LIOUVILLE THEORY: QUANTUM GEOMETRY OF RIEMANN SURFACES

Abstract

Inspired by Polyakov’s original formulation1,2 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 the 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 KPZ3 and Distler-Kawai.4