Geometry of the Grosse-Wulkenhaar Model

TL;DR

This paper explores the geometric structure of the Grosse-Wulkenhaar model on a noncommutative space, interpreting it as a scalar field coupled to curvature, and extends the discussion to four dimensions.

Contribution

It provides a geometric interpretation of the Grosse-Wulkenhaar model as a scalar field coupled to curvature on a noncommutative space, and generalizes the framework to four dimensions.

Findings

The model can be viewed as a scalar field coupled to curvature.

A natural geometric interpretation of the renormalizable action is proposed.

Extension of the geometric framework to four dimensions is discussed.

Abstract

We define a two-dimensional noncommutative space as a limit of finite-matrix spaces which have space-time dimension three. We show that on such space the Grosse-Wulkenhaar (renormalizable) action has natural interpretation as the action for the scalar field coupled to the curvature. We also discuss a natural generalization to four dimensions.