The Gauss-Bonnet Theorem for the noncommutative two torus

TL;DR

This paper proves an analogue of the Gauss-Bonnet theorem for the noncommutative two torus by showing the independence of the zeta function's value at zero from the Weyl factor, extending classical geometric results to noncommutative geometry.

Contribution

It demonstrates that the zeta function's value at zero for the Laplacian on the noncommutative two torus is independent of the conformal structure, establishing a noncommutative Gauss-Bonnet theorem.

Findings

The zeta function at zero is independent of the Weyl factor.

The result extends classical Gauss-Bonnet theorem to noncommutative geometry.

Provides detailed computation confirming the invariance.

Abstract

In this paper we show that the value at zero of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state. We had obtained, in the late eighties, in an unpublished computation, a general formula for this value at zero involving modified logarithms of the modular operator of the state. We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the Gauss-Bonnet theorem for the noncommutative two torus.