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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.
Based on the definition of the apparent horizon in a general two-dimensional dilaton gravity theory, we analyze the tunnelling phenomenon of the apparent horizon by using Hamilton-Jacobi method. In this theory the definition of the horizon is very di
We consider a higher derivative gravity theory in four dimensions with a negative cosmological constant and show that vacuum solutions of both Lifshitz type and Schr{o}dinger type with arbitrary dynamical exponent z exist in this system. Then we find
We consider a classical (string) field theory of $c=1$ matrix model which was developed earlier in hep-th/9207011 and subsequent papers. This is a noncommutative field theory where the noncommutativity parameter is the string coupling $g_s$. We const
We provide a set of chiral boundary conditions for three-dimensional gravity that allow for asymptotic symmetries identical to those of two-dimensional induced gravity in light-cone gauge considered by Polyakov. These are the most general boundary co
We demonstrate an equivalence between two integrable flows defined in a polynomial ring quotiented by an ideal generated by a polynomial. This duality of integrable systems allows us to systematically exploit the Korteweg-de Vries hierarchy and its t