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.

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.