Ultraviolet Stable Fixed Point and Scaling Relations in 2+$\epsilon$ Dimensional Quantum Gravity

TL;DR

This paper develops a renormalizable quantum gravity model in 2+ε dimensions, identifying an ultraviolet stable fixed point and deriving scaling relations that could extend to 3 and 4 dimensions.

Contribution

It generalizes the nonlinear sigma model approach to quantum gravity in 2+ε dimensions and finds conditions for an ultraviolet fixed point with implications for higher dimensions.

Findings

Existence of an ultraviolet stable fixed point for 0<c<25

Scaling relation between cosmological and gravitational constants

Potential for a consistent quantum gravity theory in 3 and 4 dimensions

Abstract

We formulate a renormalizable quantum gravity in $2+\epsilon$ dimensions by generalizing the nonlinear sigma model approach to string theory. We find that the theory possesses the ultraviolet stable fixed point if the central charge of the matter sector is in the range $0~<~c~<~25$. This may imply the existence of consistent quantum gravity theory in 3 and 4 dimensions. We compute the scaling dimensions of the relevant operators in the theory at the ultraviolet fixed point. We obtain a scaling relation between the cosmological constant and the gravitational constant, which is crucial for searching for the continuum limit in the constructive approach to quantum gravity.