Algebraic decomposition and topological fibrations of the $U(4)\to SU(2)$ 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 $\mathfrak{su}(3)$ 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 $\mathbb{C}{P}^1\cong S^2$ base space. Consequently, continuous mappings from a 3-dimensional spatial boundary are classified by the third homotopy group, ${\pi }_3(S^2)\cong \mathbb{Z}$, 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.

arXiv comment22 pages

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• Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance
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• Spontaneous symmetry breaking and Nambu-Goldstone modes in open classical and quantum systems