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Gauge transformations on quantum principal bundles

Antonio Del Donno, Emanuele Latini, Thomas Weber

TL;DR

The paper develops a comprehensive framework for gauge transformations on quantum principal bundles by embedding Hopf--Galois extensions into complete differential calculi and establishing a graded Braiding structure. It extends Brzeziński–Majid gauge transformations to differential forms, proving a correspondence with graded vertical automorphisms and enabling actions on connections and curvature. The framework yields an exact noncommutative Atiyah sequence, graded Hopf--Galois extensions of forms, and explicit examples including the noncommutative $2$-torus, the quantum Hopf fibration, and smash product algebras. This advances noncommutative differential geometry and provides concrete tools to study quantum gauge theories on noncommutative spaces. The results have potential applications to noncommutative gauge theories, quantum principal bundles, and geometric structures on quantum spaces.

Abstract

We understand quantum principal bundle as faithfully flat Hopf--Galois extensions, with a structure Hopf algebra coacting on a total space algebra and with base algebra given by the coinvariant elements. To endow such bundles with a compatible differential structure, one requires the coaction to extend as a morphism of differential graded algebras. This leads to an exact noncommutative Atiyah sequence, a graded Hopf--Galois extension of differential forms and a canonical braiding on total space forms such that the latter are graded-braided commutative. We recall this approach to noncommutative differential geometry and further discuss the extension of quantum gauge transformations, in the sense of Brzeziński, to differential forms. In this way we obtain an action of quantum gauge transformations on connections of the quantum principal bundle and their curvature. Explicit examples, such as the noncommutative 2-torus, the quantum Hopf fibration and smash product algebras are discussed.

Gauge transformations on quantum principal bundles

TL;DR

The paper develops a comprehensive framework for gauge transformations on quantum principal bundles by embedding Hopf--Galois extensions into complete differential calculi and establishing a graded Braiding structure. It extends Brzeziński–Majid gauge transformations to differential forms, proving a correspondence with graded vertical automorphisms and enabling actions on connections and curvature. The framework yields an exact noncommutative Atiyah sequence, graded Hopf--Galois extensions of forms, and explicit examples including the noncommutative -torus, the quantum Hopf fibration, and smash product algebras. This advances noncommutative differential geometry and provides concrete tools to study quantum gauge theories on noncommutative spaces. The results have potential applications to noncommutative gauge theories, quantum principal bundles, and geometric structures on quantum spaces.

Abstract

We understand quantum principal bundle as faithfully flat Hopf--Galois extensions, with a structure Hopf algebra coacting on a total space algebra and with base algebra given by the coinvariant elements. To endow such bundles with a compatible differential structure, one requires the coaction to extend as a morphism of differential graded algebras. This leads to an exact noncommutative Atiyah sequence, a graded Hopf--Galois extension of differential forms and a canonical braiding on total space forms such that the latter are graded-braided commutative. We recall this approach to noncommutative differential geometry and further discuss the extension of quantum gauge transformations, in the sense of Brzeziński, to differential forms. In this way we obtain an action of quantum gauge transformations on connections of the quantum principal bundle and their curvature. Explicit examples, such as the noncommutative 2-torus, the quantum Hopf fibration and smash product algebras are discussed.
Paper Structure (19 sections, 19 theorems, 82 equations)

This paper contains 19 sections, 19 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Noncommutative calculi on quantum principal bundles
  3. Quantum principal bundles
  4. Differential calculi
  5. Covariant calculi on Hopf algebras
  6. Complete differential calculi
  7. Total space forms, basic forms, vertical forms and horizontal forms
  8. First order completeness and the Brzeziński--Majid approach
  9. Graded Hopf--Galois extension and braidings
  10. Examples of complete calculi
  11. Hopf algebras:
  12. The noncommutative algebraic 2-torus.
  13. The quantum Hopf fibration:
  14. Smash product algebras:
  15. Gauge transformations
  16. ...and 4 more sections

Key Result

Proposition 2.3

Let $A$ be an algebra and $(\Omega^1(A),\mathrm{d})$ a FODC on $A$. Then there exists a DC $\Omega^\bullet_\mathrm{max}(A)$ on $A$ such that its truncation coincides with $(\Omega^1(A),\mathrm{d})$ and such that the following universal property holds: for all differential calculi $\tilde{\Omega}^\bu

Theorems & Definitions (33)

  • Definition 2.1: Quantum principal bundle
  • Example 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Remark 3.6
  • ...and 23 more