Orientation Bundle Research Articles

Orientations for DT invariants on quasi-projective Calabi–Yau 4-folds

For a Calabi–Yau 4-fold (X,ω), where X is quasi-projective and ω is a nowhere vanishing section of its canonical bundle KX, the (derived) moduli stack of compactly supported perfect complexes MX is −2-shifted symplectic and thus has an orientation bundle Oω→MX in the sense of Borisov–Joyce [11] necessary for defining Donaldson–Thomas type invariants of X. We extend first the orientability result of Cao–Gross–Joyce [23] to projective spin 4-folds. Then for any smooth projective compactification Y, such that D=Y﹨X is strictly normal crossing, we define orientation bundles on the stack MY×MDMY and express these as pullbacks of Z2-bundles in gauge theory, constructed using positive Dirac operators on the double of X. As a result, we relate the orientation bundle Oω→MX to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of MX. We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums.

On orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, D:Γ∞(E0)→Γ∞(E1) a real elliptic operator on X, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of all connections ∇P on P modulo gauge, as a topological stack. For each [∇P]∈BP, we can consider the twisted elliptic operator D∇Ad(P):Γ∞(Ad(P)⊗E0)→Γ∞(Ad(P)⊗E1) on X. This is a continuous family of elliptic operators over the base BP, and so has an orientation bundle OPE•→BP, a principal Z2-bundle parametrizing orientations of KerD∇Ad(P)⊕CokerD∇Ad(P) at each [∇P]. An orientation on (BP,E•) is a trivialization OPE•≅BP×Z2.In gauge theory one studies moduli spaces MPga of connections ∇P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X,g). Under good conditions MPga is a smooth manifold, and orientations on (BP,E•) pull back to orientations on MPga in the usual sense of differential geometry under the inclusion MPga↪BP. This is important in areas such as Donaldson theory, where one needs an orientation on MPga to define enumerative invariants.We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (BP,E•), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, the Kapustin–Witten equations, and the Vafa–Witten equations on 4-manifolds, and the Haydys–Witten equations on 5-manifolds.

Modified Rodrigues Parameters: An Efficient Representation of Orientation in 3D Vision and Graphics

Modified Rodrigues parameters (MRPs) are triplets in {mathbb {R}}^3 bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-of-freedom parameterization of orientation through novel solutions to prominent problems in the fields of 3D vision and computer graphics. In our primary contribution, we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Furthermore, we show that updates to unit quaternions from perturbations in parameter space can be computed without explicitly invoking the parameters in the computations. Based on the former, we introduce a novel approach for designing orientation splines by configuring their back-projections in 3D space. Finally, in the general topic of nonlinear optimization for geometric vision, we run performance analyses and provide comparisons on the convergence behavior of MRP parameterizations on the tasks of absolute orientation, exterior orientation and large-scale bundle adjustment of public datasets.

A note on the $${\mathbb {Z}}_2$$ Z 2 -cohomology algebra of oriented Grassmann manifolds

The \({\mathbb {Z}}_2\)-cohomology ring \(H^*({\widetilde{G}}_{n,k})\) of the oriented Grassmann manifold \({\widetilde{G}}_{n,k}\cong \mathrm {SO}(n)/(\mathrm {SO}(k)\times \mathrm {SO}(n-k))\) is in general unknown. This paper demonstrates that the characteristic rank of vector bundles in the sense of Korbas, Naolekar, and Thakur can be helpful in improving our knowledge. More precisely, using the full knowledge of the characteristic rank of the canonical orientable k-plane bundle \({\widetilde{\gamma }}_{n,k}\) over \({\widetilde{G}}_{n,k}\) for \(k=2\), we completely calculate the cohomology ring \(H^*({\widetilde{G}}_{n,2})\). In addition, we completely determine the generators of the cohomology ring \(H^*({\widetilde{G}}_{n,3})\) for \(n=6,\,7,\,8,\,9,\,10,\,11\).

