The Boundary Cosmological Constant in Stable 2D Quantum Gravity

TL;DR

This paper investigates the role of the boundary operator in stable 2D quantum gravity, analyzing boundary cosmological constant flows and their effects on the spectrum and phases of the theory.

Contribution

It introduces a boundary cosmological constant flow in the stable 2D quantum gravity framework and studies its impact on the spectrum and phase structure across models.

Findings

Spectrum is continuous and bounded from below by the boundary cosmological constant.

Large positive boundary cosmological constant leads to a universal topological phase.

Large negative boundary cosmological constant approaches non-perturbative physics of the unstable formulation.

Abstract

We study further the r\^ole of the boundary operator $\O_B$ for macroscopic loop length in the stable definition of 2D quantum gravity provided by the $[{\tilde P},Q]=Q$ formulation. The KdV flows are supplemented by an additional flow with respect to the boundary cosmological constant $\sigma$. We numerically study these flows for the $m=1$, $2$ and $3$ models, solving for the string susceptibility in the presence of $\O_B$ for arbitrary coupling $\sigma$. The spectrum of the Hamiltonian of the loop quantum mechanics is continuous and bounded from below by $\sigma$. For large positive $\sigma$, the theory is dominated by the `universal' $m=0$ topological phase present only in the $[{\tilde P},Q]=Q$ formulation. For large negative $\sigma$, the non--perturbative physics approaches that of the $[P,Q]=1$ definition, although there is no path to the unstable solutions of the $[P,Q]=1$ $m$-even models.