Modular curvature for toric noncommutative manifolds

Abstract

In this paper, we extend recent results on the modular geometry on noncommutative two tori to a larger class of noncommutative manifolds: toric noncommutative manifolds. We first develop a pseudo differential calculus which is suitable for spectral geometry on toric noncommutative manifolds. As the main application, we derive a general expression for the modular curvature with respect to a conformal change of metric on toric noncommutative manifolds. By specializing our results to the noncommutative two and four tori, we recovered the modular curvature functions found in the previous works. An important technical aspect of the computation is that it is free of computer assistance