The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure

TL;DR

This paper proves a version of the Gauss-Bonnet theorem for noncommutative two tori with arbitrary conformal structures, showing invariance of spectral zeta function values under changes in complex structure and Weyl factors.

Contribution

It extends the Gauss-Bonnet theorem to noncommutative tori with general conformal structures, demonstrating invariance of spectral invariants.

Findings

Spectral zeta function at zero is independent of conformal structure and Weyl factor.

The proof applies to any translation invariant complex structure on noncommutative tori.

The result generalizes previous special cases to a broader class of noncommutative geometries.

Abstract

In this paper we give a proof of the Gauss-Bonnet theorem of Connes and Tretkoff for noncommutative two tori $\mathbb{T}_{\theta}^2$ equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number $\tau$ in the upper half plane, representing the conformal class of a metric on $\mathbb{T}_{\theta}^2$, and a Weyl factor given by a positive invertible element $k \in C^{\infty}(\mathbb{T}_{\theta}^2)$, the value at the origin, $\zeta (0)$, of the spectral zeta function of the Laplacian $\triangle'$ attached to $(\mathbb{T}_{\theta}^2, \tau, k)$ is independent of $\tau$ and $k$.