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 1/|Γ|(Λ^3f(2)(0)−1/4Λf(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.