On the Scalar Curvature for the Noncommutative Four Torus

TL;DR

This paper computes the scalar curvature of the noncommutative four torus with conformal perturbations using a residue method, revealing new properties and explicit formulas related to the geometry of this noncommutative space.

Contribution

It introduces a residue-based method for calculating scalar curvature on the noncommutative four torus, simplifying the process and uncovering new functional behaviors.

Findings

Derived explicit scalar curvature formulas involving modular automorphisms.

Simplified curvature expressions for specific dilaton forms.

Calculated the gradient of the noncommutative Einstein-Hilbert action.

Abstract

The scalar curvature for the noncommutative four torus $\mathbb{T}_\Theta^4$, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form $sp$, where $s $ is a real parameter and $p \in C^\infty(\mathbb{T}_\Theta^4)$ is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by A. Connes and H. Moscovici, unbounded functions of the parameter $s$ appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated.