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 .