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Curvature and Weitzenbock formula for the Podleś quantum sphere

Bram Mesland, Adam Rennie

Abstract

We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podleś sphere $S^{2}_{q}$. We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podleś sphere using the framework of \cite{MRLC}. The scalar curvature is a constant, and as the parameter $q\to 1$, the scalar curvature converges to the classical value $2$. We prove a generalised Weitzenbock formula for the spinor bundle, which differs from the classical Lichnerowicz formula for $q\neq 1$, yet recovers it for $q\to 1$.

Curvature and Weitzenbock formula for the Podleś quantum sphere

Abstract

We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podleś sphere . We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podleś sphere using the framework of \cite{MRLC}. The scalar curvature is a constant, and as the parameter , the scalar curvature converges to the classical value . We prove a generalised Weitzenbock formula for the spinor bundle, which differs from the classical Lichnerowicz formula for , yet recovers it for .

Paper Structure

This paper contains 17 sections, 31 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Differentials and junk
  4. Hermitian torsion-free connections
  5. Curvature and Weitzenbock formula
  6. The Levi-Civita connection for the Podleś sphere
  7. Coordinates, spectral triple and inner product for Podleś sphere
  8. Frame and quantum metric for the Podleś sphere
  9. Existence of an Hermitian torsion-free connection
  10. Braiding and bimodule connection
  11. Curvature of the Podleś sphere
  12. The Riemann tensor
  13. The Ricci and scalar curvature
  14. Weitzenbock formula
  15. Dirac spectral triple
  16. ...and 2 more sections

Key Result

Theorem 1

The module of differential one-forms on the Podleś sphere $S^{2}_{q}$ equipped with the quantum metric defined in Section sec: qmetric admits a unique Levi-Civita connection whose scalar curvature equals $r=[2]_q(1+(q^{-2}-q^2)^2)$.

Theorems & Definitions (68)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 58 more