The geometric $β$-function in curved space-time under operator regularization

In this paper, I compare the generators of the renormalization group flow, or the geometric $β$-functions for dimensional regularization and operator regularization. I then extend the analysis to show that the geometric $β$-function for a scalar field theory on a closed compact Riemannian manifold is defined on the entire manifold. I then extend the analysis to find the generator of the renormalization group flow for a conformal scalar-field theories on the same manifolds. The geometric $β$-function in this case is not defined.

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