Fiber Bundle Theory Research Articles

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.

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

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

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.

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.

Colloquium : Geometric phases of light: Insights from fiber bundle theory

This Colloquium reviews the role of geometric phases in optics from the perspective of fiber bundle theory. The transformation of the polarization of light by discrete optical components illustrates many elements of the theoretical formalism, and an outlook is presented for the study and application of geometric phases in higher dimensions through analyses of higher order Gaussian modes and general vectorial fields.

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.

The Concept of Curvature in Fiber Bundles

Many basic concepts in theoretical physics can be interpreted in term of fiber bundles. A fiber bundle is a manifold that locally looks like a product manifold. The well-known examples are the tangent and the cotangent bundles. The aim of this paper is to discuss and explain the concept of curvature in fiber bundles. We followed the analytical induction mathematical method and we found that the curvature is very important concept in fiber bundles.We use Theory of fiber bundles to illustrate the structure homogenous C ∗ algebras.

Super fiber bundles, connection forms, and parallel transport

The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms, and their parallel transport, which ties together various approaches. We begin with a detailed introduction to super fiber bundles. We then introduce the concept of so-called relative supermanifolds as well as bundles and connections defined in these categories. Studying these objects turns out to be of utmost importance in order to, among other things, model anticommuting classical fermionic fields in mathematical physics. We then construct the parallel transport map corresponding to such connections and compare the results with those found by other means in the mathematical literature. Finally, applications of these methods to supergravity will be discussed, such as the Cartan geometric formulation of Poincaré supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries.

Operations and Actions of Lie Groups on Manifolds

As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on Rn. A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.

Constrained gauge-gravity duality in three and four dimensions

The equivalence between Chern–Simons and Einstein–Hilbert actions in three dimensions established by Achúcarro and Townsend (Phys Lett B 180:89, 1986) and Witten (Nucl Phys B 311:46, 1988) is generalized to the off-shell case. The technique is also generalized to the Yang–Mills action in four dimensions displaying de Sitter gauge symmetry. It is shown that, in both cases, we can directly identify a gravity action while the gauge symmetry can generate spacetime local isometries as well as diffeomorphisms. The price we pay for working in an off-shell scenario is that specific geometric constraints are needed. These constraints can be identified with foliations of spacetime. The special case of spacelike leafs evolving in time is studied. Finally, the whole set up is analyzed under fiber bundle theory. In this analysis we show that a traditional gauge theory, where the gauge field does not influence in spacetime dynamics, can be (for specific cases) consistently mapped into a gravity theory in the first order formalism.

The Geometry of Synchronization Problems and Learning Group Actions

We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group G on connected graph $$\Gamma $$ with a flat principal G-bundle over $$\Gamma $$ , thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of $$\Gamma $$ into G. We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal G-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham–Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions—partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations—and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.

A constitutive relation for the tissue composed of type-I collagen fibers under uniaxial tension

Mechanical tests for various collagen tissues showed that the stress-strain curves are composed of two regimes with different deformation mechanisms. In this paper, the constitutive relations are first studied separately for the two regimes and then merged together. For the unfurling regime, a phenomenological power-law relation is utilized to characterize the stress-strain curve, and, for the stretching regime which is modeled as a fiber bundle material, a nonlinear constitutive relation is proposed based on the Weibull distribution and Daniels’ fiber bundle theory. A technique is suggested to continuously merge the two relations for the two regimes of collagen tissues. A strategy is finally devised to quantitatively determine the critical strain for dividing the two regimes and to identify the constitutive parameters for experimental curves. The constitutive relation is first validated by comparing the identified parameters with the references, and then applied to predict the experimental stress-strain curves of collagen tissues. The present work shows that the critical strain plays an essential role in establishing the constitutive relation if experimental data are adopted.

Cartan subalgebras in C*-algebras. Existence and uniqueness

We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analyzed systematically using the theory of fiber bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear uniform Roe algebras.

Aharonov-Bohm Effect vs. Dirac Monopole: A-B ⇔ D

In the context of fiber bundle theory, we show that the existence of the Aharonov-Bohm connection implies the existence and uniqueness of the Dirac connection.

Connection and curvature in crystals with non-constant dislocation density

Given a smooth defective solid crystalline structure defined by linearly independent ‘lattice’ vector fields, the Burgers vector construction characterizes some aspect of the ‘defectiveness’ of the crystal by virtue of its interpretation in terms of the closure failure of appropriately defined paths in the material and this construction partly determines the distribution of dislocations in the crystal. In the case that the topology of the body manifold M is trivial (e.g., a smooth crystal defined on an open set in ), it would seem at first glance that there is no corresponding construction that leads to the notion of a distribution of disclinations, that is, defects with some kind of ‘rotational’ closure failure, even though the existence of such discrete defects seems to be accepted in the physical literature. For if one chooses to parallel transport a vector, given at some point P in the crystal, by requiring that the components of the transported vector on the lattice vector fields are constant, there is no change in the vector after parallel transport along any circuit based at P. So the corresponding curvature is zero. However, we show that one can define a certain (generally non-zero) curvature in this context, in a natural way. In fact, we show (subject to some technical assumptions) that given a smooth solid crystalline structure, there is a Lie group acting on the body manifold M that has dimension greater or equal to that of M. When the dislocation density is non-constant in M the group generally has a non-trivial topology, and so there may be an associated curvature. Using standard geometric methods in this context, we show that there is a linear connection invariant with respect to the said Lie group, and give examples of structures where the corresponding torsion and curvature may be non-zero even when the topology of M is trivial. So we show that there is a ‘rotational’ closure failure associated with the group structure – however, we do not claim, as yet, that this leads to the notion of a distribution of disclinations in the material, since we do not provide a physical interpretation of these ideas. We hope to provide a convincing interpretation in future work. The theory of fibre bundles, in particular the theory of homogeneous spaces, is central to the discussion.

