Bundle Theory Research Articles

Equivariant orientation of vector bundles over disconnected base spaces

AbstractIn this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group , a precise condition under which an ‐orientation of a ‐equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel–Whitney class of a “homogeneous” ‐equivariant bundle with respect to an ‐ring spectrum . As an application, we show that the 2‐fold direct sum of any homogeneous bundle is ‐orientable, where is the Burnside Mackey functor. We notice that ‐orientability is equivalent to ‐orientability when the order of is odd. When the order of is even, we show that a ‐equivariant analog of the tautological line bundle over is ‐orientable but not ‐orientable.

Frames of Group Sets and Their Application in Bundle Theory

We study fiber bundles where the fibers are not a group G but a free G-space with disjoint orbits. The fibers are then not torsors but disjoint unions of these; hence, we like to call them semi-torsors. Bundles of semi-torsors naturally generalize principal bundles, and we call these semi-principal bundles. These bundles admit parallel transport in the same way that principal bundles do. The main difference is that lifts may end up in another group orbit, meaning that the change cannot be described by group translations alone. The study of such effects is facilitated by defining the notion of a basis of a G-set, in analogy with a basis of a vector space. The basis elements serve as reference points for the orbits so that parallel transport amounts to reordering the basis elements and scaling them with the appropriate group elements. These two symmetries of the bases are described by a wreath product group. The notion of basis also leads to a frame bundle, which is principal and so allows for a conventional treatment. In fact, the frame bundle functor is found to be a retraction from the semi-principal bundles to the principal bundles. The theory presented provides a mathematical framework for a unified description of geometric phases and exceptional points in adiabatic quantum mechanics.

Homological aspects of topological gauge-gravity equivalence

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

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.

A gentle introduction to the non-abelian Hodge correspondence

We aim at giving a pedagogical introduction to the non-abelian Hodge correspondence, a bridge between algebra, geometric structures, and complex geometry. The correspondence links representations of a fundamental group, the character variety, to the theory of holomorphic bundles. We focus on motivations, key ideas, links between the concepts and applications. Among others, we discuss the Riemann–Hilbert correspondence, Goldman’s symplectic structure via the Atiyah–Bott reduction, the Narasimhan–Seshadri theorem, Higgs bundles, harmonic bundles, and hyperkähler manifolds.

The having objection to bundle theories of subjects of experience

Abstract The self or subject of experiences is often regarded as a mysterious entity, prompting approaches that seek to deflate it, metaphysically speaking. One such approach is the bundle theory, the most well-known version of which holds that each subject is a bundle of experiences. This version of the bundle theory seems vulnerable to the having objection: since subjects have experiences, they cannot be identical with bundles of experiences. I shall argue that while the having objection is intuitively plausible, its dialectical force is limited since it does not work without substantial further metaphysical assumptions.

Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

Let X be a ruled surface over a nonsingular curve C of genus g≥0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ge 0.$$\\end{document} Let MH:=MX,H(2;c1,c2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H:=M_{X,H}(2;c_1,c_2)$$\\end{document} be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes ci:=ci(E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_i:=c_i(E)$$\\end{document} for i=1,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,2.$$\\end{document} The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space MH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H$$\\end{document} in terms of its Brill–Noether locus WHk(2;c1,c2),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_H^k(2;c_1,c_2),$$\\end{document} whose points correspond to stable vector bundles in MH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H$$\\end{document} having at least k independent sections. We deal with the non-emptiness of this Brill–Noether locus, getting in most of the cases sharp bounds for the values of k such that WHk(2;c1,c2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_H^k(2;c_1,c_2)$$\\end{document} is non-empty.

Brane structures in microlocal sheaf theory

AbstractLet be an exact Lagrangian submanifold of a cotangent bundle , asymptotic to a Legendrian submanifold . We study a locally constant sheaf of ‐categories on , called the sheaf of brane structures or . Its fiber is the ‐category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from to the ‐category of sheaves of spectra on with singular support in .

An Extension of Slice Regular Functions in Terms of Fiber Bundle Theory

An Extension of Slice Regular Functions in Terms of Fiber Bundle Theory

Thomas Aquinas on Concrete Particulars

There are two competing models for how to understand Aquinas’s hylomorphic theory of material substances: the Simple Model, according to which material substances are composed of prime matter and substantial form, and the Expanded Model, according to which material substances are composed of prime matter, substantial form, and all of their accidental forms. In this paper, I first explain the main differences between these two models and show how they situate Aquinas’s theory of material substances in two different places within the contemporary debate on concrete particulars, highlighting several advantages that Aquinas’s approach has over other varieties of substratum and bundle theory along the way. I then offer some reasons to think that the Expanded Model, as a theory of concrete particulars, is preferable. I argue that the Expanded Model avoids two major concerns for the Simple Model: the problem of extrinsicality, and the problem of too-many-possessors.

