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. 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.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. 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.

Coupling of gravity to matter, spectral action and cosmic topology

“…These two classes of manifolds provide a complete classification of all the possible homogeneous compact 3-manifolds that are either positively curved or flat, hence they encompass all the possible compact cases of interest to the problem of cosmic topology (see for instance [31], [32]). It is shown in [21], [35] and [22] that the nonperturbative spectral action for spherical space forms is, up to an overall constant factor that depends on the order of the finite group Γ, the same as that of the sphere S 3 , hence so is the slow-roll potential. Similarly, the spectral action and potentials for the flat Bieberbach manifolds are a multiple of those of the flat torus T 3 .…”

“…In this paper we focus on the same topologies considered in [21], [35] and [22]. These include all the most significant candidates for a nontrivial cosmic topology, widely studied in the theoretical cosmology literature (see for example [28], [31], [32]).…”

“…In terms of the physical model, this N represents the number of fermions in the theory. We obtain this by computing the spectral action in its nonperturbative form, as in [8], [21], [22], [35], using the Poisson summation formula technique and the explicit form of the Dirac spectra derived in [6].…”

“…In [21], [35] and [22] the nonperturbative spectral action is computed for all the spherical space forms S 3 /Γ and for the flat tori and for all the flat Bieberbach 3-manifolds (for the latter case see also [29]). These two classes of manifolds provide a complete classification of all the possible homogeneous compact 3-manifolds that are either positively curved or flat, hence they encompass all the possible compact cases of interest to the problem of cosmic topology (see for instance [31], [32]).…”

“…In cases where the underlying geometry is very symmetric (space forms) and the Dirac spectrum is explicitly known, it is also possible to obtain explicit non-perturbative computations of the spectral action, computed directly as Tr(f (D/Λ)), using Poisson summation formula techniques applied to the Dirac spectrum and its multiplicities, see [8], [21], [22], [35].…”