# Cyclic Structure behind Modular Gaussian Curvature

@inproceedings{Liu2022CyclicSB, title={Cyclic Structure behind Modular Gaussian Curvature}, author={Yang Liu}, year={2022} }

We propose a systematic scheme for computing the variation of rearrangement operators arising in the recently developed spectral geometry on noncommutative tori and θ-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…

## References

SHOWING 1-10 OF 25 REFERENCES

### Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem

- Mathematics
- 1998

Abstract:In this paper we solve a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. We show that the…

### Hypergeometric function and modular curvature

- Mathematics
- 2017

We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol…

### Modular Curvature for Noncommutative Two-Tori

- Mathematics
- 2011

In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples…

### Deformation Quantization for Actions of R ]D

- Mathematics
- 1993

Oscillatory integrals The deformed product Function algebras The algebra of bounded operators Functoriality for the operator norm Norms of deformed deformations Smooth vectors, and exactness…

### Hypergeometric function and Modular Curvature II. Connes-Moscovici functional relation after Lesch's work

- Mathematics
- 2018

In this paper, we initiate a systematic approach for the variational calculus used in the study of modular geometry on noncommutative (two) tori. We introduce several transformations on the space of…

### General Rearrangement Lemma for Heat Trace Asymptotic on Noncommutative Tori

- Mathematics
- 2020

We study a technical problem arising from the spectral geometry of noncommutative tori: the small time heat trace asymptotic associated to a general second order elliptic operator. We extend the…

### The term a_4 in the heat kernel expansion of noncommutative tori

- Mathematics
- 2016

We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its…

### Noncommutative Finite-Dimensional Manifolds. I. Spherical Manifolds and Related Examples

- Mathematics
- 2001

Abstract: We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete…

### On the scalar curvature for the noncommutative four torus

- Mathematics
- 2014

The scalar curvature for noncommutative four tori T^4_Θ, where their flat geometries are conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that…