The Gauss-Bonnet Theorem for the noncommutative two torus

By Alain Connes, Paula Tretkoff. 2009

126 citations recorded by the source corpus.

In this paper we shall show that the value at the origin, �(0), 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 We had obtained, in the late eighties, in an unpublished computation, a general formula for �(0) involving modified logarithms of the modular operator of the state.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.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4953668b1abbe09aee5e3bc41db759b52c987278

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure (2010, methodology): “In this paper we extend the Gauss–Bonnet theorem of Connes and Tretkoff [8] (cf.”
Modular Curvature for Noncommutative Two-Tori (2011, methodology): “The obtained terms are, except for this nuance, similar to those of [14] and [15], but for the operator (0,1) φ there is a nontrivial term T0,1 which is given by T0,1 = ir (2)b0kδ1(k)b0kδ2(k) + ir (2)b0kδ1(k)b0δ2(k)k − ir(4)b0kδ1(k)k(2)b(2)0kδ2(k) − ir(4)b0kδ1(k)k(2)b(2)0δ2(k)k − ir(2)b0kδ2(k)b0kδ1(”
Scalar Curvature for Noncommutative Four-Tori (2013, methodology): “[17, 12]), one can derive an asymptotic expansion of the form”
Singular integrals in quantum Euclidean spaces (2017, background): “Rn πΘ(φξ)[12]B(ξ, η)[2]πΘ(φη)[23] dηdξ,”