Construction and geometric properties of the fuzzy CP3 onion
Abstract
We consider the matrix model of a seven-dimensional quantum space built from concentric fuzzy CP3's of increasing matrix size, and show that its continuum limit is the metric cone C(CP3) over CP3, not flat R7. Every ingredient is fixed before any comparison, so this geometry is a prediction rather than an input. The two geometries agree in the s wave (p=0) but separate in the higher angular sectors (harmonics on CP3 of degree p≥1): in each sector tested, the spectrum of the free Laplacian converges at second order to that of the cone, and at 20480 layers the cone is matched to 1×10−9 for p≤8, whereas flat space misses by 8-18%. The free propagator converges to that of the cone at the same order, and heat diffusion reproduces the r6 measure and the Bessel spectrum of the cone. The maps between neighbouring layers are obtained in closed form from the polar decomposition of the second-quantized Schwinger-boson lift, using a known operator identity, and coincide with the Kováčik-Tekel coefficient-copying prescription. For CP1 the construction reduces, away from the outer boundary, to the Kováčik-Tekel fuzzy onion. Modes along the twistor fibre are exactly degenerate with the harmonics on S4.
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BibTeX
@misc{uno2026construction,
title = {{Construction and geometric properties of the fuzzy $CP^3$ onion}},
author = {Kohei Uno and Tomohiro Inagaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17807},
eprint = {2610.17807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17807},
}
RIS
TY - UNPB
TI - Construction and geometric properties of the fuzzy CP^3 onion
AU - Kohei Uno
AU - Tomohiro Inagaki
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17807
UR - https://arxiv.org/abs/2610.17807
AB - We consider the matrix model of a seven-dimensional quantum space built from concentric fuzzy CP^3's of increasing matrix size, and show that its continuum limit is the metric cone C(CP^3) over CP^3, not flat R^7. Every ingredient is fixed before any comparison, so this geometry is a prediction rather than an input. The two geometries agree in the s wave (p=0) but separate in the higher angular sectors (harmonics on CP^3 of degree p≥1): in each sector tested, the spectrum of the free Laplacian converges at second order to that of the cone, and at 20480 layers the cone is matched to 1×10^−9 for p≤8, whereas flat space misses by 8-18%. The free propagator converges to that of the cone at the same order, and heat diffusion reproduces the r^6 measure and the Bessel spectrum of the cone. The maps between neighbouring layers are obtained in closed form from the polar decomposition of the second-quantized Schwinger-boson lift, using a known operator identity, and coincide with the Kováčik-Tekel coefficient-copying prescription. For CP^1 the construction reduces, away from the outer boundary, to the Kováčik-Tekel fuzzy onion. Modes along the twistor fibre are exactly degenerate with the harmonics on S^4.
ER -