Decay of Metastable Vacuum in Liouville Gravity

A decay of weakly metastable phase coupled to two-dimensional Liouville gravity is considered in the semiclassical approximation. The process is governed by the ``critical swelling'', where the droplet fluctuation favors a gravitational inflation inside the region of lower energy phase. This geometrical effect modifies the standard exponential suppression of the decay rate, substituting it with a power one, with the exponent becoming very large in the semiclassical regime. This result is compared with the power-like behavior of the discontinuity in the specific energy of the dynamical lattice Ising model. The last problem is far from being semiclassical, and the corresponding exponent was found to be 3/2. This exponent is expected to govern any gravitational decay into a vacuum without massless excitations. We conjecture also an exact relation between the exponent in this power-law suppression and the central charge of the stable phase.Comment: Extended version of a talk presented at XXXIII International Conference on High Energy Physics, Moscow, July 26 - August 02, 2006. v2: few typos corrected, a reference and an acknowledgement adde

On the Three-point Function in Minimal Liouville Gravity

The problem of the structure constants of the operator product expansions in the minimal models of conformal field theory is revisited. We rederive these previously known constants and present them in the form particularly useful in the Liouville gravity applications. Analytic relation between our expression and the structure constant in Liouville field theory is discussed. Finally we present in general form the three- and two-point correlation numbers on the sphere in the minimal Liouville gravity.Comment: Extended version of the talk delivered at the International Workshop on Classical and Quantum Integrable Systems, Dubna, January 26--29, 200

On two-dimensional quantum gravity and quasiclassical integrable hierarchies

The main results for the two-dimensional quantum gravity, conjectured from the matrix model or integrable approach, are presented in the form to be compared with the world-sheet or Liouville approach. In spherical limit the integrable side for minimal string theories is completely formulated using simple manipulations with two polynomials, based on residue formulas from quasiclassical hierarchies. Explicit computations for particular models are performed and certain delicate issues of nontrivial relations among them are discussed. They concern the connections between different theories, obtained as expansions of basically the same stringy solution to dispersionless KP hierarchy in different backgrounds, characterized by nonvanishing background values of different times, being the simplest known example of change of the quantum numbers of physical observables, when moving to a different point in the moduli space of the theory.Comment: 20 pages, based on talk presented at the conference "Liouville field theory and statistical models", dedicated to the memory of Alexei Zamolodchikov, Moscow, June 200

Structure Constants and Conformal Bootstrap in Liouville Field Theory

An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants.Comment: 31 pages, 2 Postscript figures. Important note about existing (but unfortunately previously unknown to us) paper which has significant overlap with this work is adde

N=1 SUSY Conformal Block Recursive Relations

We present explicit recursive relations for the four-point superconformal block functions that are essentially particular contributions of the given conformal class to the four-point correlation function. The approach is based on the analytic properties of the superconformal blocks as functions of the conformal dimensions and the central charge of the superconformal algebra. The results are compared with the explicit analytic expressions obtained for special parameter values corresponding to the truncated operator product expansion. These recursive relations are an efficient tool for numerically studying the four-point correlation function in Super Conformal Field Theory in the framework of the bootstrap approach, similar to that in the case of the purely conformal symmetry.Comment: 12 pages, typos corrected, reference adde

On the Thermodynamic Bethe Ansatz Equation in Sinh-Gordon Model

Two implicit periodic structures in the solution of sinh-Gordon thermodynamic Bethe ansatz equation are considered. The analytic structure of the solution as a function of complex $\theta$ is studied to some extent both analytically and numerically. The results make a hint how the CFT integrable structures can be relevant in the sinh-Gordon and staircase models. More motivations are figured out for subsequent studies of the massless sinh-Gordon (i.e. Liouville) TBA equation.Comment: 32 pages, 18 figures, myart.st