Narrow Results

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve $M$, we construct explicitly a bundle ${{𝒲}}_M$ with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations.

For a foliation ${ℱ}$ defined on a smooth complex manifold $M$ we introduce the category of vertex operator algebra $V$ bundles with sections provided by vectors of elements of the space of algebraically extended $V$-module $W$-valued differentials. An intrinsic coordinate-independent formulation for such bundles is given. Finally, we identify the cohomology of the spaces of sections for a vertex operator algebra $V$ bundle with vertex operator algebra cohomology of the holonomy groupoid $\mathrm{\text{Hol}}(M,{ℱ})$.

In this paper, we investigate the scenarios in which a holonomic versus a non-holonomic frame description of gravity theories are equivalent. It turns out that classically, the equivalence holds in a way that is independent of the particular dynamics and/or spacetime dimension. This includes general metric-affine dynamics. A global bundle-theoretical investigation is carried out, uncovering the equivalence principle as the culprit. The equivalence holds as long as the equivalence principle holds. This is not something to be expected when non-invertible configurations of the vielbein field are taken into account. In such cases, the gauge-theoretical description of gravity “unsolders” from spacetime, and one has to decide if gravity is spacetime geometry or an internal gauge theory.

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.

We study a random fiber bundle model with tips of the fibers placed on a graph having co-ordination number 3. These fibers follow local load sharing with uniformly distributed threshold strengths of the fibers. We have studied the critical behavior of the model numerically using a finite size scaling method and the mean field critical behavior is established. The avalanche size distribution is also found to exhibit a mean field nature in the asymptotic limit.

The discrete phase space and continuous time representation of relativistic quantum mechanics are further investigated here as a continuation of paper I.¹ The main mathematical construct used here will be that of an area filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as ${\bar{S}}_n^1$, a circle of radius $\sqrt{2n+1}$ where $n\in \{0,1,\dots \}$. We interpret this two-dimensional (2D) Peano circle in our framework as a phase cell inside our 2D discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell ${\bar{S}}_n^1$. The time evolution of this Peano circle sweeps out a 2D vertical cylinder analogous to the worldsheet of string theory. Extending this to 3D space, we introduce a $(2+2+2)$-dimensional phase space hyper-tori ${\bar{S}}_{n^1}^1\times {\bar{S}}_{n^2}^1\times {\bar{S}}_{n^3}^1$ as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fiber bundles.

We also study free scalar Bosons in the background $[(2+2+2)+1]$-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein–Gordon equation. The second quantized field quanta of this system can cohabit with the tiny Planck oscillators inside the ${\bar{S}}_{n^1}^1\times {\bar{S}}_{n^2}^1\times {\bar{S}}_{n^3}^1$ phase cells for eternity. Finally, a generalized free second quantized Klein–Gordon equation in a higher $[(2+2+2)N+1]$-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory.

The Composition Algebra-based Methodology (CAM) [B. Wolk, Pap. Phys.9, 090002 (2017); Phys. Scr.94, 025301 (2019); Adv. Appl. Clifford Algebras27, 3225 (2017); J. Appl. Math. Phys.6, 1537 (2018); Phys. Scr.94, 105301 (2019), Adv. Appl. Clifford Algebras30, 4 (2020)], which provides a new model for generating the interactions of the Standard Model, is geometrically modeled for the electromagnetic and weak interactions on the parallelizable sphere operator fiber bundle$B_n=(T𝕄,S^n\to {\mathcal{𝒮}}^n,SO(n+1),\pi )$ consisting of base space, the tangent bundle $T𝕄$ of space–time $𝕄$, projection operator $\pi$, the parallelizable spheres $S^n=\{S^1,S^3\}$ conceived as operator fibers $S^n\to {\mathcal{𝒮}}^n$ attaching to and operating on $T_p𝕄$$\forall p\in 𝕄$ as $p$ varies over $𝕄$, and as structure group, the norm-preserving symmetry group $SO(n+1)$ for each of the division algebras which is simultaneously the isometry group of the associated unit sphere. The massless electroweak $SU{(2)}_L\otimes U{(1)}_Y$ Lagrangian is shown to arise from $B_{3\otimes 1}$’s generation of a local coupling operation on sections of Dirac spinor and Clifford algebra bundles over $𝕄$. Importantly, CAM is shown to be a new genre of gauge theory which subsumes Yang–Mills Standard Model gauge theory. Local gauge symmetry is shown to be at its core a geometric phenomenon inherent to CAM gauge theory. Lastly, the higher-dimensional, topological architecture which generates CAM from within a unified eleven $(1,10)$-dimensional geometro-topological structure is introduced.

Rudimentary knots are invoked to generate a representation of the elementary particles, a model that endows the particles with visualizable structure. The model correlates with the basic tenets, taxonomy, and interactions of the Standard Model, but goes beyond it in a number of important ways, the most significant being that all particles (hadrons and leptons, fermions and bosons) and interactions share a common topology. Among other consequences of the modeling are the topological basis for isospin invariance and its connection to electric charge, the necessary identity of electron and proton charge magnitudes, and the existence of precisely three generations on the particle family tree. The salient feature of the model is that the elementary particles are viewed not as discrete, point-like objects in a vacuum but rather as sustainable, membrane-like distortions embodying curvature and torsion in and of an otherwise featureless continuum and that their manifest physical attributes correlate with the distortion. There are additional connections to the theories of fiber bundles, superstrings and instantons and, historically, to the work of Kelvin in the mid-nineteenth century and Cartan in the 1920s among others.

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds.

The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied.

Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed.

A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher "nonassociative" symmetries.

We formalize geometrically the idea that the (de Donder) Hamiltonian formulation of a higher derivative Lagrangian field theory can be constructed understanding the latter as a first derivative theory subjected to constraints.

In this paper, we derive the symplectic framework for field theories defined by higher order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher order system of partial differential equations to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincaré–Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher order version of the Poincaré–Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples.

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined.