Affine Left--Right Dirac Data on the Moyal Plane, with an Application to $U(1)_\star$ Gauge Coupling in Noncommutative Quantum Mechanics

arXiv:2605.10250v3 Announce Type: replace-cross Abstract: We study even non-unital Dirac data on the two-dimensional Moyal plane with fixed deformation parameter $\vartheta\neq0$, whose Dirac operator is affine in the left and right regular coordinate multipliers, with coefficient matrices $T_L$ and $T_R$. Two invariants organize this class. The right block controls algebra--Dirac transversality: local compactness holds if and only if $\det T_R\neq0$, with $|\det T_R|$ determining the local volume coefficient and $\operatorname{sgn}\det T_R$ the oriented Gaussian $K$-homology pairing. Independently, $\Gamma=\vartheta(\det T_L-\det T_R)$ controls the spectral regime and, together with $|\det T_R|$, the exact localized heat trace: $\Gamma\neq0$ gives infinitely degenerate Landau levels, while for $\Gamma=0$ in the transverse regime the spectrum is $\mathbb R$ with empty point spectrum. Applied to Wallet's Landau--Moyal family, the criterion shows that local compactness survives at the metric-degenerate point $\xi_{\mathrm W}=1$ but fails at $\xi_{\mathrm W}=-1$, thereby resolving the local-compactness behaviour at both exceptional one-sided points. We then derive a distinguished affine family from a fixed nondegenerate sector $(\hbar_0,\vartheta_0,B_0)$ of the kinematical symmetry group of two-dimensional noncommutative quantum mechanics. The sector fixes the effective Moyal scale $\vartheta_{\mathrm{eff}}=\vartheta_0/(1-\vartheta_0B_0/\hbar_0)$ and $T_R$, while an affine $U(1)_\star$ background coupling deforms only the left affine part of the Dirac operator. Consequently, local compactness, noncommutative volume and $K$-homology class remain unchanged, whereas $\Gamma=\tfrac12(1-eF_{xy}/B_0)$ drives a transition at $eF_{xy}=B_0$ from Landau levels to spectrum $\mathbb R$ with empty point spectrum.

Abstract:We study even non-unital Dirac data on the two-dimensional Moyal plane with fixed deformation parameter $\vartheta\neq0$, whose Dirac operator is affine in the left and right regular coordinate multipliers, with coefficient matrices $T_L$ and $T_R$. Two invariants organize this class. The right block controls algebra--Dirac transversality: local compactness holds if and only if $\det T_R\neq0$, with $|\det T_R|$ determining the local volume coefficient and $\operatorname{sgn}\det T_R$ the oriented Gaussian $K$-homology pairing. Independently, $\Gamma=\vartheta(\det T_L-\det T_R)$ controls the spectral regime and, together with $|\det T_R|$, the exact localized heat trace: $\Gamma\neq0$ gives infinitely degenerate Landau levels, while for $\Gamma=0$ in the transverse regime the spectrum is $\mathbb R$ with empty point spectrum. Applied to Wallet's Landau--Moyal family, the criterion shows that local compactness survives at the metric-degenerate point $\xi_{\mathrm W}=1$ but fails at $\xi_{\mathrm W}=-1$, thereby resolving the local-compactness behaviour at both exceptional one-sided points. We then derive a distinguished affine family from a fixed nondegenerate sector $(\hbar_0,\vartheta_0,B_0)$ of the kinematical symmetry group of two-dimensional noncommutative quantum mechanics. The sector fixes the effective Moyal scale $\vartheta_{\mathrm{eff}}=\vartheta_0/(1-\vartheta_0B_0/\hbar_0)$ and $T_R$, while an affine $U(1)_\star$ background coupling deforms only the left affine part of the Dirac operator. Consequently, local compactness, noncommutative volume and $K$-homology class remain unchanged, whereas $\Gamma=\tfrac12(1-eF_{xy}/B_0)$ drives a transition at $eF_{xy}=B_0$ from Landau levels to spectrum $\mathbb R$ with empty point spectrum.

40 pages. Substantially revised with a new title and reorganized presentation. The general affine left--right Dirac framework on the Moyal plane is developed first, with NCQM and (U(1)_\star) gauge coupling as applications. Analytic results and the discussion of exceptional Landau spectral triples are strengthened

arXiv:2605.10250 [math-ph]

(or arXiv:2605.10250v3 [math-ph] for this version)

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Affine Left--Right Dirac Data on the Moyal Plane, with an Application to $U(1)_\star$ Gauge Coupling in Noncommutative Quantum Mechanics

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