Pati-Salam Model in Curved Space-Time and Sheaf Quantization

This note aims to give a self-contain and detail explaination about U(k′) × U(k) Pati-Salam model in curved space-time which derived from (1 + n)-dimensional square root Lorentz manifold by self-parallel transportation principle. The concepts fundation of manifold from view point of category theory, fiber bundle and sheaf theories are reviewed. There are extra U(k′) × U(k)-principal bundle and U(k)- associated bundle than (1 + n)-dimensional Lorentz manifold. The conservative currents on square root Lorentz manifold is discussed preliminary. A detail proof of relation from sheaf quantization to path integral quantization is given.