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…