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

Abstract

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 \(G_{\mathrm{NC}}\). The sector is labelled by central parameters \((\hbar_0,\vartheta_0,B_0)\), with \(\hbar_0,\vartheta_0,B_0\neq0\) and \(\hbar_0-\vartheta_0B_0\neq0\). For this sector, the triples \((\mathcal S_{\hbar_0,\vartheta_0,B_0},\mathcal H, D^{r,s}_{\hbar_0,\vartheta_0,B_0})\) 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 \(\pi(\mathcal S_{\hbar_0,\vartheta_0,B_0})\) with the effective Moyal Fr\'echet \(\ast\)-algebra with deformation parameter \(\vartheta_{\mathrm{eff}} =\frac{\vartheta_0}{1-\vartheta_0B_0/\hbar_0}.\) For each star-product realization parameter \(\varrho\), this yields spectral triples over the involutive Moyal algebra \(\mathcal A_{\vartheta_{\mathrm{eff}},\varrho}\). External \(U(1)_{\star_{\vartheta_{\mathrm{eff}},\varrho}}\)-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 \(D_R^{\varrho,r,s}=D^{\prime\,r,s}+B_R^{(\varrho)}.\) Finally, as \(R\to\infty\), these operators converge in the strong resolvent sense to a self-adjoint limiting operator \(D_\infty\), 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 \(G_{\mathrm{NC}}\)-sector.