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…