# Renormalization group properties of the conformal mode of a torus

## TL;DR

This paper investigates how the conformal mode's renormalization group behavior on a torus is affected by its negative kinetic term, revealing size-dependent existence issues linked to inhomogeneity and moduli variations.

## Contribution

It demonstrates the impact of inhomogeneity and moduli on the renormalization group properties of the conformal mode on toroidal manifolds, highlighting size-dependent existence constraints.

## Key findings

- Existence of a universal shape function that influences theory viability.
- Inhomogeneity in toroidal geometries triggers size-dependent failures.
- Rich phenomenology emerges from moduli variations affecting the conformal mode.

## Abstract

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.