An invariant approach to dynamical fuzzy spaces with a three-index variable

TL;DR

This paper develops an invariant framework for dynamical fuzzy spaces using a three-index variable, exploring solutions with Lie group symmetries and constructing scalar field theories on fuzzy spheres.

Contribution

It introduces a new invariant approach to dynamical fuzzy spaces with a three-index variable and demonstrates solutions based on Lie group invariants.

Findings

Solutions constructed from Lie group invariant tensors.

Analysis of SO(3) symmetric solutions.

Framework for scalar field theory on fuzzy spheres.

Abstract

A dynamical fuzzy space might be described by a three-index variable C_{ab}^c, which determines the algebraic relations f_a f_b =C_{ab}^c f_c among the functions f_a on the fuzzy space. A fuzzy analogue of the general coordinate transformation would be given by the general linear transformation on f_a. I study equations for the three-index variable invariant under the general linear transformation, and show that the solutions can be generally constructed from the invariant tensors of Lie groups. As specific examples, I study SO(3) symmetric solutions, and discuss the construction of a scalar field theory on a fuzzy two-sphere within this framework.