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

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 ϕ^2 of the Higgs field is shifted to ϕ* ϕ+ const X_μ* X^μ, where X_μis 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.

LaTeX, 23 pages. Preface added which explains the mathematical achievements since v1; these are presented in arXiv:1108.2184. Incorrect section about field equations removed

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Cited by in corpus (25)

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• Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
• Vacuum configurations for renormalizable non-commutative scalar models
• On the vacuum states for noncommutative gauge theory
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• Scalar and gauge translation-invariant noncommutative models
• Quantum gauge theories on noncommutative 3-d space
• Loop Calculations for the Non-Commutative U(1) Gauge Field Model with Oscillator Term
• Noncommutative gauge theories on ${\mathbb{R}}_{\theta }^2$ as matrix models
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• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics
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• Why Renormalizable NonCommutative Quantum Field Theories?
• Spectral theorem in noncommutative field theories: Jacobi dynamics
• Spinors on a curved noncommutative space: coupling to torsion and the Gross-Neveu model
• First numerical approach to a Grosse-Wulkenhaar model