5 SO(16) Decomposition
The decomposition \(E_8\supset \mathrm{SO}(16)\) reveals how GIFT topological invariants encode gauge bosons and fermions separately.
5.1 SO(n) Dimension
\(\mathrm{dim}(\mathrm{SO}(n)) = \frac{n(n-1)}{2}\)
\(\mathrm{dim}(\mathrm{SO}(16)) = \frac{16 \times 15}{2} = 120\)
\(\mathrm{dim}(\mathrm{SO}(7)) = \frac{7 \times 6}{2} = 21 = b_2\)
5.2 Spinor Representation
The chiral spinor of \(\mathrm{SO}(16)\) has dimension \(2^8/2 = 128\).
\(2^{|\mathrm{Im}(\mathbb {O})|} = 2^7 = 128\)
5.3 Geometric and Spinorial Parts
The geometric part encodes K\(_7\) topology:
\(b_2+ b_3+ \mathrm{dim}(G_2) + \mathrm{rank}(E_8) = 120 = \mathrm{dim}(\mathrm{SO}(16))\)
The spinorial part: \(2^{|\mathrm{Im}(\mathbb {O})|} = 128\)
The spinorial part equals the \(\mathrm{SO}(16)\) spinor dimension.
5.4 Master Decomposition
Physical interpretation:
120 = topology + holonomy + Cartan \(\to \) gauge bosons
128 = \(2^7\) from octonions \(\to \) fermions