Thermodynamic Properties, Phases and Classical Vacua of Two Dimensional $R^2$-Gravity

TL;DR

This paper investigates the thermodynamic properties and classical vacua of two-dimensional R$^2$-gravity, analyzing solutions with different curvatures and phases, while considering topological and area constraints in a semiclassical framework.

Contribution

It provides a detailed analysis of classical solutions and phases of 2D R$^2$-gravity, incorporating topology and area constraints with new insights into dual solutions.

Findings

Classical vacua characterized by positive and negative constant curvature.

Identification of dual pairs of solutions with different branches.

Comprehensive phase characterization of 2D R$^2$-gravity.

Abstract

Two dimensional quantum R$^2$-gravity is formulated in the semiclassical method. The thermodynamic properties,such as the equation of state, the temperature and the entropy, are explained. The topology constraint and the area constraint are properly taken into account. A total derivative term and an infrared regularization play important roles. The classical solutions (vacua) of R$^2$-Liouville equation are obtained by making use of the well-known solution of the ordinary Liouville equation. The positive and negative constant curvature solutions are 'dual' each other. Each solution has two branches($\pm$). We characterize all phases. The topology of a sphere is mainly considered.