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Yang-Mills-Connes Theory and Quantum Principal Bundles

Gustavo Amilcar Saldaña Moncada

TL;DR

The paper develops a bridge between Connes' noncommutative Dirac/Yang–Mills framework and Durdevich's quantum principal bundles by formulating Yang–Mills functionals for quantum principal connections and gauge Dirac operators on arbitrary spectral triples and quantum groups. It constructs the gauge theory on associated quantum vector bundles, defines curvature operators, and derives noncommutative Yang–Mills equations, including a dual formulation for quantum principal connections. A key contribution is showing that, under suitable regularity and representation conditions, the Connes YM functional and the quantum principal bundle YM functional can be compared and their critical points analyzed within the same geometric setting, with explicit treatment of toric noncommutative manifolds and the canonical $SU(N)$-type quantum groups. The work thus provides a versatile, physically motivated framework to study noncommutative gauge theories beyond the purely spectral-triple approach, integrating differential calculi, gauge states, and Dirac operators in a unified formalism.

Abstract

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.

Yang-Mills-Connes Theory and Quantum Principal Bundles

TL;DR

The paper develops a bridge between Connes' noncommutative Dirac/Yang–Mills framework and Durdevich's quantum principal bundles by formulating Yang–Mills functionals for quantum principal connections and gauge Dirac operators on arbitrary spectral triples and quantum groups. It constructs the gauge theory on associated quantum vector bundles, defines curvature operators, and derives noncommutative Yang–Mills equations, including a dual formulation for quantum principal connections. A key contribution is showing that, under suitable regularity and representation conditions, the Connes YM functional and the quantum principal bundle YM functional can be compared and their critical points analyzed within the same geometric setting, with explicit treatment of toric noncommutative manifolds and the canonical -type quantum groups. The work thus provides a versatile, physically motivated framework to study noncommutative gauge theories beyond the purely spectral-triple approach, integrating differential calculi, gauge states, and Dirac operators in a unified formalism.

Abstract

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.

Paper Structure

This paper contains 19 sections, 33 theorems, 664 equations.

Table of Contents

  1. Introduction
  2. Compact Quantum Groups and Quantum Differential Forms
  3. Basic Topics
  4. Example: Compact Matrix Lie Groups with the Standard Differential Calculus
  5. Example: The Group $U(1)$ with Non--Standard Differential Calculus
  6. Quantum Principal Bundles
  7. Basic Topics
  8. Example: Toric Non--Commutative Manifolds
  9. Associated Quantum Vector Bundles and Gauge Quantum Linear Connections
  10. Yang--Mills--Connes Theory
  11. Generalities
  12. The Gauge Dirac Operator
  13. Yang--Mills Theory
  14. Example: Electromagnetism in the Quantum $n$--Torus
  15. A quantum principal $\mathcal{U}(1)$--bundle over the Quantum $n$--Torus.
  16. ...and 4 more sections

Key Result

Theorem 2.1

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 (74)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Theorem 3.4
  • ...and 64 more