Dirac Spectra, Summation Formulae, and the Spectral Action

Noncommutative geometry is a source of particle physics models with matter Lagrangians coupled to gravity. One may associate to any noncommutative space ( A,H,D ) its spectral action, which is defined in terms of the Dirac spectrum of its Dirac operator D . When viewing a spin manifold as a noncommutative space, D is the usual Dirac operator. In this paper, we give nonperturbative computations of the spectral action for quotients of SU(2), Bieberbach manifolds, and SU(3) equipped with a variety of geometries. Along the way we will compute several Dirac spectra and refer to applications of this computation.