Algebraic decomposition and topological fibrations of the U(4) → SU(2) quotient space
Abstract
We investigate the algebraic and topological properties of the unitary group U(4) subjected to a non-standard continuous symmetry breaking cascade down to an SU(2)H subgroup. By explicitly embedding the unbroken SU(2)H symmetry into the lower-right quadrant of the fundamental matrix representation, we demonstrate that the resulting 12-dimensional quotient space, SU(4)/SU(2)H, undergoes a strictly asymmetric geometric partition. This specific breaking pattern uniquely isolates an 8-dimensional sector that retains closed commutation relations, forming a rigorous su(3) Lie subalgebra. Concurrently, it isolates a 4-dimensional off-diagonal sector that fails to close algebraically, generating a purely non-linear topological coset. By analyzing the global spherical fibrations of the manifold, we show that this non-linear territory reduces to a ℂP1 ≅ S2 base space. Consequently, continuous mappings from a 3-dimensional spatial boundary are classified by the third homotopy group, π3(S2) ≅ ℤ, guaranteeing absolute vacuum stability via the integer-quantized Hopf invariant. Ultimately, this framework demonstrates that a non-simple unitary group can organically provide a mathematically rigorous geometric template capable of simultaneously accommodating both continuous gauge symmetries and stable, non-perturbative topological sectors.
Versions
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BibTeX
@misc{mastoridis2026algebraic,
title = {{Algebraic decomposition and topological fibrations of the $U(4) \to SU(2)$ quotient space}},
author = {Dimitris Mastoridis and Konstantinos Kalogirou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23859},
eprint = {2610.23859},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23859},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Algebraic decomposition and topological fibrations of the U(4) → SU(2) quotient space
AU - Dimitris Mastoridis
AU - Konstantinos Kalogirou
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.23859
UR - https://arxiv.org/abs/2610.23859
AB - We investigate the algebraic and topological properties of the unitary group U(4) subjected to a non-standard continuous symmetry breaking cascade down to an SU(2)_H subgroup. By explicitly embedding the unbroken SU(2)_H symmetry into the lower-right quadrant of the fundamental matrix representation, we demonstrate that the resulting 12-dimensional quotient space, SU(4)/SU(2)_H, undergoes a strictly asymmetric geometric partition. This specific breaking pattern uniquely isolates an 8-dimensional sector that retains closed commutation relations, forming a rigorous su(3) Lie subalgebra. Concurrently, it isolates a 4-dimensional off-diagonal sector that fails to close algebraically, generating a purely non-linear topological coset. By analyzing the global spherical fibrations of the manifold, we show that this non-linear territory reduces to a ℂP^1 ≅ S^2 base space. Consequently, continuous mappings from a 3-dimensional spatial boundary are classified by the third homotopy group, π_3(S^2) ≅ ℤ, guaranteeing absolute vacuum stability via the integer-quantized Hopf invariant. Ultimately, this framework demonstrates that a non-simple unitary group can organically provide a mathematically rigorous geometric template capable of simultaneously accommodating both continuous gauge symmetries and stable, non-perturbative topological sectors.
ER -