THE EFFECT OF THE 2D GRAVITY ON THE BETA-FUNCTIONS OF A NONLINEAR SIGMA-MODEL

We study the beta-functions of a non-linear sigma-model coupled to two -dimensional gravity. Keeping the two-dimensional geometry classical, we present, in the light cone gauge, a simple diagrammatic derivation of the induced 2D gravitational action from the non-linear sigma-model . Quantizing the two-dimensional geometry, we discuss the gravitationa l dressing of the beta-functions, and establish the equivalence of the conformal and light cone gauges in a consistent dimensional regulariz ation scheme. By a direct calculation in the light cone gauge we find that the beta-functions receive gravitational corrections in conformit y with that obtained from a (D + 1)-dimensional sigma-model in the con formal gauge.