Scalar Curvature for Noncommutative Four-Tori

TL;DR

This paper investigates the geometry of noncommutative 4-tori, computing scalar curvature and related geometric invariants using pseudodifferential calculus, and explores the noncommutative analogue of Einstein-Hilbert action.

Contribution

It provides explicit formulas for scalar curvature, establishes noncommutative geometric analogues of classical theorems, and analyzes the Einstein-Hilbert action on noncommutative 4-tori.

Findings

Explicit scalar curvature formulas derived

Noncommutative Weyl law established

Metrics with constant scalar curvature identified

Abstract

In this paper we study the curved geometry of noncommutative 4-tori $\mathbb{T}_\theta^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of $\mathbb{T}_\theta^4$. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe the scalar curvature of $\mathbb{T}_\theta^4$. We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action.