A cohomological bundle theory for sheaf cohomology

A cohomological bundle theory for sheaf cohomology

Supply-demand bundles and ecological function management of urban ecosystem services: Taking central urban area of Qiqihar as an example

The comprehensive management of multiple ecosystem services is the focus of current research in the field of ecosystem service. The ecological space in the central urban area is scarce, but ecological needs are diverse. How to scientifically use urban land while considering various ecological functions has gradually become an academic hotspot. Taking the central urban area of Qiqihar as an example, we applied the ecosystem service bundle theory to analyze the supply and demand relationship of five ecosystem services in the study area, including flood regulation, air purification, carbon storage and sequestration, biodiversity conservation, and cultural recreation. Based on the supply-demand ratio, we identified the supply-demand bundles of ecosystem services in the study area, completed the division of ecological functional areas, and proposed corresponding spatial control strategies. The results showed that the supply of those five services presented a spatial pattern of low in the city center and south side, while high in the northwest and southeast side, and that the demand for those five services presented a spatial pattern of high in the city center and low around the city. Air purification, biodiversity conservation, and cultural recreation ser-vices were in short supply, with a supply-demand ratio of -0.160, -0.071, and -0.008, respectively. The supply to demand ratios of flood regulation and carbon storage and sequestration services were in excess, with a supply to demand ratio of 0.089 and 0.008, respectively. The spatial imbalance between supply and demand of various services was significant. The research area could be divided into six types of ecosystem service bundles. After comprehensive analysis, the research area would be further divided into four types of zones: urban ecological restoration zone, urban ecological strengthening zone, urban development transition zone, and urban ecological conservation zone. We proposed targeted functional optimization strategies based on the spatial relationship, ecological background characteristics, and supply and demand ratios of various services within the zone.

Theory and calculation of abelian and non-abelian geometric phase factors with SpinDynamica

Cyclic quantum evolution is accompanied by a systematic change in the phase of the initial state vector. This change only depends upon the path traced out by the system itself. Such effects are collectively known as geometric phase factors. The geometric foundations of these phase factors are most elegantly formulated in terms of fibre bundle theory and differential forms, both of which can represent a significant hurdle to master. We present a derivation of the abelian and non-abelian Berry phase in terms of embedded manifolds of linear vector spaces. Embedded manifolds offer the advantage of being less abstract than fibre bundles, and are well-suited for explicit calculations. Essential features of the derivation reduce to matrix–vector manipulations. We further discuss a numerical strategy for the calculation of abelian and non-abelian phase factors. Our approach is based upon Hungarian method and the polar decomposition, and is made freely available as a SpinDynamica addon. Additionally, all derivations and analytic calculations are supported by Mathematica notebooks.

The principal Floquet bundle and the dynamics of fast diffusing communities

We consider, for N ≥ 2 N \geq 2 , the system of N N competing species which are ecologically identical and having distinct diffusion rates { D i } i = 1 N \{D_i\}_{i=1}^N , in an environment with the carrying capacity m ( x , t ) m(x,t) . For a generic class of m ( x , t ) m(x,t) that varies with space and time, we show that there is a positive number D ∗ D_* independent of N N so that if D i ≥ D ∗ D_i \geq D_* for all 1 ≤ i ≤ N 1\le i\le N , then the slowest diffusing species is able to competitively exclude all other species. In the case when the environment is temporally constant or temporally periodic, our result provides some further evidence in the affirmative direction regarding the conjecture by Dockery et al. [J. Math. Biol. 37 (1998), pp. 61–83]. The main tool is the theory of the principal Floquet bundle for linear parabolic equations.

Intuitionistic Modal Algebras

Recent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus. Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.

Lagrangian cobordism functor in microlocal sheaf theory I

AbstractLet be Legendrian submanifolds in the cosphere bundle . Given a Lagrangian cobordism of Legendrians from to , we construct a functor between sheaf categories of compact objects with singular support on and its right adjoint on sheaf categories of proper objects, using Nadler–Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high‐dimensional Legendrian submanifolds.

