Scalar curvature for the noncommutative two torus

Abstract

We give a local expression for the scalar curvature of the noncommutative two torus $A_{\theta }=C({\mathbb{T}}_{\theta }^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor. This is achieved by evaluating the value of the (analytic continuation of the) spectral zeta functional ${\zeta }_a(s):=\mathrm{Trace}(a{△}^{-s})$ at $s=0$ as a linear functional in $a\in C^{\infty }({\mathbb{T}}_{\theta }^2)$. A new, purely noncommutative, feature here is the appearance of the modular automorphism group from the theory of type III factors and quantum statistical mechanics in the final formula for the curvature. This formula coincides with the formula that was recently obtained independently by Connes and Moscovici in their paper [15].