Principal Fiber Bundle Research Articles

Breakdown of Inter-Hemispheric Connectivity Is Associated with Posttraumatic Symptomatology and Memory Impairment.

Altered brain anatomy in specific gray-matter regions has been shown in patients with posttraumatic stress disorder (PTSD). Recently, white-matter tracts have become a focus of research in PTSD. The corpus callosum (CC) is the principal white-matter fiber bundle, crucial in relaying sensory, motor and cognitive information between hemispheres. Alterations in CC fibers have been reported in PTSD and might be assumed to underlie substantial behavioral and cognitive sequelae; however most diffusion tensor imaging (DTI) studies in adult-onset PTSD failed to address the clinical correlates between imaging and PTSD symptoms severity, behavioral manifestation and cognitive functions. In the current study we examined (a) to what extent microstructural integrity of the CC is associated with memory performance and (b) whether imaging and cognitive parameters are associated with PTSD symptom severity. DTI data were obtained and fractional anisotropy (FA) values were computed for 16 patients and 14 controls. PTSD symptom severity was assessed by employing the clinician administered PTSD scale (CAPS) and memory was tested using a task probing item and associative memory for words and pictures. Significant correlations were found between PTSD symptoms severity, memory accuracy and reaction-time to CC FA values in the PTSD group. This study demonstrates meaningful clinical and cognitive correlates of microstructural connectivity. These results have implications for diagnostic tools and future studies aimed at identifying individuals at risk for PTSD.

Geometric Controllability of The Purcell’s Swimmer and its Symmetrized Cousin

Abstract: We analyse weak and strong controllability notions for the locomotion of the 3-link Purcell’s swimmer, the simplest possible swimmer at low Reynolds number, from a geometric framework. After revisiting a purely kinematic form of the equations, we apply an extension of Chow’s theorem to analyze controllability in the strong and weak sense. Further, the connection form for the symmetric version of the Purcell’s swimmer is derived, based on which the controllability analysis utilizing the Abelian nature of the structure group is presented. The novelty in our approach is the usage of geometry and the principal fiber bundle structure of the configuration manifold of the system to arrive at strong and weak controllability results.

ALGEBRA AND STRUCTURES OF SPINORS FIBER BUNDLES

The aim of this paper is to investigate the mathematics of spinor bundles and their classification. We devote the methods of principal fiber bundles allows through a coherent treatment of Pseudo-Riemannian manifolds and spinor structures with Clifford algebras which couple to Dirac operator to study important applications in cohomology theory.

Topological invariance of the Hall conductance and quantization

It is shown that the Kubo equation for the Hall conductance can be expressed as an integral which implies quantization of the Hall conductance. The integral can be interpreted as the first Chern class of a [Formula: see text] principal fiber bundle on a two-dimensional torus. This accounts for the conductance given as an integer multiple of [Formula: see text]. The formalism can be extended to deduce the fractional conductivity as well.

Homogeneous generalized holomorphic structures

Homogeneous generalized holomorphic structures in the context of homogeneous principal fiber bundles are investigated. They are characterized in terms of Lie algebra data, and the generalized Dolbeault cohomology groups associated to a homogeneous generalized holomorphic vector bundle are identified with certain relative Lie algebra cohomology groups. We also provide some examples, using generalized flag manifolds.

Reflection symmetry in higher dimensional black hole spacetimes

In four spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein’s equation there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in spacetime dimensions such that the action of the isometry group is trivial. (Here ‘trivial’ means that if the ‘axes’—i.e., the points where the axial Killing fields are linearly dependent—are removed, the action of the isometry group is that of a trivial principal fiber bundle. This excludes actions like that found in the Sorkin monopole.) We prove that there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of each axial Killing vector field. The proof relies in an essential way on the first law of black hole mechanics.

