Nonperturbative Spectral Action of Round Coset Spaces of SU(2)

TL;DR

This paper calculates the spectral action for certain quotient spaces of SU(2), revealing explicit formulas and spectral properties for these nonperturbative geometric objects with applications to spectral geometry.

Contribution

It provides explicit computation of the spectral action for SU(2)/Γ with trivial spin structure, including cases with Berger metrics and various finite subgroups, extending spectral analysis to nonperturbative regimes.

Findings

Spectral action formula for SU(2)/Γ with trivial spin structure.

Explicit spectrum of Dirac operator for these spaces.

Results for Berger metrics and specific finite subgroups.

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 $\frac{1}{|\Gamma|} (\Lambda^3 \hat{f}^{(2)}(0) - 1/4\Lambda \hat{f}(0))+ O(\Lambda^{-\infty})$. 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.