Cyclic Structure behind Modular Gaussian Curvature

TL;DR

This paper introduces a systematic categorical framework for analyzing the variation of rearrangement operators in spectral geometry on noncommutative tori and deformed manifolds, linking it to cyclic category relations.

Contribution

It develops a new categorical scheme for spectral function variations, connecting rearrangement operators with Connes's cyclic category and partial derivatives.

Findings

Categorical scheme for rearrangement operator variations

Relation to Connes's cyclic category established

Inclusion of partial derivatives in the framework

Abstract

We propose a systematic scheme for computing the variation of rearrangement operators arising in the recently developed spectral geometry on noncommutative tori and $\theta$-deformed Riemannian manifolds. It can be summarized as a category whose objects consists of spectral functions of the rearrangement operators and morphisms are generated by transformations associated to basic operations of the variational calculus. The generators of the morphisms fulfil most of the relations in Connes's cyclic category, but also include all the partial derivatives. Comparison with Hopf cyclic theory has also been made.