Quantisation of 2D-gravity with Weyl and area-preserving diffeomorphism invariances

TL;DR

This paper investigates the constraint structure of 2D gravity with Weyl and area-preserving diffeomorphism invariances, exploring gauge choices and quantization methods, and addressing issues with the Virasoro algebra's central charge.

Contribution

It introduces a gauge-independent SL(2,R) current approach for consistent quantization of 2D gravity with specific invariances, avoiding anomalies.

Findings

The conformal gauge does not exist with area-preserving diffeomorphisms.

A quasi-light-cone gauge with SL(2,R) symmetry is feasible.

A consistent quantum theory is achieved using SL(2,R) currents, avoiding the Virasoro central charge anomaly.

Abstract

The constraint structure of 2D-gravity with the Weyl and area-preserving diffeomorphism invariances is analysed in the ADM formulation. It is found that when the area-preserving diffeomorphism constraints are kept, the usual conformal gauge does not exist, whereas there is the possibility to choose the so-called ``quasi-light-cone'' gauge, in which besides the area-preserving diffeomorphism invariance, the reduced Lagrangian also possesses the SL(2,R) residual symmetry. The string-like approach is applied to quantise this model, but a fictitious non-zero central charge in the Virasoro algebra appears. When a set of gauge-independent SL(2,R) current-like fields is introduced instead of the string-like variables, a consistent quantum theory is obtained.