The 3D Spin Geometry of the Quantum Two-Sphere

TL;DR

This paper develops a detailed 3D spin geometry framework for the quantum two-sphere S^2_q, utilizing a spectral triple approach derived from quantum group calculus, advancing the understanding of noncommutative geometry.

Contribution

It introduces an explicit 3D differential calculus and spin geometry on S^2_q based on a spectral triple with a real structure satisfying key properties up to infinitesimals.

Findings

Explicit description of forms on S^2_q

Construction of a spectral triple with real structure

Verification of the commutant property and first order condition

Abstract

We study a three-dimensional differential calculus on the standard Podles quantum two-sphere S^2_q, coming from the Woronowicz 4D+ differential calculus on the quantum group SU_q(2). We use a frame bundle approach to give an explicit description of the space of forms on S^2_q and its associated spin geometry in terms of a natural spectral triple over S^2_q. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.