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

Abstract

The constraint structure of the induced 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 socalled ''quasi-light-cone'' gauge, in which besides the areapreserving diffeomorphism invariance, the reduced Lagrangian also possesses the SL(2, R) residual symmetry. This observation indicates that the claimed correspondence between the SL(2, R) residual symmetry and the area-preserving diffeomorphism invariance in both regularisation approaches does not hold. The string-like approach is then 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, which means that the area-preserving diffeomorphism invariance can be maintained at the quantum level.The induced 2D-gravity with Weyl and area-preserving diffeomorphism invariances has attracted much attention recently [1-7]. As is well known, in the conventional regularisation approach to 2D gravity [8][9][10][11], the diffeomorphism invariance is preserved, while the Weyl invariance is lost. In the path integral formulation, this can be accomplished by choosing the diffeomorphism-invariant, but not Weyl-invariant measures for the functional integrations [10,11]. Nevertheless, one can adopt an alternative regularization approach in which part of the diffeomorphism invariance is sacrificed so as to obtain a Weyl-invariant theory. This alternative approach is motivated by the observation that the Lagrangian, invariant classically with respect to reparametrization and Weyl transformation, depends on the metric only through the Weyl-invariant combination [1][2][3]. From the conformal geometry's point of view, this idea is based on the fact that the amount of gauge degrees of freedom provided by the diffeomorphism group DiffM of manifold M is equivalent to the amount provided by the combination of the Weyl rescaling and the area-preserving diffeomorphism subgroup SDiffM of DiffM [7]. Recently, the anomalous Ward identities and part of the constraint structure for the induced 2D-gravity with Weyl and area-preserving diffeomorphism invariances have been discussed in [3]. Especially, some physical results associated with areapreserving diffeomorphism invariance, like 2D Hawking radiation [12], have been studied in [4,5]. However, there are still some questions that need to be answered. For example, in the induced 2D-gravity with reparametrization invariance, when the diffeomorphism constraints are kept, i.e., no gauge fixings are chosen for them, one has the freedom to choose the conformal or light-cone gauge. A natural question arises, in the induced 2D-gravity with area-preserving diffeomorphism invariance, when the Virasoro generators are required to annihila...