Circle Bundle Research Articles

$2$-nodal domain theorems for higher-dimensional circle bundles

We prove that the real parts of equivariant (but non-invariant) eigenfunctions of generic bundle metrics on non-trivial principal S^{1} bundles over manifolds of any dimension have connected nodal sets and exactly 2 nodal domains. This generalizes earlier results of the authors in the 3 -dimensional case. The failure of the results on for non-free S^{1} actions is illustrated on even-dimensional spheres by one-parameter subgroups of rotations whose fixed point set consists of two antipodal points.

The Ricci flow for circle bundles over surfaces

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

Classification of superpotentials for cohomogeneity one Ricci solitons

We classify superpotentials for the Hamiltonian system corresponding to the cohomogeneity one gradient Ricci soliton equations. Aside from recovering known examples of superpotentials for steady solitons, we find a new superpotential on a specific case of the Bérard Bergery–Calabi ansatz. The latter is used to obtain an explicit formula for a steady complete soliton with an equidistant family of hypersurfaces given by circle bundles over S2×S2. There are no superpotentials in the non-steady case in dimensions greater than 2, even if polynomial coefficients are allowed. We also briefly discuss generalised first integrals and the limitations of some known methods of finding them.

A new approach to topological T-duality for principal torus bundles

In this paper, we introduce a new ‘Thom class’ formulation of topological T-duality for principal torus bundles. This definition is equivalent to the established one of Bunke, Rumpf, and Schick but has the virtue of removing the global assumptions on the H-flux required in the old definition. With the new definition, we provide easier and more transparent proofs of the classification of T-duals and generalize the local formulation of T-duality for circle bundles by Bunke, Schick, and Spitzweck to the torus case.

Equivariant Topological T-Duality

Topological T-duality is a relationship between pairs (E, P) over a fixed space X, where E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document} is a principal torus bundle and P→E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \\rightarrow E$$\\end{document} is a twist, such as a gerbe for a principal PU(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$PU({\\mathcal {H}})$$\\end{document}-bundle. This is of interest to topologists because of the T-duality transformation: a T-duality relation between pairs (E, P) and (E^,P^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\hat{E}}, {\\hat{P}})$$\\end{document} comes with an isomorphism (with degree shift) between the twisted K-theory of E and the twisted K-theory of E^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hat{E}}$$\\end{document}. We formulate topological T-duality for circle bundles in the equivariant setting, following the definition of Bunke, Rumpf, and Schick. We define the T-duality transformation in equivariant K-theory and show that it is an isomorphism for all compact Lie groups, equal to its own inverse and uniquely characterized by naturality and a normalization for trivial situations.

Lifting Directional Fields to Minimal Sections

Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement.

The symplectic structure of a toric conic transform

Suppose that a compact r-dimensional torus Tr acts in a holomorphic and Hamiltonian manner on polarized complex d-dimensional projective manifold M, with nowhere vanishing moment map Φ. Assuming that Φ is transverse to the ray through a given weight ν, associated to these data there is a complex (d−r+1)-dimensional polarized projective orbifold Mˆν (referred to as the ν-th conic transform of M). Namely, Mˆν is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polarization of M. With the aim to clarify the geometric significance of this construction, we consider the special case where M is toric, and show that Mˆν is itself a Kähler toric orbifold, whose (marked) moment polytope is obtained from the one of M by a certain ‘transform’ operation (depending on Φ and ν).

Mapping Class Groups of Circle Bundles over a Surface

Mapping Class Groups of Circle Bundles over a Surface

A generalized 4d Chern-Simons theory

A generalization of the 4d Chern-Simons theory action introduced by Costello and Yamazaki is presented. We apply general arguments from symplectic geometry concerning the Hamiltonian action of a symmetry group on the space of gauge connections defined on a 4d manifold and construct an action functional that is quadratic in the moment map associated to the group action. The generalization relies on the use of contact 1-forms defined on non-trivial circle bundles over Riemann surfaces and mimics closely the approach used by Beasley and Witten to reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also show that the path integral of the generalized theory associated to integrable field theories of the PCM type, takes the canonical form of a symplectic integral over a subspace of the space of gauge connections, turning it a potential candidate for using the method of non-Abelian localization. Alternatively, this new quadratic completion of the 4d Chern-Simons theory can also be deduced in an intuitive way from manipulations similar to those used in T-duality. Further details on how to recover the original 4d Chern-Simons theory data, from the point of view of the Hamiltonian formalism applied to the generalized theory, are included as well.

