Heat trace and spectral action on the standard Podles sphere

TL;DR

This paper introduces a new approach to the dimension spectrum in non-regular spectral triples, computes the exact heat trace and spectral action for standard Podles spheres, and analyzes their behavior as q approaches 1.

Contribution

It provides a novel definition of dimension spectrum for non-regular spectral triples and exact calculations of heat trace and spectral action for Podles spheres.

Findings

Exact heat trace computed for $S^2_q$

Spectral action explicitly calculated for specific cutoff functions

Behavior of heat trace analyzed as q approaches 1

Abstract

We give a new definition of dimension spectrum for non-regular spectral triples and compute the exact (i.e. non only the asymptotics) heat-trace of standard Podles spheres $S^2_q$ for $0<q<1$, study its behavior when $q\to 1$ and fully compute its exact spectral action for an explicit class of cut-off functions.