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Pati-Salam model in curved space-time from square root Lorentz manifold

There is a U(4')\times U(4)-bundle on four-dimensional square root Lorentz manifold. Then a Pati-Salam model in curved space-time (Lagrangian) and a gravity theory (Lagrangian) are constructed on square root Lorentz manifold based on self-parallel transportation principle. An explicit formulation of Sheaf quantization on this square root Lorentz manifold is shown. Sheaf quantization is based on superposition principle and construct a linear Sheaf space in curved space-time. The transition amplitude in path integral quantization is given which is consistent with Sheaf quantization. All particles and fields in Standard Model (SM) of particle physics and Einstein gravity are found in square root metric and the connections of bundle. The interactions between particles/fields are described by Lagrangian explicitly. There are few new physics in this model. The gravity theory is Einstein-Cartan kind with torsion. There are new particles, right handed neutrinos, dark photon, Fiona, X^{\pm C} and Y^0,Y^1,Y^2,Y_*^1,Y_*^2.