Liouville Theory: Quantum Geometry of Riemann Surfaces

Inspired by Polyakov's original formulation of quantum Liouville theory through functional integral, we analyze perturbation expansion around a classical solution. We show the validity of conformal Ward identities for puncture operators and prove that their conformal dimension is given by the classical expression. We also prove that total quantum correction to the central charge of Liouville theory is given by one-loop contribution, which is equal to 1. Applied to the bosonic string, this result ensures the vanishing of total conformal anomaly along the lines different from those presented by KPZ and Distler-Kawai.Comment: 8 pages, plain LaTex, submitted Mod. Phys. Lett.

Equivalence of Geometric h<1/2 and Standard c>25 Approaches to Two-Dimensional Quantum Gravity

We show equivalence between the standard weak coupling regime c>25 of the two-dimensional quantum gravity and regime h<1/2 of the original geometric approach of Polyakov [1,2], developed in [3,4,5].Comment: 10 pages, late

On real projective connections, V.I. Smirnov's approach, and black hole type solutions of the Liouville equation

We consider real projective connections on Riemann surfaces and corresponding solutions of the Liouville equation. It is shown that these solutions have singularities of special type (of a black hole type) on a finite number of simple analytical contours. The case of the Riemann sphere with four real punctures, considered in V.I. Smirnov's thesis (Petrograd, 1918), is analyzed in detail.Comment: 13 pages, final versio