Aspects of the q-deformed Fuzzy Sphere

TL;DR

This paper reviews the q-deformed fuzzy sphere, a noncommutative geometric structure covariant under quantum groups, exploring its algebraic properties and connections to gauge theories and string theory models.

Contribution

It provides a comprehensive overview of the algebraic and geometric aspects of the q-deformed fuzzy sphere, including real structure, differential calculus, and gauge actions.

Findings

Construction of the q-deformed fuzzy sphere and its properties.

Development of gauge theories on the noncommutative sphere.

Connection to D-branes in the SU(2)_k WZW model.

Abstract

These notes are a short review of the q-deformed fuzzy sphere S^2_{q,N}, which is a ``finite'' noncommutative 2-sphere covariant under the quantum group U_q(su(2)). We discuss its real structure, differential calculus and integration for both real q and q a phase, and show how actions for Yang-Mills and Chern- Simons-like gauge theories arise naturally. It is related to D-branes on the SU(2)_k WZW model for q = exp(\frac{i \pi}{k+2}).