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 $\triangle\mkern-.5mu '$ attached to $(\mathbb{T}_{\theta}^2, \tau, k)$ is independent of $\tau$ and $k$.