CYCLIC STRUCTURE BEHIND THE MODULAR GAUSSIAN CURVATURE

The modular Gaussian curvature on noncommutative two-tori introduced by Connes and Moscovici leads to a functional relation derived from its variational nature, which is a novel feature purely due to the noncommutativity. I will begin with the differential calculus behind the curvature computation, focusing on the variational aspect, and then report some new interpretations of the functional relation in connection with cyclic (co)homology. Moreover, I will also try to explain the reason why the standard cyclic theory is not sufficient to describe the whole differential calculus by making comparison with Hopf-cyclic theory discovered by Connes and Moscovici. The talk is based on the recent preprint <arXiv:2201.08730>.