Renormalization of multicritical scalar models in curved space

We consider the leading order perturbative renormalization of the multicritical ${\varphi }^{2n}$ models and some generalizations in curved space. We pay particular attention to the nonminimal interaction with the scalar curvature $\frac{1}{2}\xi {\varphi }^{2}R$ and discuss the emergence of the conformal value of the coupling $\xi$ as the renormalization group fixed point of its beta function at and below the upper critical dimension as a function of $n$. We also examine our results in relation with Kawai and Ninomiya’s formulation of two dimensional gravity.