Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group

Gustavo Amilcar Saldaña Moncada

TL;DR

The paper constructs a robust non--commutative generalization of principal bundle gauge theory by defining associated quantum vector bundles via $\textsc{Mor}(\delta^{V},\Delta_P)$, establishing their finite projectivity, and equipping them with induced quantum linear connections derived from quantum principal connections. It introduces canonical Hermitian structures on these quantum vector bundles and proves their compatibility with the induced connections, providing a parallel to the classical Hermitian setting. A novel quantum gauge group $\mathfrak{qGG}$ is defined with a quantum translation map, and its action on spaces of quantum principal connections and induced connections is explored, including how gauge transformations propagate to curvatures. The framework supports a foundation for quantum Yang--Mills type theories within Durdevich's non--commutative geometry, and it integrates the bicovariant differential calculus for quantum groups with Serre--Swan style projective modules to mirror classical differential geometric constructions.

Abstract

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.

Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group

TL;DR

The paper constructs a robust non--commutative generalization of principal bundle gauge theory by defining associated quantum vector bundles via , establishing their finite projectivity, and equipping them with induced quantum linear connections derived from quantum principal connections. It introduces canonical Hermitian structures on these quantum vector bundles and proves their compatibility with the induced connections, providing a parallel to the classical Hermitian setting. A novel quantum gauge group is defined with a quantum translation map, and its action on spaces of quantum principal connections and induced connections is explored, including how gauge transformations propagate to curvatures. The framework supports a foundation for quantum Yang--Mills type theories within Durdevich's non--commutative geometry, and it integrates the bicovariant differential calculus for quantum groups with Serre--Swan style projective modules to mirror classical differential geometric constructions.

Abstract

It is well--known that if one is given a principal --bundle with a principal connection, then for every unitary finite--dimensional linear representation of 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.

Paper Structure

This paper contains 16 sections, 34 theorems, 667 equations.

Table of Contents

  1. Introduction
  2. Preparatory Material
  3. Quantum Groups
  4. Basic Theory of Quantum Principal Bundles
  5. Associated Quantum Vector Bundles, Induced Quantum Linear Connections and Hermitian Structures
  6. Associated Quantum Vector Bundles and Induced Quantum Linear Connections
  7. The Canonical Hermitian Structure
  8. The Quantum Gauge Group and its Action on Quantum Connections
  9. The Quantum Translation Map and The Quantum Gauge Group
  10. Action on Quantum Connections
  11. On Induced Quantum Linear Connections
  12. Examples
  13. Trivial Quantum Principal Bundles
  14. Homogeneous Quantum Principal Bundles
  15. Concluding Comments
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\mathcal{T}$ be a complete set of mutually non--equivalent, irreducible, finite--dimensional $\mathcal{G}$--corepresentations. For any $\delta^V$$\in$$\mathcal{T}$ that coacts on $V$, where $\{ e_i\}^{n_{V}}_{i=1}$ is an orthonormal basis of $V$ (with respect to the inner product that makes $\delta^V$ unitary) and $\{g^{V}_{ij}\}^{n_{V}}_{i,j=1}$$\subseteq$$H$. Then $\{g^{V}_{ij}\}_{\delta^V

Theorems & Definitions (76)

  • Theorem 2.1
  • Example 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Example 2.6
  • Definition 2.7: Embedded differential
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 66 more