Asymmetric noncommutative torus has vanishing Einstein tensor

TL;DR

This paper computes geometric quantities for a noncommutative torus spectral triple, revealing that both torsion and Einstein tensor vanish, indicating a form of geometric flatness in this noncommutative setting.

Contribution

It provides explicit calculations of spectral metric, torsion, and Einstein tensors for a specific noncommutative torus spectral triple, demonstrating their vanishing.

Findings

Spectral metric and Einstein tensor are explicitly computed.

Torsion tensor vanishes for the considered spectral triple.

Einstein tensor also vanishes, indicating a flat noncommutative geometry.

Abstract

We explicitly compute the spectral metric, torsion and Einstein tensors for a nontrivial spectral triple on a noncommutative torus, with the Dirac operator related to the fully equivariant Dirac by a partial conformal rescaling (as introduced in [1]). The results show that the spectral triple has vanishing torsion and the Einstein tensor also identically vanishes.