The Lorentzian DeWitt Metric and the Pati-Salam Gauge Group

Paper 1 of 5 in the Metric Bundle Programme. We study the DeWitt supermetric on the space of Lorentzian metrics over a four-dimensional spacetime manifold X. While the DeWitt metric evaluated on a Riemannian background is well known to have signature (9,1) on the ten-dimensional fibre of symmetric two-tensors, we show that on a \emph{Lorentzian} background g = \diag(-1,1,1,1) the signature changes to (6,4). The normal bundle of a metric section g\colon X → Y^{14} = \Met(X) therefore carries structure group \SO(6,4), whose maximal compact subgroup is \SO(6)×\SO(4) ≅ \SU(4)×\SU(2)_L×\SU(2)_R---the Pati-Salam gauge group. The Dynkin indices of \SU(4), \SU(2)_L, and \SU(2)_R in the fundamental representation of \SO(6,4) are all equal to one, giving the tree-level prediction g_4 = g_L = g_R and hence \sin^2\theta_W = 3/8 at the unification scale, which runs to ≈ 0.231 at M_Z. We further identify the four negative-norm DeWitt modes as the (1,2,2) Pati-Salam bidoublet---the Higgs sector---and show that the tree-level scalar potential is flat, realising a Gauge-Higgs Unification mechanism with a computable quartic coupling \lambda(M_{\mathrm{PS}}) = g^2/4. The Gauss equation for the embedding g\colon X ↪ Y yields the Einstein-Hilbert, Yang-Mills, and torsion terms with the correct relative signs. 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).