Noncommutative Geometric Spaces with Boundary: Spectral Action

TL;DR

This paper extends spectral action principles to noncommutative spaces with boundary, analyzing boundary conditions, and shows how the Einstein-Hilbert action is modified by extrinsic curvature and dilaton effects in the noncommutative standard model.

Contribution

It generalizes spectral triples to noncommutative spaces with boundary and evaluates the spectral action, including boundary and dilaton effects, for the noncommutative standard model.

Findings

Boundary conditions consistent with hermiticity established

Spectral action includes extrinsic curvature terms with correct sign

Dilaton effects incorporated into the spectral action

Abstract

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 dilaton field.