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

Gustavo Amilcar Saldaña Moncada

TL;DR

This work develops a comprehensive noncommutative geometric formulation of Yang–Mills–scalar-matter theory via quantum principal bundles in Durdevich's framework. The authors dualize the classical bundle-based gauge theory to define a NC Lagrangian, curvature-based actions, and coupled field equations, including both gauge and scalar fields, while ensuring adjointability of the differential and connection operators. They establish the mathematical backbone with quantum differential calculi, Mor(δ^V,Δ_Hor) modules, and induced quantum linear connections, and then demonstrate the theory on three representative classes of qpbs: over classical manifolds, trivial Moyal–Weyl spaces, and the U_q(n) rrow SU_q(n+1) rrow CP_q^n homogeneous bundle. The results pave a geometrical path toward NC versions of the Standard Model, with potential extensions to spin geometry and spectral triples, and reveal intriguing physical features such as nontrivial contributions from the S^ω operator (including charges in vacuum) in certain NC settings.

Abstract

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

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

TL;DR

This work develops a comprehensive noncommutative geometric formulation of Yang–Mills–scalar-matter theory via quantum principal bundles in Durdevich's framework. The authors dualize the classical bundle-based gauge theory to define a NC Lagrangian, curvature-based actions, and coupled field equations, including both gauge and scalar fields, while ensuring adjointability of the differential and connection operators. They establish the mathematical backbone with quantum differential calculi, Mor(δ^V,Δ_Hor) modules, and induced quantum linear connections, and then demonstrate the theory on three representative classes of qpbs: over classical manifolds, trivial Moyal–Weyl spaces, and the U_q(n) rrow SU_q(n+1) rrow CP_q^n homogeneous bundle. The results pave a geometrical path toward NC versions of the Standard Model, with potential extensions to spin geometry and spectral triples, and reveal intriguing physical features such as nontrivial contributions from the S^ω operator (including charges in vacuum) in certain NC settings.

Abstract

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 --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

Paper Structure

This paper contains 16 sections, 24 theorems, 443 equations.

Table of Contents

  1. Introduction
  2. Notation and Basic Concepts
  3. Compact Quantum Groups and their Associated Spaces of Quantum Differential Forms
  4. Quantum Principal Bundles
  5. Associated Quantum Vector Bundles, Induced Quantum Linear Connections and Hermitian Structures
  6. Formal Adjointability
  7. Formal Adjointability of the Differential
  8. Formal Adjointability of Quantum Linear Connections
  9. Yang–Mills Scalar Matter Fields in Non–Commutative Geometry
  10. Yang--Mills Theory
  11. Non--Commutative Geometrical Yang--Mills Scalar Matter Theory
  12. Examples
  13. Quantum Principal Bundles over Smooth Manifolds
  14. Trivial Quantum Principal Bundles over the Moyal--Weyl Algebra
  15. The Quantum Principal Homogeneous Bundle $U_q(n)\lhook\joinrel\relbar\joinrel\rightarrow SU_q(n+1)\longrightarrow \mathbbm{CP}_q^n$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathcal{T}$ be a complete set of mutually non--equivalent irreducible (necessarily finite--dimensional) $\mathcal{G}$--corepresentations with $\delta^\mathbb{C}_\mathrm{triv}$$\in$$\mathcal{T}$ (the trivial corepresentation on $\mathbb{C}$). For any $\delta^V$$\in$$\mathcal{T}$ that coacts on where $\{ e_i\}^{n_{V}}_{i=1}$ is an orthonormal basis of $V$ and $\{g^{V}_{ij}\}^{n_{V}}_{i,j=1}$$

Theorems & Definitions (56)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Definition 7
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 46 more