Spontaneous reduction of noncommutative gauge symmetry and model building

We propose a mechanism for the spontaneous (gauge-invariant) reduction of noncommutative $\mathcal{U}(n)$ gauge theories down to SU(n). This can be achieved through the condensation of composite $\mathcal{U}(n)$ gauge invariant fields that involves half-infinite Wilson lines in trace-U(1) noninvariant and SU(n) preserving direction. Based on this mechanism we discuss anomaly-free fully gauge invariant noncommutative Standard Model based on the minimal gauge group $\mathcal{U}(3)\times \mathcal{U}(2)\times \mathcal{U}(1)$, previously proposed, and show how it can be consistently reduced to the Standard Model with the usual particle spectrum. Charge quantization for quarks and leptons naturally follows from the model.