The Hausdorff Dimension of Surfaces in Two-Dimensional Quantum Gravity Coupled to Ising Minimal Matter

TL;DR

This paper investigates the Hausdorff dimension of surfaces in two-dimensional quantum gravity coupled with Ising matter, addressing discrepancies between theoretical predictions and numerical results by analyzing the relationship between area and boundary length.

Contribution

It provides a detailed analysis of how boundary length and area relate in models of 2D quantum gravity with Ising matter, proposing modifications to the Hausdorff dimension based on perturbations.

Findings

Numerical simulations find d_H approximately 4 for certain models.

String field theory predicts d_H = 2m, conflicting with simulations.

Modified relationships between area and boundary length suggest revised Hausdorff dimensions.

Abstract

Within the framework of string field theory the intrinsic Hausdorff dimension d_H of the ensemble of surfaces in two-dimensional quantum gravity has recently been claimed to be 2m for the class of unitary minimal models (p = m+1,q = m). This contradicts recent results from numerical simulations, which consistently find d_H approximatly 4 in the cases m = 2, 3, 5 and infinity. The string field calculations rely on identifying the scaling behavior of geodesic distance and area with respect to a common length scale l. This length scale is introduced by formulating the models on a disk with fixed boundary length l. In this paper we study the relationship between the mean area and the boundary length for pure gravity and the Ising model coupled to gravity. We discuss how this relationship is modified by relevant perturbations in the Ising model. We discuss how this leads to a modified value for the Hausdorff dimension.