  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.
