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Noncommutative principal bundles and central extensions

Stefan Wagner

TL;DR

This work extends the classical lifting problem for principal bundles to the noncommutative setting by studying free C$^*$-dynamical systems and their lifts along central extensions. It develops a general framework based on factor systems and the equivariant Picard group to determine existence, classify lifts, and describe obstructions via cohomology groups $H^3_ riangle$ and $H^2_ riangle$. The main contributions include a four-step constructive approach to building ${oldsymbol{ ilde{G}}}$-structures, a cohomological classification of such structures, and a suite of illuminating examples (quantum torus coverings, noncommutative Spin(3) structures, noncommutative frame bundles, and Heisenberg-group algebras) that unify geometric, cohomological, and operator-algebraic perspectives. The results pave the way for integrating noncommutative spin geometry, gauge theory, and higher-gerbe concepts, with potential implications for noncommutative Riemannian geometry and quantum field theory frameworks.

Abstract

Motivated by the classical theory of spin structures, we develop a theory for lifting free C$^*$-dynamical systems, a.k.a. noncommutative principal bundles, along central extensions. This theory extends the bundle-theoretic notion of spin structures and yields a complete existence and classification result for such lifts. Using factor system techniques and Picard formalism, our approach introduces new invariants and obstruction classes, thereby unifying geometric, cohomological, and operator-algebraic perspectives. A range of examples demonstrates the scope of the theory.

Noncommutative principal bundles and central extensions

TL;DR

This work extends the classical lifting problem for principal bundles to the noncommutative setting by studying free C-dynamical systems and their lifts along central extensions. It develops a general framework based on factor systems and the equivariant Picard group to determine existence, classify lifts, and describe obstructions via cohomology groups and . The main contributions include a four-step constructive approach to building -structures, a cohomological classification of such structures, and a suite of illuminating examples (quantum torus coverings, noncommutative Spin(3) structures, noncommutative frame bundles, and Heisenberg-group algebras) that unify geometric, cohomological, and operator-algebraic perspectives. The results pave the way for integrating noncommutative spin geometry, gauge theory, and higher-gerbe concepts, with potential implications for noncommutative Riemannian geometry and quantum field theory frameworks.

Abstract

Motivated by the classical theory of spin structures, we develop a theory for lifting free C-dynamical systems, a.k.a. noncommutative principal bundles, along central extensions. This theory extends the bundle-theoretic notion of spin structures and yields a complete existence and classification result for such lifts. Using factor system techniques and Picard formalism, our approach introduces new invariants and obstruction classes, thereby unifying geometric, cohomological, and operator-algebraic perspectives. A range of examples demonstrates the scope of the theory.

Paper Structure

This paper contains 20 sections, 21 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Principal bundles
  4. Representations of groups
  5. C$^*$- dynamical systems
  6. Freeness
  7. The equivariant Picard group
  8. The lifting problem
  9. Analyzing a given $\mathbf{{\hat{G}}}$-structure
  10. Classification of $\mathbf{{\hat{G}}}$-structures
  11. Constructing $\mathbf{{\hat{G}}}$-structures
  12. Imposing the continuity of the lifted action
  13. Examples
  14. A simple example
  15. Coverings of quantum tori
  16. ...and 5 more sections

Key Result

Theorem 1.2

A principal $\mathop{\mathrm{SO}}\nolimits(n)$-bundle $P$ with base space $X$ admits a spin structure if and only if its second Stiefel--Whitney class$w_2(P) \in H^2(X,\mathbb{Z}_2)$ vanishes. In that case, the inequivalent spin structures are parametrized, up to equivalence, by $H^1(X,\mathbb{Z}_2)

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2: see, e. g., Ply86 and friedrich2000
  • Remark 1.3
  • Lemma 2.1: cf.~SchWa15
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 26 more