QUANTIZATION OF 2D-GRAVITY WITH WEYL AND AREA-PRESERVING DIFFEOMORPHISM INVARIANCES

The constraint structure of the induced 2D-gravity with the Weyl and a rea-preserving diffeomorphism invariances is analysed in the ADM formu lation. It is found that when the area-preserving diffeomorphism const raints are kept, the usual conformal gauge does not exist, whereas the re is the possibility to choose the so-called ''quasi-light-cone'' gau ge, in which besides the area-preserving diffeomorphism invariance, th e reduced Lagrangian also possesses the SL(2,R) residual symmetry. Thi s observation indicates that the claimed correspondence between the SL (2,R) residual symmetry and the area-preserving diffeomorphism invaria nce in both regularisation approaches does not hold. The string-like a pproach is then applied to quantise this model, but a fictitious non-z ero central charge in the Virasoro algebra appears. When a set of gaug e-independent SL(2, R) current-like fields is introduced instead of th e string-like variables, a consistent quantum theory is obtained, whic h means that the area-preserving diffeomorphism invariance can be main tained at the quantum level.