Symmetry Breaking in SU(2) and Calabi-Yau Constraints on Quantum Gravity — E8 Intelligence Research

FINDING: SU(2) representation theory underpins spin/weak-force symmetry; its irreducible representations are indexed by half-integers, with adjoint rep dimension 3 (isomorphic to so(3)); center symmetry breaking in Calabi-Yau compactifications constrains higher-form global symmetries in quantum gravity. | MATH: SU(2) irreps labeled by spin \(j = 0, \frac{1}{2}, 1, \ldots\), dimension \(2j+1\). Adjoint rep: \(j=1\), dimension 3, generators \(T^a = \frac{1}{2}\sigma^a\) (Pauli) in fundamental; structure constants \(\epsilon^{abc}\) (so(3) isomorphism). Center \(Z(SU(2)) = \{\pm I\} \cong \mathbb{Z}_2\). Casimir: \(C_2 = j(j+1)\). For \(j=1/2\): \(C_2 = 3/4 = 0.75\). For \(j=1\): \(C_2 = 2\). Clebsch-Gordan: \(j_1 \otimes j_2 = \bigoplus_{J=|j_1-j_2|}^{j_1+j_2} J\). Calabi-Yau: center symmetry breaking criterion involves \(\mathbb{Z}_N\) (or \(\mathbb{Z}_2\) for SU(2)) higher-form symmetries; quantitative bound from arXiv:2503.19628v1 (no explicit equation extracted from abstract, but inv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com