8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory

TL;DR

This paper constructs an 8-dimensional spectral triple on 4D Moyal space, analyzing the resulting noncommutative gauge theory's vacuum structure and revealing non-constant vacuum expectation values for Higgs and gauge fields.

Contribution

It extends the spectral triple framework to 8 dimensions on Moyal space and computes the spectral action for a noncommutative gauge-Higgs model, revealing novel vacuum solutions.

Findings

The Higgs potential is modified by a constant shift involving covariant coordinates.

The vacuum is characterized by non-constant Higgs and gauge field configurations.

The spectral action leads to new insights into noncommutative gauge theory vacua.

Abstract

Observing that the Hamiltonian of the renormalisable scalar field theory on 4-dimensional Moyal space A is the square of a Dirac operator D of spectral dimension 8, we complete (A,D) to a compact 8-dimensional spectral triple. We add another Connes-Lott copy and compute the spectral action of the corresponding U(1)-Yang-Mills-Higgs model. We find that in the Higgs potential the square \phi^2 of the Higgs field is shifted to \phi * \phi + const X_\mu * X^\mu, where X_\mu is the covariant coordinate. The classical field equations of our model imply that the vacuum is no longer given by a constant Higgs field, but both the Higgs and gauge fields receive non-constant vacuum expectation values.