Spectral Action of round Coset

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 1 |Γ| ( Λf̂ (0)− 1 4 Λf̂(0) ) +O(Λ). 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.