Random surfaces in conformal gauge

Properties of random surfaces are derived using conformal gauge. The fixed-area partition function for manifolds with the topology of the torus is calculated. We find a divergence for d > 1 arising from large values of the imaginary part of the modular parameter. For manifolds with the topology of the sphere an expression for the partition function is derived for certain 'magic' values of the central extension d; we find that it vanishes for d = 1. Correlation functions on random surfaces of fixed area are also considered. (orig.)