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

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

2010-05-26

arxiv.org/abs/1005.4947v2

Mathematics

Operator Algebras

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

Scientific paper

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 $k$.

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