Effective actions, cutoff regularization, quasi-locality, and gluing of partition functions

The paper studies a regularization of the quantum (effective) action for a scalar field theory in a general position on a compact smooth Riemannian manifold. As the main method, we propose the use of a special averaging operator, which leads to a quasi-locality and is a natural generalization of a cutoff regularization in the coordinate representation in the case of a curved metric. It is proved that the regularization method is consistent with a process of gluing of manifolds and partition functions, that is, with the transition from submanifolds to the main manifold using an additional functional integration. It is shown that the method extends to other models, and is also consistent with the process of multiplicative renormalization. Additionally, we discuss issues related to the correct introduction of regularization and the locality.