Problems of network simplification by edge bundling

Node occlusion and edge congestion problems,which are caused by the increment of network scale and complexity,had become a hot spot in network visualization research. To solve the visual clutter problem in network,edges close to each other in network were bundled by curving them. The bundling started from node position and group division,laying emphasis on edge bundling technique through edge convergence,edges bundled close to each other in network to reduce visual complexity. A segmental forced directed algorithm( FDA) simplification model and a group based consistent edge bundling network model were proposed and improved. In segmental FDA bundling model,quadratic spline was used to display line in network,control points of spline were produced by iteration to implement edge bundling. To solve the problem of excessive bending of some edges in segmental FDA bundling model,network was divided into different groups by CNM cluster algorithm. Lines in the same group were applied with edge consistent principle on basement of network group structure,and bundling level was calculated by the matching coefficient of edges. After edge bundling,the phenomena of curve's distortion decreased and curves became smoother. Domestic airline network data was chosen as experimental subject,and the network was simplified by two bundling models,and then the simplification result was analyzed. Experimental result shows that positions of nodes in the same group stand close to each other,and airline network shows distinct crossing skeleton. Airlines have north-south and westeast orientation bundle independently,which reveals the whole trends of airline network construction in these two directions. The bundling simplification algorithm introduced mentioned above has a wide applicability,and network visualized by this algorithm has good visual effect and readability.

The moduli space of maps with crosscaps: Fredholm theory and orientability

Just as a symmetric surface with separating fixed locus halves into two oriented bordered surfaces, an arbitrary symmetric surface halves into two oriented symmetric halfsurfaces, i.e. surfaces with crosscaps. Motivated in part by the string theory view of real Gromov-Witten invariants, we introduce moduli spaces of maps from surfaces with crosscaps, develop the relevant Fredholm theory, and resolve the orientability problem in this setting. In particular, we give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators over Riemann surfaces with crosscaps. Special cases of our formulas are closely related to the orientability question for the space of real maps from symmetric Riemann surfaces to an almost complex manifold with an anti-complex involution and in fact resolve this question in genus 0. In particular, we show that the moduli space of real J-holomorphic maps from the sphere with a fixed-point free involution to a simply connected almost complex manifold with an even canonical class is orientable. In a sequel, we use the results of this paper to obtain a similar orientability statement for genus 1 real maps.

Quantizing models of (2+1)-dimensional gravity on non-orientable manifolds

Motivated by the relation between the Chern–Simons gauge theory and (2+1)-dimensional gravity, we find a formulation of gauge theories which applies to both orientable and non-orientable manifolds, using orientation bundles and density-valued forms. We show that on a non-orientable manifold, (2+1)D gravity is equivalent to BF theory, which is still topological and can be mapped in turn to Chern–Simons theory on the orientable double cover. By quantizing U(1) BF theory on a non-orientable manifold, we find that non-orientability introduces additional constraints on the quantized BF theory beyond those present for an orientable manifold, such that the coupling constant can only adopt a small number of discrete values. Specifically, for both the Klein bottle of demigenus 2 (N2) and the compact surface of non-orientable genus 3 (Dyck’s surface or N3), we find explicit representations for the holonomy, large gauge, and mapping class groups, as well as the Hilbert space; here the above symmetries along with the non-orientability of the surface constrain the coupling constant k to only take values 1/2, 1, or 2.

The orientability problem in open Gromov–Witten theory

We give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators. A special case of this formula resolves the orientability question for spaces of maps from Riemann surfaces with Lagrangian boundary condition. As a corollary, we show that the local system of orientations on the moduli space of J-holomorphic maps from a bordered Riemann surface to a symplectic manifold is isomorphic to the pull-back of a local system defined on the product of the Lagrangian and its free loop space. As another corollary, we show that certain natural bundles over these moduli spaces have the same local systems of orientations as the moduli spaces themselves (this is a prerequisite for integrating the Euler classes of these bundles). We will apply these conclusions in future papers to construct and compute open Gromov-Witten invariants in a number of settings.

