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

Abstract

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