# Nonperturbative spectral action of round coset spaces of SU(2)

### Kevin Teh

Caltech, Pasadena, USA

## Abstract

We compute the spectral action of SU(2)/$\Gamma$ with the trivial spin structure and the round metric and find it in each case to be equal to $\frac{1}{\vert \Gamma \vert}(\Lambda^3 \hat{f}^{(2)}(0) - \frac{1}{4}\Lambda \hat{f}(0) )+ O(\Lambda^{-\infty})$. We do this by explicitly computing the spectrum of the Dirac operator for SU(2)/$\Gamma$ equipped with the trivial spin structure and a selection of metrics. Here $\Gamma$ is a finite subgroup of SU(2). In the case where $\Gamma$ is cyclic, or dicyclic, we consider the one-parameter family of Berger metrics, which includes the round metric, and when $\Gamma$ is the binary tetrahedral, binary octahedral or binary icosahedral group, we only consider the case of the round metric.

## Cite this article

Kevin Teh, Nonperturbative spectral action of round coset spaces of SU(2). J. Noncommut. Geom. 7 (2013), no. 3, pp. 677–708

DOI 10.4171/JNCG/131