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

摘要

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 φ ~2 of the Higgs field is shifted to φ{star operator}φ+const{dot operator}X _μ{star operator}X ~μ, where X _μ 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.