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Deformation quantization and intrinsic noncommutative differential geometry

Haoyuan Gao, Xiao Zhang

Abstract

We provide an intrinsic formulation of the noncommutative differential geometry developed earlier by Chaichian, Tureanu, R. B. Zhang and the second author. This yields geometric definitions of covariant derivatives of noncommutative metrics and curvatures, as well as the noncommutative version of the first and the second Bianchi identities. Moreover, if a noncommutative metric and chiral coefficients satisfy certain conditions which hold automatically for quantum fluctuations given by isometric embedding, we prove that the two noncommutative Ricci curvatures are essentially equivalent. For (pseudo-) Riemannian metrics given by certain type of spherically symmetric isometric embedding, we compute their quantum fluctuations and curvatures. We find that they have closed forms, which indicates that the quantization of gravity is renormalizable in this case. Finally, we define quasi-connections and their curvatures with respect to general associative star products constructed by Kontsevich on Poisson manifolds. As these star products are not compatible with the Leibniz rule, we can only prove the first Bianchi identity.

Deformation quantization and intrinsic noncommutative differential geometry

Abstract

We provide an intrinsic formulation of the noncommutative differential geometry developed earlier by Chaichian, Tureanu, R. B. Zhang and the second author. This yields geometric definitions of covariant derivatives of noncommutative metrics and curvatures, as well as the noncommutative version of the first and the second Bianchi identities. Moreover, if a noncommutative metric and chiral coefficients satisfy certain conditions which hold automatically for quantum fluctuations given by isometric embedding, we prove that the two noncommutative Ricci curvatures are essentially equivalent. For (pseudo-) Riemannian metrics given by certain type of spherically symmetric isometric embedding, we compute their quantum fluctuations and curvatures. We find that they have closed forms, which indicates that the quantization of gravity is renormalizable in this case. Finally, we define quasi-connections and their curvatures with respect to general associative star products constructed by Kontsevich on Poisson manifolds. As these star products are not compatible with the Leibniz rule, we can only prove the first Bianchi identity.
Paper Structure (7 sections, 18 theorems, 199 equations)

This paper contains 7 sections, 18 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Basic knowledge and main theorem
  3. Curvature operators and Bianchi identities
  4. Equivalence of noncommutative Ricci curvatures
  5. Spherically symmetric isometric embedding
  6. Quasi-connections and curvatures
  7. Moyal products of trigonometric functions

Key Result

Theorem 2.1

Let $M$ be an $n$-dimensional smooth manifold and $U \subset M$ a coordinate chart. Let $\nabla$, $\tilde{\nabla}$ be the canonical connections with respect to noncommutative metric $g$ and chiral coefficients $\Upsilon_{ijk}$ on $U$. If $g_{ij}$ satisfy and $\Upsilon_{ijk}$ satisfy then two Ricci curvatures are equivalent in the sense that and

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Definition 2.5
  • Theorem 2.1
  • Proposition 3.1
  • Corollary 3.1
  • Remark 3.1
  • ...and 29 more