Spectral action in noncommutative geometry: An example

TL;DR

This paper computes the spectral action on noncommutative tori using zeta functions and discusses the role of Diophantine conditions in extending these results, highlighting analytical challenges and holomorphic series properties.

Contribution

It provides explicit calculations of the spectral action on noncommutative tori and explores the analytical conditions necessary for extending these computations.

Findings

Spectral action explicitly computed for noncommutative tori

Diophantine condition is crucial for extending results

Holomorphic continuation properties of related series analyzed

Abstract

This is a report on a joint work [12] with D. Essouabri, C. Levy and A. Sitarz. The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined as far as the difficulties to go beyond. Some results on holomorphic continuation of series of holomorphic functions are presented.