Renormalization group properties of the conformal mode of a torus

The Wilsonian renormalization group properties of the conformal factor of the metric are profoundly altered by the fact that it has a wrong-sign kinetic term. If couplings are chosen so that the quantum field theory exists on ${\mathbb{R}}^4$, it fails to exist on manifolds below a certain size, if a certain universal shape function turns negative. We demonstrate that this is triggered by inhomogeneity in the cases of ${\mathbb{T}}^4$ and ${\mathbb{T}}^3\times \mathbb{R}$, including twisted versions. Varying the moduli, we uncover a rich phenomenology.