Controlled Mather–Thurston theorems

Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor–Wood inequality is for circle bundles over surfaces, whereas the Mather–Thurston theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern–Weil theory and other smooth obstructions such as the Bott classes and the Godbillion–Vey invariant. Contradiction is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g., \operatorname{PSL}(2,\mathbb{R}) versus \operatorname{U}(1) in the former case and C^1 versus C^2 in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather–Thurston cobordism to a semi- s -cobordism (ssc), and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one. The motivation is to lay mathematical foundations for a physical program. The philosophy is that living in the IR we cannot expect to know, for a given bundle, if it has curvature or is flat, because we cannot resolve the fine scale topology which may be present in the base, introduced by an ssc, nor minute symmetry violating distortions of the fiber. Small scale, UV, “distortions“ of the base topology and structure group allow flat connections to simulate curvature at larger scales. The goal is to find a duality under which curvature terms, such as Maxwell's F \wedge F^\ast and Hilbert's \int R\, d\mathrm{vol} are replaced by an action which measures such “distortions“. In this view, curvature results from renormalizing a discrete, group theoretic, structure.

Calculation of Nielsen periodic numbers of fibre preserving maps

In 2000 Heath and Keppelmann proved the naïve type addition formulae expressing the Nielsen periodic numbers NPn(f) and NFn(f) of the fibre preserving map (f,f¯) with the base map f¯n weakly Jiang, as simple sums of the Nielsen periodic numbers on the fibres. In this paper we extend these formulae to the general case. Our formulae are particularly useful for fibre preserving maps f with the base map f¯n non-weakly Jiang. We illustrate these formulae with examples of fibre preserving maps on a circle bundle over the Klein bottle.

Uniformizations of Compact Sasakian Manifolds

Abstract We give a criterion for compact Sasakian manifolds to be deformed to Sasakian manifolds, which are locally isomorphic to circle bundles of anti-canonical bundles over Hermitian symmetric spaces as a Sasakian analogue of Simpson’s uniformization results related to variations of Hodge structure and Higgs bundles.

On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

Kähler-Einstein metrics and obstruction flatness of circle bundles

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential −log⁡u such that u is C∞-smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles S(L) of negative Hermitian line bundles (L,h) over Kähler manifolds (M,g). We prove that if (M,g) has constant Ricci eigenvalues, then S(L) is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and (M,g) is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of S(L) when (M,g) is a Kähler surface (dim⁡M=2) with constant scalar curvature.

The heterotic 𝐺₂ system on contact Calabi–Yau 7-manifolds

We obtain non-trivial approximate solutions to the heterotic G 2 \mathrm {G}_2 system on the total spaces of non-trivial circle bundles over Calabi–Yau 3 3 -orbifolds, which satisfy the equations up to an arbitrarily small error, by adjusting the size of the S 1 S^1 fibres in proportion to a power of the string constant α ′ \alpha ’ . Each approximate solution provides a cocalibrated G 2 \mathrm {G}_2 -structure, the torsion of which realises a non-trivial scalar field, a constant (trivial) dilaton field and an H H -flux with non-trivial Chern–Simons defect. The approximate solutions also include a connection on the tangent bundle which, together with a non-flat G 2 \mathrm {G}_2 -instanton induced from the horizontal Calabi–Yau metric, satisfy the anomaly-free condition, also known as the heterotic Bianchi identity. The approximate solutions fail to be genuine solutions solely because the connections on the tangent bundle are only G 2 \mathrm {G}_2 -instantons up to higher order corrections in α ′ \alpha ’ .

