Quantum $R^2$ Gravity in Two Dimensions

TL;DR

This paper explores two-dimensional quantum gravity with an added $R^2$ term, revealing suppression of small-area configurations and smooth surfaces at short distances despite positivity issues.

Contribution

It provides a continuum analysis of $R^2$ quantum gravity in two dimensions, highlighting the effects on the partition function and surface smoothness.

Findings

Partition function is highly suppressed for small areas.

Surfaces remain smooth at short distances despite positivity violation.

Avoidance of branched polymer problem at small scales.

Abstract

Two-dimensional quantum gravity with an $R^2$ term is investigated in the continuum framework. It is shown that the partition function for small area $A$ is highly suppressed by an exponential factor $exp \{ -2\pi (1-h)^2/(m^2A) \}$, where $1/m^2$ is the coefficient (times $32\pi$) of $R^2$ and $h$ is the genus of the surface. Although positivity is violated, at a short distance scale ( $\ll 1/m$) surfaces are smooth and the problem of the branched polymer is avoided.