In this work we explore the geometrical interpretation of gauge theories through the formalism of fiber bundles. Moreover, we conduct an investigation in the topology of fiber bundles, providing a proof of the Classification Theorem. In the last chapter we present some applications, such as electromagnetism and generalized Kaluza-Klein Theory.
This note is a continuation of our earlier articles arXiv:1612.08897 and arXiv:1709.09030, where using the dependent coordinates the local Lagrange-Poincaré equations were obtained for a mechanical system with symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and vector space), on which a free proper and isometric action of a compact semisimple Lie group is given. Assuming the existence of the parametric representations for local sections in the principal bundle, we make the transiti…
I articulate and discuss a geometrical interpretation of Yang-Mills theory. Analogies and disanalogies between Yang-Mills theory and general relativity are also considered.
We unify three approaches within the vast body of gauge-theory research that have independently developed distinct representations of a geometrical surface-like structure underlying the vector-potential. The three approaches that we unify are: those who use the compactified dimensions of Kaluza-Klein theory, those who use Grassmannian models (also called gauge theory embedding or $CP^{N-1}$ models) to represent gauge fields, and those who use a hidden spatial metric to replace the gauge fields. In this paper we identify a correspondence between the geometrical representations of the three scho…
A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations.…
The usual derivation of the Lagrangian of a model for massive vector bosons, by spontaneous symmetry breaking of a gauge theory, implies that the prefactors of the various interaction terms are uniquely determined functions of the coupling constant(s) and the masses. Since, under the renormalization group (RG) flow, different interaction terms get different loop-corrections, it is uncertain, whether these functions remain fixed under this flow. We investigate this question for the $U(1)$-Higgs-model to 1-loop order in the framework of Epstein-Glaser renormalization. Our main result reads: choo…
This work uses fiber bundles as a framework to describe some effects of number scaling on gauge theory and some geometric quantities. A description of number scaling and fiber bundles over a flat space time manifold, M, is followed by a description of gauge theory. A fiber at point x of M contains a pair of scaled complex number and vector space structures, $C^{c}_{x}\times V^{c}_{x} $ for each c in GL(1,C). A space time dependent scalar field, g, determines, for each x, the scaling value of the vector space structures that contain the values of a vector valued matter field at x. The vertical …
We derive an interacting quintessence model on the framework of a recently introduced new class of geometrical scalar-tensor theories of gravity formulated on a Weyl-Integrable geometry, where the gravitational sector is described by both a scalar and a tensor metric field. By using a Palatini variational principle we construct a scalar-tensor action invariant under the Weyl symmetry group of the background geometry, which in the Einstein-Riemann frame leads to a gravitoelectromagnetic theory. We use the gauge freedom of the theory and the fact that the Weyl scalar field couples with matter fi…
Taking only the characteristics as absolute, in the spirit of Arnold's "Geometrical Methods in the Theory of Ordinary Differential Equations" (Springer, 1988), we give an independent of coordinates formulation of general variational entropy inequalities for quasilinear equations of first order, that locally read as Kruzhkov inequalities, in terms of certain "entropy densities", and in the case of the equation of 2D flat projective structure we get the expression of the general entropy density from its abstract Rankine-Hugoniot rule for shocks using the projective geometry of the plane.
The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show t…
The purpose of this paper is to put the description of number scaling and its effects on physics and geometry on a firmer foundation, and to make it more understandable. A main point is that two different concepts, number and number value are combined in the usual representations of number structures. This is valid as long as just one structure of each number type is being considered. It is not valid when different structures of each number type are being considered. Elements of base sets of number structures, considered by themselves, have no meaning. They acquire meaning or value as elements…
Quiver 5D $\mathcal{N}=1$ gauge theories describe the low-energy dynamics on webs of $(p,q)$-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping $(p,q)$-branes to branes of charge $(-q,p)$, and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the …
Geometric foundations of thermodynamics in the quantum regime
Universidad de Granada（格拉纳达大学）；Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada（格拉纳达大学卡洛斯一世理论与计算物理研究所）；Universidad de Córdoba（科尔多瓦大学）
We show that the suspended pinch point can be seen as an elliptically fibered variety with singular fibers of type I$_2$ over codimension-one points of the base and a torsionless Mordell--Weil group of rank one. In the F-theory algorithm, this corresponds to a Lie group $SU(2)\times U(1)$. We also identify the matter content as given by the direct sum of the adjoint representation (with zero U($1$)-charge) and the fundamental representation with U($1$)-charge $\pm 1$. We then study the geometry of an SU($2$)$\times$U($1$)-model given by a compact elliptically fibered variety with the singulari…