Algebraic Structures, Physics and Geometry from a Unified Field Theoretical Framework

Starting from a Unified Field Theory (UFT) proposed previously by the author, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in this UFT general context that the underlying basis of the single geometrical structure P(G, M) (the principal fiber bundle over the real spacetime manifold M with structural group G) reflecting the symmetries of the different fields carry naturally a biquaternionic structure instead of a complex one. This fact allows us to analyze algebraically and to interpret physically in a straighforward way the Majorana and Dirac representations and the relation of such structures with the spacetime signature and non-hermitian (CP) dynamic operators. Also, from the underlying structure of the tangent space, the existence of hidden (super) symmetries and the possibility of supersymmetric extensions of these UFT models are given showing that Rothstein’s theorem is incomplete for that description. The importance of the Clifford algebras in the description of all symmetries, mainly the interaction of gravity with the other fields, is briefly discussed.

Geometric Arbitrage and Spectral Theory

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the ''instantaneous arbitrage capability'' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the no-free-lunch-with vanishing-risk condition if it is only if 0 is in the spectrum.

On some flat connection associated with locally symmetric surface

Abstract For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P(M,G). Associated with - satisfying some particular conditions - local basis of TM local connection form of such a connection is an R(G)-valued 1-form build from the dual basis ω1, ω2 and from the local connection form ω of ▽. The structural equations of (M,∇) are equivalent to the condition dΩ-Ω∧Ω=0. This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.

Classical higgs fields

We consider a classical gauge theory on a principal fiber bundle P → X in the case where its structure group G is reduced to a subgroup H in the presence of classical Higgs fields described by global sections of the quotient fiber bundle P/H → X. We show that matter fields with the exact symmetry group H in such a theory are described by sections of the composition fiber bundle Y → P/H → X, where Y → P/H is the fiber bundle with the structure group H, and the Lagrangian of these sections is factored by virtue of the vertical covariant differential determined by a connection on the fiber bundle Y → P/H.

Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Frechet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γ_c(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γ_c(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C^∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗A, where A is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)_0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SU_n(A)_0 with A a complete involutive commutative continuous inverse algebra.

On globally non-trivial almost-commutative manifolds

Within the framework of Connes’ noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almost-commutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a “principal module,” which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of “gauge modules,” and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.

Geometric phases of water waves

Recently, Banner et al. (Phys. Rev. Lett., 112 (2014) 114502) highlighted a new fundamental property of open ocean wave groups, the so-called crest slowdown. For linear narrow-band waves, this is related to the geometric and dynamical phase velocities Ud and Ug associated with the parallel transport through the principal fiber bundle of the wave dynamics with U(1) symmetry. The theoretical predictions are shown to be in fair agreement with ocean field observations, from which the average crest speed with and .

Algebraic aspects of gauge theories

Gauge theories are primary tools in modern elementary particle physics. The generally recognized mathematical foundations of these theories are in differential geometry, namely, in the theory of connections in a principal fiber bundle. We propose another approach to the mathematical description of gauge theories based on a combination of algebraic and geometric methods.

Geometry of quantum evolution for mixed quantum states

The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert–Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.

Geometric uncertainty relation for mixed quantum states

In this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical Robertson-Schrödinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schrödinger, and vice versa.

A Characterization of Invariant Connections

Given a principal fibre bundle with structure group $S$, and a fibre transitive Lie group $G$ of automorphisms thereon, Wang's theorem identifies the invariant connections with certain linear maps $\psi\colon \mathfrak{g}\rightarrow \mathfrak{s}$. In the present paper, we prove an extension of this theorem which applies to the general situation where $G$ acts non-transitively on the base manifold. We consider several special cases of the general theorem, including the result of Harnad, Shnider and Vinet which applies to the situation where $G$ admits only one orbit type. Along the way, we give applications to loop quantum gravity.

Geometry of matrix product states: Metric, parallel transport, and curvature

We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e., the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a Kähler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.

Wong’s equations in Yang-Mills theory

AbstractWong’s equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The equations obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations, we obtain Wong’s equations in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields.

Deformations of generalized holomorphic structures

A deformation theory of generalized holomorphic structures in the setting of (generalized) principal fibre bundles is developed. It allows the underlying generalized complex structure to vary together with the generalized holomorphic structure. We study the related differential graded Lie algebra, which controls the deformation problem via the Maurer–Cartan equation. As examples, we check the content of the Maurer–Cartan equation in detail in the special cases where the underlying generalized complex structure is symplectic or complex. A deformation theorem, together with some non-obstructed examples, is also included.