Conformal Field Theoretical Approach to Two-Dimensional Quantum Gravity

Abstract

A conformal (or Weyl) mode of two-dimensional gravity coupled to conformally invariant matter, which is a gauge degree of freedom in the classical level, becomes dynamical through the conformal (or Weyl) anomaly. We consider the BRS quantization of the above system in the conformal gauge. As a practical approach which enables us to do explicit calculations, we reduce the BRS invariant path integral measure for the conformal mode to the naive one by including a one-loop renormalization effect. Since the two-dimensional metric is locally conformally flat, the resulting effective theory is constructed in a flat space-time. We also construct tensorial operators (the BRS, the ghost number current and the energy momentum tensor) in the effective theory in a flat space-time based on anomalous Ward identities, i.e., the trace, the BRS and the ghost number anomalies. Then we show that the effective theory has desired properties by calculating the operator product expansions. For example, we see that the total central charge of the Virasoro algebra vanishes. Our result forms a basis of a conformal field theoretical approach to two-dimensional quantum gravity and it has applications to a theory of non-critical strings.