ray fluoroscopy. The degeneration of lumbar IVDs with DLBP leads to the occurrence of fat infiltration of paraspinal muscles, which is related to the expression of TNF-α.

THE HITCHIN CONNECTION IN ARBITRARY CHARACTERISTIC

AbstractWe give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan–Atiyah–Bott Kähler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson–Schechtman and Bloch–Esnault.

Extension groups of tautological bundles on symmetric products of curves

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if Ene mathcal O_X is simple, then the natural map {{,mathrm{mathsf {Ext}},}}^1(E,E)rightarrow {{,mathrm{mathsf {Ext}},}}^1(E^{[n]},E^{[n]}) is injective for every n. Along with previous results, this implies that Emapsto E^{[n]} defines an embedding of the moduli space of stable bundles of slope mu notin [-1,n-1] on the curve X into the moduli space of stable bundles on the symmetric product X^{(n)}. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X^{(n)} where the dimension of the tangent space jumps. We also prove that E^{[n]} is simple if E is simple.

Einstein's quantum elevator: Hermitization of non-Hermitian Hamiltonians via a generalized vielbein formalism

The formalism for non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbein-like formalism which transforms the Hilbert space bundles of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be "transformed" into a Hermitian one without altering the physics. Thus we show how to find a reference frame (corresponding to Einstein's quantum elevator) in which a non-Hermitian system, equipped with a non-trivial Hilbert space metric, reduces to a Hermitian system within the standard formalism of quantum mechanics.

On birational Torelli theorems

Let G be a simple simply-connected connected linear algebraic group over \({\mathbb {C}}\). We prove a 2-birational Torelli theorem for the moduli space of semistable principal G-bundles over a smooth curve of genus \(\ge 3\), which says that if two such moduli spaces are 2-birational, then the curves are isomorphic. We also prove a 3-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus \(\ge 4\).

25 open questions about vector bundles and their moduli

We present 25 open questions about moduli spaces of vector bundles and related topics, and discuss some longstanding conjectures. We hope to inspire young researchers to engage in this area of re- search.

Disconnected moduli spaces of stable bundles on surfaces

We use hypersurfaces containing unexpected linear spaces to construct interesting vector bundles on complete intersection surfaces in projective space. We discover examples of moduli spaces of rank 2 stable bundles on surfaces of Picard rank one with arbitrarily many connected components.

Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the $n$-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an $n$-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits.

On the cohomology rings of partially projective quaternionic Stiefel manifolds

Abstract The quaternionic Stiefel manifold is the total space of a fibre bundle over the corresponding Grassmannian . The group acts freely on the fibres of this bundle. The quotient space is called the quaternionic projective Stiefel manifold. Its real and complex analogues were actively studied earlier by a number of authors. A finite group acting freely on the three-dimensional sphere also acts freely and discretely on the fibres of the quaternionic Stiefel bundle. The corresponding quotient spaces are called partially projective Stiefel manifolds. The cohomology rings of partially projective quaternionic Stiefel manifolds with coefficients in , where is prime, are calculated. Bibliography: 14 titles.

A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve

Let C C be an elliptic curve, w ∈ C w\in C , and let S ⊂ C S\subset C be a finite subset of cardinality at least 3 3 . We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle O C ( w ) \mathcal O_C(w) over ( C , S ) (C,S) which are semistable with respect to a weight vector ( 1 2 , … , 1 2 ) \big (\frac {1}{2}, \dots , \frac {1}{2}\big ) .

Mirror symmetry for Nahm branes

The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than $1$. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank $n$ and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.

Automorphic Bloch theorems for hyperbolic lattices

Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (2D) hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in an experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps correspond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.

Homotopy decompositions of PU(n)-gauge groups over Riemann surfaces

In this paper, we show that the gauge group of a principal [Formula: see text]-bundle over a compact Riemann surface decomposes up to homotopy as the product of factors, one of which is a corresponding gauge group for [Formula: see text] and the others are immediately recognizable spaces. Further, when [Formula: see text] is a prime [Formula: see text], the gauge group for [Formula: see text] decomposes as a product of immediately recognizable factors. These gauge groups have strong connections to moduli spaces of stable vector bundles.

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve