Renormalization of interacting scalar field theories in curved space-time

The renormalization of interacting scalar field theories in general curved space-times is discussed. The background-field method is used to calculate the effective action. Divergences are analyzed using heat-kernel techniques and dimensional regularization. Renormalization of a scalar field theory with cubic and quartic self-interactions is shown at the two-loop level in a four-dimensional space-time. Counterterms, including the gravitational ones, are computed to this order. Renormalization of the one-loop effective action is examined for a scalar field with a cubic self-interaction in a general six-dimensional space-time. As a result of the asymptotic freedom of this theory, the coupling constant appearing in the $R{\phi }^2$ term is shown using the renormalization group to have an ultraviolet fixed point given by its conformal value of $\frac{1}{5}$.