From Noncommutative Kinematics to $U(1{)}_{\star }$ Gauge Theory: A Family of Spectral Triples with Localized Gauge Perturbations

We construct locally compact non-unital spectral triples for a noncommutative planar system determined by a fixed nondegenerate irreducible unitary sector of the kinematical symmetry group $G_{\mathrm{N}\mathrm{C}}$. The sector is labelled by central parameters $({\mathrm{\hslash }}_0,{\vartheta }_0,B_0)$, with ${\mathrm{\hslash }}_0,{\vartheta }_0,B_0\mathrm{\ne }0$ and ${\mathrm{\hslash }}_0-{\vartheta }_0{B}_0\mathrm{\ne }0$. For this sector, the triples $({\mathcal{S}}_{{\mathrm{\hslash }}_0,{\vartheta }_0,B_0},\mathcal{H},D_{{\mathrm{\hslash }}_0,{\vartheta }_0,B_0}^{r,s})$ form an even two-parameter family indexed by $(r,s)$, and different choices of $(r,s)$ give unitarily equivalent realizations. The unperturbed Dirac operators have Landau-type spectral levels of infinite multiplicity; hence local compactness, rather than compact resolvent, is the relevant analytic condition. We then identify the represented algebra $\pi ({\mathcal{S}}_{{\mathrm{\hslash }}_0,{\vartheta }_0,B_0})$ with the effective Moyal Fréchet $\ast$-algebra with deformation parameter ${\vartheta }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{{\vartheta }_0}{1-{\vartheta }_0{B}_0\mathrm{/}{\mathrm{\hslash }}_0}\mathrm{.}$ For each star-product realization parameter $\varrho$, this yields spectral triples over the involutive Moyal algebra ${\mathcal{A}}_{{\vartheta }_{\mathrm{e}\mathrm{f}\mathrm{f}},\varrho }$. External $U(1{)}_{{\star }_{{\vartheta }_{\mathrm{e}\mathrm{f}\mathrm{f}},\varrho }}$-gauge potentials are incorporated by localizing the affine gauge potentials with smooth cutoffs. The resulting bounded self-adjoint perturbations $B_R^{(\varrho )}$ define Dirac operators $D_R^{\varrho ,r,s}=D^{\mathrm{\prime }r,s}+B_R^{(\varrho )}\mathrm{.}$ Finally, as $R\to \mathrm{\infty }$, these operators converge in the strong resolvent sense to a self-adjoint limiting operator $D_{\mathrm{\infty }}$, the closure of the formal minimally coupled operator. Thus the finite-cutoff triples rigorously approximate the limiting minimally coupled Dirac operator associated with the fixed nondegenerate $G_{\mathrm{N}\mathrm{C}}$-sector.