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

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 ${△}^{\prime }$ attached to $({\mathbb{T}}_{\theta }^2,\tau ,k)$ is independent of $\tau$ and $k$.

To appear in Journal of Noncommutative Geometry. The long formula for b_2 on pages 7 to 15 will be removed