Classical Solution of Two Dimensional $R^2$-Gravity and Cross-Over Phenomenon

TL;DR

This paper analytically solves two-dimensional R^2-gravity, characterizes its phase structure, and demonstrates that classical solutions can explain the crossover phenomena observed in numerical simulations.

Contribution

It provides a classical solution to the R^2-Liouville equation and analytically derives the partition function, linking classical solutions to phase transitions in 2D quantum gravity.

Findings

Classical solutions reproduce the crossover transition in surface properties.

Three distinct phases are characterized by the effective action.

Analytical partition function confirms the role of surface terms.

Abstract

Two dimensional quantum R$^2$-gravity and its phase structure are examined in the semiclassical approach and compared with the results of the numerical simulation. Three phases are succinctly characterized by the effective action. A classical solution of R$^2$-Liouville equation is obtained by use of the solution of the ordinary Liouville equation. The partition function is obtained analytically. A toatal derivative term (surface term) plays an important role there. It is shown that the classical solution can sufficiently account for the cross-over transition of the surface property seen in the numerical simulation.