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The Levi-Civita connection and Chern connections for cocycle deformations of Kähler manifolds

Jyotishman Bhowmick, Bappa Ghosh

TL;DR

The paper develops a noncommutative analogue of the classical relation between Levi-Civita and Chern connections on Kahler manifolds via unitary 2-cocycle twists in covariant *-differential calculi. Using bar-category and monoidal equivalence techniques, it shows that complex structures, holomorphic bimodules, and Chern connections deform coherently and that the Levi-Civita connection on the deformed space of one-forms $Ω^1_γ$ decomposes as a direct sum of twisted Chern connections on the holomorphic and anti-holomorphic parts when the complex structure is factorizable and the metric is compatible. The results apply to both classical Kahler manifolds under group actions and noncommutative spaces such as the Heckenberger-Kolb calculi, providing a unified deformation-theory toolkit with broad applicability to toric and quantum homogeneous space deformations. This work thereby extends foundational Riemannian and complex-geometry relations to a wide class of quantum geometries via explicit cocycle twisting.

Abstract

We consider unitary cocycle deformations of covariant $\ast$-differential calculi. We prove that complex structures, holomorphic bimodules and Chern connections on the deformed calculus are twists of their untwisted counterparts. Moreover, for cocycle deformations of a class of classical Kähler manifolds, the Levi-Civita connection on the space of one-forms of the deformed calculus is shown to be a direct sum of the Chern connections on the twisted holomorphic and the anti-holomorphic bimodules. Our class of examples also includes cocycle deformations of the Heckenberger-Kolb calculi.

The Levi-Civita connection and Chern connections for cocycle deformations of Kähler manifolds

TL;DR

The paper develops a noncommutative analogue of the classical relation between Levi-Civita and Chern connections on Kahler manifolds via unitary 2-cocycle twists in covariant *-differential calculi. Using bar-category and monoidal equivalence techniques, it shows that complex structures, holomorphic bimodules, and Chern connections deform coherently and that the Levi-Civita connection on the deformed space of one-forms decomposes as a direct sum of twisted Chern connections on the holomorphic and anti-holomorphic parts when the complex structure is factorizable and the metric is compatible. The results apply to both classical Kahler manifolds under group actions and noncommutative spaces such as the Heckenberger-Kolb calculi, providing a unified deformation-theory toolkit with broad applicability to toric and quantum homogeneous space deformations. This work thereby extends foundational Riemannian and complex-geometry relations to a wide class of quantum geometries via explicit cocycle twisting.

Abstract

We consider unitary cocycle deformations of covariant -differential calculi. We prove that complex structures, holomorphic bimodules and Chern connections on the deformed calculus are twists of their untwisted counterparts. Moreover, for cocycle deformations of a class of classical Kähler manifolds, the Levi-Civita connection on the space of one-forms of the deformed calculus is shown to be a direct sum of the Chern connections on the twisted holomorphic and the anti-holomorphic bimodules. Our class of examples also includes cocycle deformations of the Heckenberger-Kolb calculi.

Paper Structure

This paper contains 25 sections, 40 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments:
  3. Preliminaries
  4. Bar Categories
  5. Differential calculi and metrics
  6. Connections
  7. Cocycle deformation of relative Hopf modules and differential calculi
  8. Cocycle deformation of relative Hopf modules
  9. Cocycle deformation of Hopf -algebras
  10. Twisting Covariant Differential Calculi
  11. Hermitian metrics and their cocycle deformations
  12. Cocycle twists of Hermitian metric
  13. Hermitian metrics on the bimodule of one-forms
  14. Chern connections on twisted holomorphic bimodules
  15. Complex structures
  16. ...and 10 more sections

Key Result

Proposition 2.1

(Ulb90, aschieriweber) Let $B$ be a left $A$-comodule algebra. If $\mathcal{E}$ and $\mathcal{F}$ are objects in ${ {}^{\space \hbox{${A}$}}_{\space \hbox{${B}$}} \mathsf{Mod}^{}_{B}}$ and $\mathcal{E}$ is finitely generated as a left $B$-module, then ${}_{B}\mathrm{Hom}(\mathcal{E}, \mathcal{F}

Theorems & Definitions (97)

  • Proposition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 3.1
  • ...and 87 more