Gauge Fields from the Gauss Equation of the Metric Bundle

Paper 3 of 5 in the Metric Bundle Programme. In earlier work, we showed that the DeWitt supermetric on the space of Lorentzian metrics over a four-dimensional spacetime X has signature (6,4), yielding the Pati-Salam gauge group from the maximal compact subgroup of \SO(6,4). In this paper, we carry out the Gauss-Codazzi-Ricci decomposition of the scalar curvature of the fourteen-dimensional metric bundle Y^{14} = \Met(X) restricted to a metric section g\colon X ↪ Y. We show that the Einstein-Hilbert, Yang-Mills, and extrinsic curvature terms emerge with the correct relative signs: (i) +R_X for gravity, (ii) -|II|^2 for the extrinsic curvature (torsion), and (iii) -(h/4)|F|^2 with h > 0 for the Yang-Mills gauge fields. The fibre \GL^+(4,\R)/\SO(4) is a symmetric space of non-compact type with scalar curvature R_{\mathrm{fibre}} = -36. We compute the gauge kinetic metric h_{ab} for the \SO(4) = \SU(2)_L × \SU(2)_R gauge sector via the Kaluza-Klein mechanism and find h_L = h_R = 6\cdot I_3, confirming left-right symmetry g_L = g_R at the unification scale and the absence of ghosts in the gauge sector. The non-vanishing commutators of the shape operators, [A_m, A_n] ≠ 0 for 24 of the 45 normal-direction pairs, provide the non-abelian field strength via the Ricci equation. Part of a five-paper series deriving the Pati-Salam gauge group, fermion content, gauge dynamics, anomaly cancellation, and the three-generation structure from the geometry of the metric bundle Y14 = Met(X4).