In this paper, we construct the Brownian motion of Liouville Quantum gravity when the underlying conformal field theory has a c = 1 central charge. Liouville quantum gravity with c = 1 corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with a O(n = 2) loop model or a Q = 4-state Potts model embedded in a two dimensional surface in a c...

In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. We also show that a fractional Brownian motion and the related Riemann–Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of o...

We show that the spectrum of conical defects in three-dimensional de Sitter space is in one-toone correspondence with the spectrum of vertex operators in Liouville conformal field theory. The classical conformal dimensions of vertex operators are equal to the masses of the classical point particles in dS3, that cause the conical defect. The quantum dimensions instead are shown to coincide with ...

We study instanton contribution to the partition function of the one matrix model in the k-th multicritical region which corresponds to the (2, 2k − 1) minimal model coupled to Liouville theory. The instantons in the one matrix model are given by local extrema of the effective potential for a matrix eigenvalue and identified with the ZZ branes in Liouville theory. We show that the 2-instanton c...

We develop a general technique for solving the Riemann-Hilbert problem in presence of a number of ”heavy charges” and a small one thus providing the exact Green functions of Liouville theory for various non trivial backgrounds. The non invariant regularization suggested by Zamolodchikov and Zamolodchikov gives the correct quantum dimensions; this is shown to one loop in the sphere topology and ...

Liouville theory with a negative norm boson and no screening charge corresponds to an exact classical solution of closed bosonic string theory describing time-dependent bulk tachyon condensation. A simple expression for the two point function is proposed based on renormalization/analytic continuation of the known results for the ordinary (positive-norm) Liouville theory. The expression agrees e...

We analyse boundary conformal field theories on random surfaces using the conformal gauge approach of David, Distler and Kawai. The crucial point is the choice of boundary conditions on the Liouville field. We discuss the Weyl anomaly cancellation for Polyakov’s non-critical open bosonic string with Neumann, Dirichlet and free boundary conditions. Dirichlet boundary conditions on the Liouville ...

The divergent part of the one-loop off-shell effective action is computed for a single scalar field coupled to the Ricci curvature of 2D gravity (cφR), and self interacting by an arbitrary potential term V (φ). The Vilkovisky-DeWitt effective action is used to compute gauge-fixing independent results. In our background field/covariant gauge we find that the Liouville theory is finite on shell. ...

In 1975, Yau [Y] proved, by way of a gradient estimate, that a complete manifold M with non-negative Ricci curvature must satisfy the strong Liouville property for harmonic functions. The strong Liouville property (Liouville property) asserts that any positive (bounded) harmonic function defined on M must be identically constant. In 1980, Cheng [C] generalized the gradient estimate to harmonic ...

We study the connection between Zamolodchikov operator-valued relations in Liouville field theory and in the SL(2,R)k WZNW model. In particular, the classical relations in SL(2,R)k can be formulated as a classical Liouville hierarchy in terms of the isotopic coordinates, and their covariance is easily understood in the framework of the AdS3/CFT2 correspondence. Conversely, we find a closed expr...