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

TL;DR

This paper develops a spectral-triple framework for noncommutative planar systems, incorporating localized U(1) gauge fields and demonstrating convergence of finite-cutoff models to a limiting Dirac operator.

Contribution

It introduces a family of spectral triples with localized gauge perturbations in a noncommutative setting, extending the spectral triple framework to include gauge fields and their limits.

Findings

Constructed spectral triples with isospectral Dirac operators in non-unital, noncompact settings.

Implemented localized U(1)_ ext{star} gauge fields via smooth cutoff functions.

Proved convergence of finite-cutoff spectral triples to a limiting minimally coupled Dirac operator.

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.