Noncommutative geometric spaces with boundary: Spectral action

We study spectral action for Riemannian manifolds with boundary, and then generalize this to noncommutative spaces which are products of a Riemannian manifold times a finite space. We determine the boundary conditions consistent with the hermiticity of the Dirac operator. We then define spectral triples of noncommutative spaces with boundary. In particular we evaluate the spectral action corresponding to the noncommutative space of the standard model and show that the Einstein-Hilbert action gets modified by the addition of the extrinsic curvature terms with the right sign and coefficient necessary for consistency of the Hamiltonian. We also include effects due to the addition of a dilaton field.