Liouville Theory: Quantum Geometry of Riemann Surfaces

TL;DR

This paper investigates the quantum geometry of Riemann surfaces via Liouville theory, confirming classical conformal dimensions, one-loop central charge correction, and implications for string theory anomaly cancellation.

Contribution

It provides a perturbative analysis validating conformal Ward identities and calculating quantum corrections to the central charge in Liouville theory.

Findings

Conformal Ward identities hold for puncture operators.

Quantum correction to the central charge is exactly 1 at one-loop.

Results support anomaly cancellation in bosonic string theory.

Abstract

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.