Geometry of the Grosse-Wulkenhaar model
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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.
KeywordsNon-Commutative Geometry Nonperturbative Effects Differential and Algebraic Geometry