Related papers: Generalised Connections and Curvature
Frame bundles equipped with a principal connection have their local structure characterised by a 1-form, called the Cartan connection 1-form, which gathers the principal connection form and the soldering form. We introduce generalised frame…
The first and shorter part of this thesis deals with the structural assumption of invertibility in a Lie groupoid. When this assumption is dropped, we obtain the notion of a Lie category: a small category, endowed with a compatible…
Any leafwise connection on a fibre bundle over a foliated manifold is proved to come from a connection on this fibre bundle.
Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and…
This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of…
Higher bundles are homotopy coherent generalisations of classical fibre bundles. They appear in numerous contexts in geometry, topology and physics. In particular, higher principal bundles provide the geometric framework for higher-group…
Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental…
This paper presents a brief study on connections on fiber, principal and vector smooth bundles as well as some relations with their curvatures.
Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $\mathsf{Bun}_{G,M}$ of principal…
Generalized Yang-Mills theories are constructed, that can use fields other than vector as gauge fields. Their geometric interpretation is studied. An application to the Glashow-Weinberg-Salam model is briefly review, and some related…
The generalized covariant derivative, that uses both scalar and vector bosons, is defined. It is shown how a grand unified theory of the Standard Model can be constructed using a generalized Yang-Mills theory.
It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of…
In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open…
Non-abelian gauge theories in the context of generalized complex geometry are discussed. The generalized connection naturally contains standard gauge and scalar fields, unified in a purely geometric way. We define the corresponding…
The theory of general Galois-type extensions is presented, including the interrelations between coalgebra extensions and algebra (co)extensions, properties of corresponding (co)translation maps, and rudiments of entwinings and…
Perhaps the most important contribution of gauge theory to general mathematics is to point out the importance of association functors. Emphasizing category theory we characterize association functors by two of their natural properties and…
We consider two-dimensional Yang-Mills theories on arbitrary Riemann surfaces. We introduce a generalized Yang-Mills action, which coincides with the ordinary one on flat surfaces but differs from it in its coupling to two-dimensional…
We give a modern account of the construction and structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.
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…
A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…