Implementation of a Peer-Developed Orientation Bundle to Improve Intern Satisfaction in the Neonatal Intensive Care Unit

Implementation of a Peer-Developed Orientation Bundle to Improve Intern Satisfaction in the Neonatal Intensive Care Unit

Arbeitsgemeinschaft mit aktuellem Thema: Twisted K-Theory

Ordinary (co)homology theories always come along with twisted versions of themselves; the most basic example is cohomology twisted by the orientation bundle, which shows up when one discusses Poincar\'e duality (and more general push-forward maps) for non-orientable manifolds.

The Eta-Invariant and Pontryagin Duality in K -Theory

The topological significance of the spectral Atiyah--Patodi--Singer \(\eta \)-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. Pontryagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.

The absolute degree and Nielsen root number of a fibre-preserving map

Let p :M→X and q :N→Y be locally trivial bundles, with fibres F and G, respectively, where all the spaces are connected closed manifolds, but neither the manifolds nor the bundles need be orientable. Assume further that dim X=dim Y and dim F=dim G so dim M=dim N. A fibre-preserving map f :M→N induces a map f ̄ :X→Y of the base and maps f x :F x→G f ̄ (x) of the fibres. The purpose of this paper is to relate the Nielsen root number NR( f) of f with that of f ̄ and f x and to do the same for the absolute degree A(f) . If both f ̄ and f x are orientable maps or if f is both orientable and root-essential (that is, if NR( f)>0), then the multiplicative property A(f)= A( f ̄ )· A(f x) is shown to be valid. Applying this property to selfmaps of compact solvmanifolds produces a computational result for the absolute degree of such a map. If f is a root-essential nonorientable map, the multiplicative property for the absolute degree must be modified in a way that includes a factor κ( f) that describes a relationship between the root class structure of f x and that of f. The Nielsen root numbers of any root-essential fibre-preserving map are found to satisfy κ(f)·NR(f)=NR( f ̄ )·NR(f x) . A fibre-preserving version FG(f) of the classical geometric degree G(f) is defined and it is shown that FG(f)= G(f) if and only if the absolute degree has the multiplicative property. Letting MR[ f] denote the minimum number of points in the pre-image of a given point of N among all maps homotopic to f and FMR[ f] the same with regard to fibre-preserving maps and homotopies, every fibre-preserving map satisfies FMR[f]=MR[f x]·MR[ f ̄ ] . Moreover, if none of the manifolds M, X and F are two-dimensional, then FMR[ f]= MR[ f] if and only if NR(f)=NR( f ̄ )·NR(f x) . A new bundle and pairing, the fibrewise orientation bundle and the orientation bundle pairing, are introduced in order to relate the orientation bundles of base, fibre, and total space of a locally trivial bundle.

Nonorientable Manifolds, Complex Structures, and Holomorphic Vector Bundles

A generalization of the notion of almost complex structure is defined on a nonorientable smooth manifold M of even dimension. It is defined by giving an isomorphism J from the tangent bundle TM to the tensor product of the tangent bundle with the orientation bundle such that J○J=−Id TM . The composition J○J is realized as an automorphism of TM using the fact that the orientation bundle is of order two. A notion of integrability of this almost complex structure is defined; also the Kahler condition has been extended. The usual notion of a complex vector bundle is generalized to the nonorientable context. It is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These classes take value in the cohomology of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle is extended to the context under consideration. Stable vector bundles and Einstein–Hermitian connections are also generalized. It is shown that a generalized holomorphic vector bundle on a compact nonorientable Kahler manifold admits an Einstein–Hermitian connection if and only if it is polystable.