Subcritical crack growth models for SiC fiber in air and steam: Low temperature data and possible effects of residual stress

Failure times (tf) and brittle creep strain rates (ἑ) for static fatigue of Hi-Nicalon™-S fiber tows were measured at 500 and 600 °C. These measurements added to those done previously at 700–1100 °C. Subcritical crack growth (SCG) based models developed to predict tf and ἑ from the 700–1100 °C data were modified to include the additional data. Parameters for two SCG laws were determined for air and Si(OH)4 saturated steam environments, and an additional parameter for effective room-temperature residual stress (σRo) along the crack path was introduced. Tow failure was modelled as sequential filament failure by SCG from weakest to strongest in a Weibull distribution. The increase in stress on intact filaments as they oxidized and as other filaments failed caused brittle creep strain. SCG-based models for both surface and interior cracks were considered. Orthogonal direction regression (ODR) was used to find the best parameter fits to data. Faster numerical methods to find the global parameter-space minima and avoid local minima were developed. New model parameters were determined using the entire 500–1100 °C tf and ἑ data set, done at initial applied stresses from 260 to 1526 MPa. In air, measured tf and ἑ at 500 and 600 °C follow the trends previously observed for higher temperatures. However, beneath 700 °C in steam, tf are longer and ἑ are much slower than expected. They are similar to values measured for air, suggesting an SCG mechanism change between 700° and 600 °C in steam. An atomistic SCG law-based model with activation energy (Q) of 310 kJ/mol and σRo of ∼200 MPa compression is suggested to be appropriate for static fatigue in air. An atomistic SCG law-based model with Q of 360 kJ/mol and σRo of ∼0 MPa best fit data measured in steam. However, differences in data fit quality between models were small, so firm conclusions about the correct model, residual stress effects, and associated crack paths could not be made. As stress approaches zero, the model converges to tow failure times that are the times for complete fiber oxidation. As failure times approach zero and temperatures decrease, the model converges to tow strengths predicted by fiber bundle theory at room temperature. The sources of error in data fit, the merit of different SCG models, and the possible roles of residual stress in SCG for Hi-Nicalon™-S fibers are discussed.

Are the Realist Bundle Theorists Committed to the Principle of Constituent Identity?

Abstract One of the key questions in the contemporary analytic ontology concerns the relation between the Principle of Identity of Indiscernibles (PII) and the Bundle Theory (BT). The majority of authors believe that BT implies PII. Therefore, it is widely believed that the world violating PII presented by Max Black (1952. “The Identity of Indiscernibles.” Mind 61 (242): 153–64) is also devastating for BT. However, this has been questioned by Rodriguez-Pereyra (2004. “The Bundle Theory is Compatible with Distinct but Indiscernible Particulars.” Analysis 64 (1): 72–81), who formulated an interpretation of BT with instances. Recently Robert (2019. “Can the Realist Bundle Theory Account for the Numerical Difference between Qualitavely Non-discernible Concrete Particulars?” Theorema 38 (1): 25–39) argued that this version of BT is not a constituent ontology and, therefore, Rodriguez-Pereyra’s solution comes at a price of excluding bundle theory from the domain of constituent ontologies, and, in this sense, it fails. I question Robert’s point by claiming that his account of constituent ontologies is too demanding. In particular, I show that the instance version of BT is compatible with the constrains defining constituent ontologies in general, and therefore Rodriguez-Pereyra’s argument is correct.

Characterization method of core pore structure based on truncated Gaussian and its application in shale cores

The core pore structure is mainly described by fractal theory. Conventional single fractal models cannot express the multi-peak distribution characteristics of coal rock, shale, carbonate rock, and other cores. A multifractal model has numerous dimensions and discontinuous functions, making the model parameter selection difficult. Based on the capillary bundle theory and the truncated Gaussian distribution model, a model for characterizing the multi-peak distribution structure of cores is developed in this study. The model's accuracy is verified using the tested NMR experimental data of shale bimodal distribution and the cited coal trimodal distribution. This study further analyzes the influence of tortuosity and tortuosity fractal dimensions on this model. The results show that tortuosity and tortuosity fractal dimensions are conducive to improving the accuracy of the multi-peak pore structure model. The structure model, which can characterize the multi-scale and multi-peak distribution characteristics of shale pore diameter, provides a new concept for describing the pore structure of porous media.

Logarithmic moduli of roots of line bundles on curves

We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a ‘double ramification cycle’ measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho–Pandharipande–Schmitt and Holmes–Schwarz).