The Lorentzian DeWitt Metric and the Pati-Salam Gauge Group

Paper 1 of 6 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 six-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). v2 changes (March 2026): Corrected eigenvalue listing in Theorem 3.1 (values are {-2,-2,-2,-1,+1,+1,+1,+2,+2,+2}, not ±1; signature (6,4) unchanged). Fixed SO(4) description in Higgs section. Corrected Nesti-Percacci author order. Added Paper 6 cross-reference. v3 changes (March 2026): v3: Fixed eigenvalue multiplicities in Theorem 3.1 ({-2,-2,-2,-1} not {-2,-1,-1,-1}; signature (6,4) unchanged). Simplified Dynkin index proof. Updated Discussion with forward references to Papers IV-VI (anomaly cancellation, three generations, coupling factor 1.4x, cosmological constant). Added Paper4/Paper5/Paper6 bib entries. v4 changes (March 2026): v4: Recompiled with updated cross-references. Post-publication correction (May 2026): Post-publication corrected preprint (May 2026). The three-generation mechanism in the Outlook/Discussion is corrected: the published version (PLB 860, 140447) attributed three generations to a quaternionic structure on R^6 and an N_G = 0 (mod 3) anomaly constraint; both are withdrawn (R^6 admits no quaternionic structure; the D=4 pure gravitational anomaly vanishes identically, so no mod-3 constraint exists). The conclusion N_G = 3 is unchanged -- the correct mechanism is the spin^c index theorem on a K3 base (Paper V). The central results (the (6,4) DeWitt signature and the Pati-Salam gauge group) are unaffected. Companion-paper numbering aligned to the current scheme.