Invertible Apertured Orientation Filters in Image Analysis

A unitary approach for locally apertured orientation analysis of 2D and 3D scalar images is proposed. The size of the local aperture (the scale) needed for the orientation representation induces in general a lost of spatial acuity, or blur. Our construction permits a compensation of the blur by a reconstruction procedure. For this purpose, a special scale-dependent orientation bundle (map of the visual space into function of both position and orientation) is build from the local Gaussian-derivatives jet of a scalar image. In this construction there is an invertible relation between the orientation bundle and the original image. This invertible transformation is used to regain the original acuity in the spatial domain after analyzing orientation features at any given scale. The approach turns out to be highly effective for the detection of elongated structures and for removal of elongated artifacts in 2D images.

ON FRAMED LINK PRESENTATIONS OF SURFACE BUNDLES

In this paper, we show that every closed orientable surface bundle over the circle is represented by a fibered link in the 3-sphere with framings induced by the fibration of the complement.

Harmonic Morphisms as Sphere Bundles Over Compact Riemann Surfaces

We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.

$S^2$-bundles over aspherical surfaces and 4-dimensional geometries

Melvin has shown that closed 4-manifolds that arise as S2-bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries S2 ×E2 or S2 ×H2 [depending on whether χ(M) = 0 or χ(M) < 0]. Conversely a geometric closed, connected 4-manifold M of type S2 × E2 or S2 × H2 is the total space of an S2-bundle over a closed, connected aspherical surface precisely when its fundamental group Π1(M) is torsion free. Furthermore the total spaces of RP-bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifoldM ′ is the total space of an RP-bundle if and only if Π1(M ′) ∼= Z/2Z×K where K is torsion free. Introduction Melvin [Me] has shown that closed 4-manifolds that arise as S-bundles over closed aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. Moreover one may construct ξ from w(ξ). See [Me, Structure Lemma]. In particular there are two orientable 4-manifolds that arise as total spaces of S-bundles over each closed surface and are distinguished by whether or not w2(ξ) = 0. Ue [Ue] has shown that the total spaces M of such orientable bundles over the surfaces of Euler characteristic χ = 0 have the structures of Seifert 4-manifolds and are geometric. There are just two S-bundles over S, the product bundle and S2×S2; the latter is not geometric. There are four S-bundles over RP, each of which admits an S2×S2 geometry and is distinguishable by its Stiefel-Whitney classes [H1]. There are four S-bundles over each closed orientable surface B with Euler characteristic χ(B) ≤ 0, six S-bundles over the Klein bottle Kb and eight S-bundles over each closed non-orientable surface B with χ(B) ≤ 0, [Me]. The aim of this paper is to prove the following: Main Theorem. (1) A closed S2×E2or S2×H2-manifold M is the total space of an S-bundle over some aspherical surface precisely when its fundamental group Π1(M) is torsion free. Conversely all total spaces of such bundles are geometric. Received by the editors May 10, 1996. 1991 Mathematics Subject Classification. Primary 57N50; Secondary 57N13, 55R25.

New generalizations of Poincar�'s geometric theorem

In this work we show that the number of closed trajectories of the field of kernels of a closed, nondegenerate, and center of gravity preserving 2-form on the total space of an oriented fiber bundle with fiber the circle over an orientable compact two-dimensional base is not less than the minimal number of critical points of a smooth function on the basis, under the assumption that the field of kernels is C1-close to a vertical field. Counting multiplicities, this number is not smaller than the minimal number of critical points of a Morse function on the base. We also give a lower estimate for the number of closed trajectories in the case of a higher-dimensional base. A form preserves.the center of gravity if its cohomology class is the lift of a class from the base. As an application we prove that the number of closed trajectories of a particle on a surface under the action of a strong and little varying magnetic field perpendicular to the surface is not smaller than the minimal number of critical points of a function on the given surface. The author is grateful to V. I. Arnol'd for formulating the problem on the closed trajectories of fields of kernels of differential forms preserving the center of gravity and for helping in work. I. Condition of Preservation of the Center of Gravity. Let S I + M ~ B be an orientable bundle over a closed (compact and boundaryless) orientable manifold B.