ON WEYL INVARIANCE CONDITIONS IN STRING THEORY COUPLED WITH 2-DIMENSIONAL GRAVITY

The unified theory of string and two-dimensional quantum gravity is co nsidered. We introduce nontrivial dynamics for the two-dimensional met ric g(mu v) from the very beginning and calculate the path integral ov er the string coordinates and g(mu v) without taking into account the order of integrations. Throughout the paper we use two different kinds of gauges - the covariant one of the harmonic type and also the confo rmal gauge, where the original (D+1)-dimensional sigma model with quan tum gravity becomes the (D+2)-dimensional sigma model on the classical background of g(mu v). The general symmetries of the theory consist i n the reparametrizations of the target space coordinates, in the confo rmal transformations of the metric and in the usual 2d diffeomorphisms . These symmetries do not disturb the structure of the background fiel ds in the (D+2)-dimensional formulation. On the other hand the related arbitrariness of the renormalization does not affect the qualitative structure of the loop contributions to the Weyl anomaly. In the theory with quantum gravity the parameter alpha ' does not play as the param eter of the loop expansion. That is why the one-loop conditions of the Weyl invariance differs from the well known effective equations which arise in the standard approach when g(mu v) is not quantized simultan eously with the string coordinates. Therefore, despite the new conditi ons of the Weyl invariance for the background fields are different fro m the standard effective equations, our result does not contradict to the standard approach. The new one-loop conditions of the Weyl invaria nce are much more complicated and contain the higher derivatives in th e dilaton sector.