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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.10250 |
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| _version_ | 1866910208019660800 |
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| author | Jim, Md. Rafsanjany Sarkar, Tanmoy Kumar Chowdhury, S. Hasibul Hassan |
| author_facet | Jim, Md. Rafsanjany Sarkar, Tanmoy Kumar Chowdhury, S. Hasibul Hassan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10250 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Noncommutative Kinematics to \(U(1)_{\star}\) Gauge Theory: A Family of Spectral Triples with Localized Gauge-induced Perturbations Jim, Md. Rafsanjany Sarkar, Tanmoy Kumar Chowdhury, S. Hasibul Hassan Mathematical Physics High Energy Physics - Theory Quantum Physics 58B34, 46L87, 81R60, 46L65, 47A10 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. |
| title | From Noncommutative Kinematics to \(U(1)_{\star}\) Gauge Theory: A Family of Spectral Triples with Localized Gauge-induced Perturbations |
| topic | Mathematical Physics High Energy Physics - Theory Quantum Physics 58B34, 46L87, 81R60, 46L65, 47A10 |
| url | https://arxiv.org/abs/2605.10250 |