Affine Left--Right Dirac Data on the Moyal Plane, with an Application to U(1)⋆ Gauge Coupling in Noncommutative Quantum Mechanics
Abstract
We study even non-unital Dirac data on the two-dimensional Moyal plane with fixed deformation parameter ϑ≠0, whose Dirac operator is affine in the left and right regular coordinate multipliers, with coefficient matrices TL and TR. Two invariants organize this class. The right block controls algebra--Dirac transversality: local compactness holds if and only if det TR≠0, with |det TR| determining the local volume coefficient and sgndet TR the oriented Gaussian K-homology pairing. Independently, Γ=ϑ(det TL−det TR) controls the spectral regime and, together with |det TR|, the exact localized heat trace: Γ≠0 gives infinitely degenerate Landau levels, while for Γ=0 in the transverse regime the spectrum is ℝ with empty point spectrum. Applied to Wallet's Landau--Moyal family, the criterion shows that local compactness survives at the metric-degenerate point ξW=1 but fails at ξW=−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 (ℏ0,ϑ0,B0) of the kinematical symmetry group of two-dimensional noncommutative quantum mechanics. The sector fixes the effective Moyal scale ϑeff=ϑ0/(1−ϑ0B0/ℏ0) and TR, while an affine U(1)⋆ background coupling deforms only the left affine part of the Dirac operator. Consequently, local compactness, noncommutative volume and K-homology class remain unchanged, whereas Γ=1/2(1−eFxy/B0) drives a transition at eFxy=B0 from Landau levels to spectrum ℝ with empty point spectrum.
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BibTeX
@misc{sarkar2026affine,
title = {{Affine Left--Right Dirac Data on the Moyal Plane, with an Application to $U(1)_\star$ Gauge Coupling in Noncommutative Quantum Mechanics}},
author = {Tanmoy Kumar Sarkar and Md. Rafsanjany Jim and S. Hasibul Hassan Chowdhury},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.10250},
eprint = {2605.10250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10250},
}
RIS
TY - UNPB
TI - Affine Left--Right Dirac Data on the Moyal Plane, with an Application to U(1)_⋆ Gauge Coupling in Noncommutative Quantum Mechanics
AU - Tanmoy Kumar Sarkar
AU - Md. Rafsanjany Jim
AU - S. Hasibul Hassan Chowdhury
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2605.10250
UR - https://arxiv.org/abs/2605.10250
AB - We study even non-unital Dirac data on the two-dimensional Moyal plane with fixed deformation parameter ϑ≠0, 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≠0, with |det T_R| determining the local volume coefficient and sgndet T_R the oriented Gaussian K-homology pairing. Independently, Γ=ϑ(det T_L−det T_R) controls the spectral regime and, together with |det T_R|, the exact localized heat trace: Γ≠0 gives infinitely degenerate Landau levels, while for Γ=0 in the transverse regime the spectrum is ℝ with empty point spectrum. Applied to Wallet's Landau--Moyal family, the criterion shows that local compactness survives at the metric-degenerate point ξ_W=1 but fails at ξ_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 (ℏ_0,ϑ_0,B_0) of the kinematical symmetry group of two-dimensional noncommutative quantum mechanics. The sector fixes the effective Moyal scale ϑ_eff=ϑ_0/(1−ϑ_0B_0/ℏ_0) and T_R, while an affine U(1)_⋆ background coupling deforms only the left affine part of the Dirac operator. Consequently, local compactness, noncommutative volume and K-homology class remain unchanged, whereas Γ=1/2(1−eF_xy/B_0) drives a transition at eF_xy=B_0 from Landau levels to spectrum ℝ with empty point spectrum.
ER -