Scalar Curvature for the Noncommutative Two Torus

TL;DR

This paper derives a local formula for scalar curvature on the noncommutative two torus using spectral zeta functions, revealing a novel role for the modular automorphism group in noncommutative geometry.

Contribution

It provides the first local expression for scalar curvature on the noncommutative two torus incorporating the modular automorphism group.

Findings

Derived a local scalar curvature formula for the noncommutative two torus.

Connected the curvature formula with the spectral zeta functional at s=0.

Confirmed the formula matches recent independent results by Connes and Moscovici.

Abstract

We give a local expression for the {\it scalar curvature} of the noncommutative two torus $ A_{\theta} = C(\mathbb{T}_{\theta}^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor. This is achieved by evaluating the value of the (analytic continuation of the) {\it spectral zeta functional} $\zeta_a(s): = \text{Trace}(a \triangle^{-s})$ at $s=0$ as a linear functional in $a \in C^{\infty}(\mathbb{T}_{\theta}^2)$. A new, purely noncommutative, feature here is the appearance of the {\it modular automorphism group} from the theory of type III factors and quantum statistical mechanics in the final formula for the curvature. This formula coincides with the formula that was recently obtained independently by Connes and Moscovici in their recent paper.