Strengths of ceramic fiber bundles: Theory and practice

AbstractThe Weibull modulus and reference strength of ceramic fibers can be inferred from measurements of the tensile stress‐strain response of a bundle of such fibers. The goal of the present article is to address issues in the fidelity of results stemming from fiber bundle tests and strategies to optimize outcomes. The issues are addressed through established theorems in uncertainty propagation, Monte Carlo simulations of fiber bundle fracture, and experimental measurements on bundles of SiC fibers of various length and surface condition. The study shows that optimal results are obtained when: (i) tests are performed on fiber bundles with a gauge length that exceeds a critical value (specifically, that needed to prevent mechanical instabilities in the post‐load‐maximum domain); (ii) bundles are lubricated with a low‐viscosity oil, to mitigate both inter‐fiber friction and dynamic coupling associated with release waves following fiber fracture; and (iii) the Weibull parameters are obtained by directly fitting the measured stress‐strain curves with the function predicted by fiber bundle theory, rather than using methods based on either linear regression analysis of Weibull probability plots or fitting of the peak stress and strain alone.

A geometric approach to modeling of four- and five-link planar snake-like robot

The article deals with the issue of use of geometric mechanics tools in modelling nonholonomic systems. The introductory part of the article contains fiber bundle theory that we use at creating mathematical model of nonholonomic locomotion system with undulatory movement. Further the determination of general mathematical model for n-link snake-like robot is presented, where we used nonholonomic constraints. The relation between changes of shape and position variables was expressed using the local connection that was used to analyze and control system movement by vector fields. The effect of links number of snake-like robot on its mathematical model was investigated. The last part of this article consists of detailed description of modeling reconstruction equation for four- and five-link snake-like robot.

Weyl correspondence for a charged particle in the field of a magnetic monopole

We construct a generalized Weyl correspondence for an electrically charged particle in the field of the Dirac magnetic monopole. Our starting points are a global Lagrangian description of this system as a constrained system with U(1) gauge symmetry given in terms of the fiber bundle theory and a reduction of the presymplectic structure arising on the constraint surface. In contrast to the recently proposed quantization scheme based on using a quaternionic Hilbert module, the quantum operators corresponding to classical observables in our construction act in the complex Hilbert space of U(1)-equivariant functions introduced by Greub and Petry. These functions are defined on the total space of a fiber bundle that is topologically equivalent to the Hopf fibration.

A New Formalism for Classifying Spin-Orbit Systems Using Tools Distilled from the Theory of Bundles

We give an informal summary of ongoing work which uses tools distilled from the theory of fibre bundles to classify and connect invariant fields associated with spin motion in storage rings. We mention four major theorems. One ties invariant fields with the notion of normal form, the second allows comparison of different invariant fields and the two others tie the existence of invariant fields to the existence of certain invariant sets. We explain how the theorems apply to the spin dynamics of spin-[Formula: see text] and spin-[Formula: see text] particles. Our approach elegantly unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics and suggests an avenue for addressing the question of the existence of the invariant spin field.

基于微分几何的蛇形机器人动力学与控制统一模型

Whereas the inputs for a snakelike robot are torques, the dynamics system is a nonlinear control system. With increasing modules in a snakelike robot, its nonlinear control system becomes complex and inconvenient for regulation and control. In this paper, the differential geometry method is used, and the Euler-Lagrange equations are extended to equations under any base. Thus, the dynamics equations are reduced to the standard affine control system, and the dynamics-control unified model is derived; this simplifies the regulation and control of the snakelike robot. Based on the unified model, a partial feedback linearization method is developed, and the head trajectory controller is designed. The configuration space of a snakelike robot corresponds to the manifold space, the velocity corresponds to the tangent space, the torque space corresponds to the cotangent space, and the kinematic energy provides a Riemann measure on the manifold. Thus, the dynamics of a snakelike robot can be described by Riemann geometry. Additionally, the passive wheels installed under the snakelike robot introduce the velocity constraint, which constrains the velocity space to a subspace of the tangent space. That is, the velocity space forms a distribution, and the dynamics system becomes a nonholonomic dynamics system. For a snakelike robot with passive wheels, the configuration is a Riemann manifold with a distribution. In the distribution, the appropriate base can be chosen to simplify the dynamics. In this paper, a base model is built based on the fiber bundle theory. Any set base is only a section in the fiber bundle. The orthogonal normalization technique is adopted to derive a set base that can simplify the dynamics calculation, and the dynamics-control unified model is derived. Finally, a nine-module snakelike robot is used as an example of the partial feedback linearization method.