On 2d gravity and parallel transport field equation on quantum sphere

TL;DR

This paper derives exact quantum geometric quantities for a noncommutative sphere using the Moyal star product and explores quantum effects on geodesic flow, providing solutions at different quantum orders.

Contribution

It introduces explicit quantum expressions for geometric tensors on a quantum sphere and formulates equations for quantum effects on geodesic flow, extending prior work to noncommutative geometry.

Findings

Exact quantum Levi-Civita connection components derived.

Quantum Ricci tensor and scalar curvature expressions obtained.

Solutions for geodesic equations at zero and first quantum orders.

Abstract

In this work we have obtained the exact quantum expressions for the compenents of the Levi Cevita connection, the Ricci tensor and the scalar curvature, generalizing those of \cite{CTZX08} for a spherical surface via the noncommutative Moyal star product, and we have established equations describing quantum effect on the geodesic flow equation or auto parallel fields equations. These later are solved for the zero and the first orders of the quantum parameter $\alpha$ when $y$-symmetry is assumed. We expressed the general system of equations in terms of Fourier modes indexed by the integer $p=1,2,...,\infty$ to understand the interdependence of $y$ modes oscillations.