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Gustavo Amilcar Saldaña Moncada

Publications and source records attributed to Gustavo Amilcar Saldaña Moncada.

12 recordsLinked to original sources
arXiv2026

Some Constructions on Quantum Principal Bundles

This paper works as an appendix of the paper titled Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group and for paper titled Yang-Mills-Connes Theory and Quantum Principal SU(N)-Bundles. Here, we are going to prove four statements in the theory of quantum principal bundles:: 1) The universal differential envelope $\ast$--calculus of a matrix (compact) Lie group, for the classical bicovariant $\ast$--First Order Differential Calculus, is the algebra of differential forms. 2) An example of a quantum principal bundle in which the space of base forms is not generated by the base space. 3) The group isomorphism between convolution-invertible maps and covariant left module isomorphisms at the level of differential calculus 4) The way the maps $\{T^V_k \}$ from Remark 3.1 look in differential geometry.

math.QA↗
arXiv2026

On the Non-Commutative Geometry of the Electroweak Interaction

In differential geometry, the underlying mathematical structure behind the electroweak theory is a trivial principal $(SU(2)\times U(1))$--bundle over $\R^4$ with the Minkowski metric. The aim of this paper is to show that, using non--commutative geometry, the electroweak theory can be described using a trivial quantum principal $SU(2)$--bundle over $C^\infty_\C(\R^4)$ with the Minkowski metric.

math-ph↗
arXiv2026

Yang--Mills--Connes Theory and Quantum Principal $SU(N)$--Bundles over Toric Non--Commutative Manifolds

This paper has two main objectives. The first one is to show that the Connes formulation of Dirac theory can be applied in the framework of quantum principal bundles for any n dimensional spectral triple, any quantum group, any quantum principal connection and any finite dimensional corepresentation of the quantum group. The second objective is to demonstrate that, under certain conditions, one can define a Yang Mills functional that measures the squared norm of the curvature of a quantum principal connection, in contrast to the Yang Mills functional proposed by Connes, which measures the squared norm of the curvature of a compatible quantum linear connection. An illustrative example based on the noncommutative n torus is presented, highlighting the differences and similarities between the two functionals.

math.QA↗
arXiv2026

Quantum Principal Bundles and Yang-Mills-Scalar-Matter Fields

This paper aims to develop a non-commutative geometrical version of the theory of Yang--Mills--Scalar--Matter fields. To accomplish this purpose, we will dualize the geometrical formulation of this theory, in which principal $G$--bundles, principal connections, and linear representations play the most important role. In addition, we will present the non-commutative geometrical Lagrangian of the system as well as non-commutative geometrical associated field equations. At the end of this work, we show some examples

math.QA↗
arXiv2026

Functoriality of Quantum Principal Bundles and Quantum Connections

Within the framework of Category Theory, we study the association between finite--dimensional representations of a compact quantum group and quantum vector bundles endowed with quantum linear connections, for a fixed quantum principal bundle equipped with a regular quantum principal connection. In particular, we establish a categorical equivalence between such quantum association functors and quantum principal bundles with a particular regular and multiplicative quantum principal connection.

math.QA↗
arXiv2026

Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group

It is well--known that if one is given a principal $G$--bundle with a principal connection, then for every unitary finite--dimensional linear representation of $G$ one can induce a linear connection and a Hermitian structure on the associated vector bundles which are compatible. Furthermore, the gauge group acts on the space of principal connections and on the space of linear connections defined on the associated vector bundles. This paper aims to present the {\it non--commutative geometrical} counterpart of all of these {\it classical} facts in the theory of quantum bundles and quantum connections.

math.QA↗
arXiv2024

Electromagnetic Theory with Quantum Principal Bundles

The aim of this paper is to formulate a {\it non--commutative geometrical} version of the classical electromagnetic field theory in the vacuum with the Moyal--Weyl algebra as the space--time by using the theory of quantum principal bundles and quantum principal connections. As a result we will present the correct Maxwell equations in the vacuum of the model, in which we can appreciate the existence of electric and magnetic charges and currents. Finally, in the fourth section we are going to present a {\it mathematical model} for which there are instantons that are not solutions of the corresponding Yang--Mills equation.

math.QA↗
arXiv2022

Functoriality of Principal Bundles and Connections

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 use this characterization to establish an equivalence between the category of principal bundles and a suitably defined category of functors. From the point of view of differential geometry we detail the specialization of non-linear or Ehresmann to principal and linear connections and discuss the widely known and very useful universality of principal curvature in order to characterize the vector bundles in the image of a given association functor. This is the second version.

math.DG↗
arXiv2022

On Connections and their Curvatures

This paper presents a brief study on connections on fiber, principal and vector smooth bundles as well as some relations with their curvatures.

math.DG↗
arXiv2021

Yang-Mills-Scalar-Matter Fields in the Two-Point Space

The purpose of this work is to present a Non--Commutative Geometrical version of the Yang--Mills Theory and the Yang--Mills Scalar Matter Theory by constructing a concrete example using M. Durdevich's theory of quantum principal bundles.

math-ph↗