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

• URL: http://arxiv.org/abs/2605.10250v1
• Date: Mon, 11 May 2026 09:20:11 GMT
• Title: From Noncommutative Kinematics to \(U(1)_{\star}\) Gauge Theory: A Family of Spectral Triples with Localized Gauge-induced Perturbations
• Abstract summary: We construct a noncommutative planar system associated with a fixed nondegenerate irreducible unitary sector of the kinematical symmetry group $G_mathrmNC$.<n>For each $varrho$, this yields locally compact non-unital base spectral triples over the involutive Moyal algebra.<n>We prove strong resolvent convergence to a self-adjoint limiting operator, the closure of the formal minimally coupled operator.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We construct a spectral-triple framework for a noncommutative planar system associated with a fixed nondegenerate irreducible unitary sector of the kinematical symmetry group $G_{\mathrm{NC}}$, labelled by central parameters $(\hbar_0,\vartheta_0, B_0)$ with $\hbar_0,\vartheta_0, B_0\neq 0$ and $\hbar_0 - \vartheta_0 B_0\neq 0$. For the corresponding two-parameter family $(r,s)$ of unitarily equivalent concrete realizations, we construct even spectral triples whose Dirac operators are isospectral and have compact resolvent despite the non-unital and noncompact setting. Passing to the Moyal-side description, a linear Darboux normalization and the Stone-von Neumann theorem identify the represented smooth operator algebra with the effective Moyal-side Frechet *-algebra at $\vartheta_{\mathrm{eff}} =\vartheta_0/(1 -\vartheta_0 B_0/\hbar_0)$. For each $\varrho$, this yields locally compact non-unital base spectral triples over the involutive Moyal algebra $\mathcal{A}_{\vartheta_{\mathrm{eff}},\varrho}$, with $(r,s)$ as kinematical presentation parameters and $\varrho$ as an independent star-gauge parameter. To incorporate an external $U(1)_\star$ gauge field, we replace the linear gauge potentials by smooth cutoff localizations; the resulting bounded self-adjoint perturbations define, for every $R > 0$, locally compact non-unital spectral triples. Finally, as $R\rightarrow\infty$, we prove strong resolvent convergence to a self-adjoint limiting operator, the closure of the formal minimally coupled operator. Thus the finite-cutoff spectral triples approximate, at the level of spectral triples, the limiting minimally coupled Dirac operator over a fixed nondegenerate $G_{\mathrm{NC}}$-background.

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