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Comparison of Levi-Civita connections in noncommutative geometry

Alexander Flamant, Bram Mesland, Adam Rennie

TL;DR

This work unifies multiple noncommutative Levi-Civita constructions by embedding them into centred Hermitian differential calculi with strongly non-degenerate inner products. It develops a duality between differential forms and vector fields, establishing precise conditions under which the AWcurvature and MRLC formalisms coincide and yield the same Hermitian torsion-free Levi-Civita connection. A key result is an existence and uniqueness theorem for Levi-Civita connections on centred bimodules, together with explicit duality formulas that translate connections on forms into affine connections on vector fields. The curvature frameworks are connected via duality, and the analysis extends to isospectral deformations, including well-known theta-deformations of $\mathbb{T}^2$ and $\mathbb{S}^3$, thereby providing a robust algebraic foundation for pseudo-Riemannian noncommutative geometry with broad applicability.

Abstract

We compare the constructions of Levi-Civita connections for noncommutative algebras developed in arXiv:1505.07330, arXiv:1809.06721, arXiv:2403.13735. The assumptions in these various constructions differ, but when they are all defined, we provide direct translations between them. An essential assumption is that the (indefinite) Hermitian inner product on differential forms/vector fields provides an isomorphism with the module dual. By exploiting our translations and clarifying the simplifications that occur for centred bimodules, we extend the existence results for Hermitian torsion-free connections in arXiv:1505.07330, arXiv:1809.06721.

Comparison of Levi-Civita connections in noncommutative geometry

TL;DR

This work unifies multiple noncommutative Levi-Civita constructions by embedding them into centred Hermitian differential calculi with strongly non-degenerate inner products. It develops a duality between differential forms and vector fields, establishing precise conditions under which the AWcurvature and MRLC formalisms coincide and yield the same Hermitian torsion-free Levi-Civita connection. A key result is an existence and uniqueness theorem for Levi-Civita connections on centred bimodules, together with explicit duality formulas that translate connections on forms into affine connections on vector fields. The curvature frameworks are connected via duality, and the analysis extends to isospectral deformations, including well-known theta-deformations of and , thereby providing a robust algebraic foundation for pseudo-Riemannian noncommutative geometry with broad applicability.

Abstract

We compare the constructions of Levi-Civita connections for noncommutative algebras developed in arXiv:1505.07330, arXiv:1809.06721, arXiv:2403.13735. The assumptions in these various constructions differ, but when they are all defined, we provide direct translations between them. An essential assumption is that the (indefinite) Hermitian inner product on differential forms/vector fields provides an isomorphism with the module dual. By exploiting our translations and clarifying the simplifications that occur for centred bimodules, we extend the existence results for Hermitian torsion-free connections in arXiv:1505.07330, arXiv:1809.06721.

Paper Structure

This paper contains 17 sections, 37 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Pseudo-Riemannian calculi on noncommutative vector fields
  4. Modules of differential forms
  5. Relating the vector field and differential form formalisms
  6. Dualising noncommutative differential forms
  7. Derivations and first-order differential calculi
  8. Centred bimodules of differential forms
  9. Relating connections on vector fields and forms
  10. Affine connections on modules of vector fields
  11. Connections on differential forms
  12. Dualising noncommutative connections
  13. Dual Levi-Civita connections
  14. Existence of the Levi-Civita connection
  15. Comparing curvatures
  16. ...and 2 more sections

Key Result

Proposition 3.4

The musical maps $\overrightarrow{\sharp}, \overleftarrow{\sharp}$ are isomorphisms of left- and right-${\mathcal{A}}$-modules respectively, that are compatible with the dagger maps on forms and vector fields $\dagger \circ \overrightarrow{\sharp} = \overleftarrow{\sharp} \circ \dagger$. We denote b

Theorems & Definitions (107)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 97 more