Geometry of the Grosse-Wulkenhaar model

Abstract

We analyze properties of a family of finite-matrix spaces obtained by a truncation of the Heisenberg algebra and we show that it has a three-dimensional, noncommutative and curved geometry. Further, we demonstrate that the Heisenberg algebra can be described as a two-dimensional hyperplane embedded in this space. As a consequence of the given construction we show that the Grosse-Wulkenhaar (renormalizable) action can be interpreted as the action for the scalar field on a curved background space. We discuss the generalization to four dimensions. © 2010 SISSA.