Anti-quasi-Sasakian manifolds

We introduce and study a special class of almost contact metric manifolds, which we call anti-quasi-Sasakian (aqS). Among the class of transversely Kähler almost contact metric manifolds (M,varphi , xi ,eta ,g), quasi-Sasakian and anti-quasi-Sasakian manifolds are characterized, respectively, by the varphi -invariance and the varphi -anti-invariance of the 2-form textrm{d}eta . A Boothby–Wang type theorem allows to obtain aqS structures on principal circle bundles over Kähler manifolds endowed with a closed (2, 0)-form. We characterize aqS manifolds with constant xi -sectional curvature equal to 1: they admit an Sp(n)times 1-reduction of the frame bundle such that the manifold is transversely hyperkähler, carrying a second aqS structure and a null Sasakian eta -Einstein structure. We show that aqS manifolds with constant sectional curvature are necessarily flat and cokähler. Finally, by using a metric connection with torsion, we provide a sufficient condition for an aqS manifold to be locally decomposable as the Riemannian product of a Kähler manifold and an aqS manifold with structure of maximal rank. Under the same hypothesis, (M, g) cannot be locally symmetric.

Gabor frames from contact geometry in models of the primary visual cortex

We analyze the interplay between contact geometry and Gabor filters signal analysis in geometric models of the primary visual cortex. We show in particular that a specific framed lattice and an associated Gabor system is determined by the Legendrian circle bundle structure of the $3$-manifold of contact elements on a surface (which models the V1-cortex), together with the presence of an almost-complex structure on the tangent bundle of the surface (which models the retinal surface). We identify a scaling of the lattice, also dictated by the manifold geometry, that ensures the frame condition is satisfied. We then consider a $5$-dimensional model where receptor profiles also involve a dependence on frequency and scale variables, in addition to the dependence on position and orientation. In this case we show that a proposed profile window function does not give rise to frames (even in a distributional sense), while a natural modification of the same generates Gabor frames with respect to the appropriate lattice determined by the contact geometry.

Some applications of Menke’s JSJ decomposition for symplectic fillings

We apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over S 2 S^2 , we reduce the problem of classifying exact symplectic fillings to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings.

Effects of tube cross-sectional shapes on flow pattern, liquid film and heat transfer of n-pentane across tube bundles

The heat transfer of hydrocarbon refrigerant across tube bundles have been widely used in refrigeration. Three-dimensional simulation model using volume of fluid (VOF) was presented to study the effects of tube shapes on flow pattern, film thickness and heat transfer of n-pentane across tube bundles, including circle, ellipse-shaped, egg-shaped and cam-shaped tube bundles. Simulation results agree well with experimental data in the literature. The liquid film thickness of sheet flow and heat transfer for different tube shapes were obtained numerically. The flow pattern transition occurs lower vapor quality for ellipse-shaped tube than other tube shapes. For sheet flow, the liquid film on circle tube and ellipse-shaped tube is symmetrically distributed along the circumferential direction. However, the liquid film on egg-shaped tube at circumferential angles (θ) = 15°–60° is thicker than θ = 135°–165°. The liquid film on cam tube at θ = 15°–60° is slightly thinner than θ = 135°–165°. The liquid film thickness varies from thinner to thicker for ellipse-shaped, cam-shaped, egg-shape and circle within θ = 15°–60°. The effect of tube shape is insignificant on thin liquid film thickness. Ellipse-shaped tube has largest heat transfer coefficient for sheet flow. In practical engineering, the tube shape could be designed as ellipse to promote heat transfer.

Partition functions and fibering operators on the Coulomb branch of 5d SCFTs

We study 5d mathcal{N} = 1 supersymmetric field theories on closed five-manifolds mathcal{M} 5 which are principal circle bundles over simply-connected Kähler four-manifolds, mathcal{M} 4, equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on mathcal{M} 5 through the insertion of a fibering operator, which introduces a non-trivial fibration over mathcal{M} 4, in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such mathcal{M} 5, which is conjectured to be the holomorphic ‘integrand’ of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When mathcal{M} 4 is toric, we also reconstruct our Coulomb branch partition function by appropriately gluing Nekrasov partition functions. As a by-product of our analysis, we provide strong new evidence for the validity of the Lockhart-Vafa formula for the five-sphere partition function.