Liouville Theory: Quantum Geometry of Riemann Surfaces

Inspired by Polyakov’s original formulation [1, 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 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 [3] and Distler-Kawai [4]. 1 According to Polyakov [1, 2], basic properties of quantum Liouville theory can be read from the correlation function of puncture operators, which is depicted by the following functional integral < X >= ∫ C(X) Dφ e. (1) Here X is an n-punctured sphere, i.e. a Riemann sphere Ĉ with n removed distinct points, called punctures; C(X) is “domain of integration”, consisting of all smooth conformal metrics ds = e|dw| on X satisfying asymptotics e ∼= 1 r i log 2 ri , i = 1, . . . , n, (2)