The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure

In this paper we give a proof of the Gauss-Bonnet theorem of Connes andTretkoff for noncommutative two tori $\mathbb{T}_{\theta}^2$ equipped with anarbitrary translation invariant complex structure. More precisely, we show thatfor any complex number $\tau$ in the upper half plane, representing theconformal class of a metric on $\mathbb{T}_{\theta}^2$, and a Weyl factor givenby a positive invertible element $k \in C^{\infty}(\mathbb{T}_{\theta}^2)$, thevalue at the origin, $\zeta (0)$, of the spectral zeta function of theLaplacian $\triangle'$ attached to $(\mathbb{T}_{\theta}^2, \tau, k)$ isindependent of $\tau$ and $k$.