Gravitationally Dressed Conformal Field Theory and Emergence of Logarithmic Operators

We study correlation functions in two-dimensional conformal field theory coupled to induced gravity in the light-cone gauge. Focussing on the fermion four-point function, we display an unexpected non-perturbative singularity structure: coupling to gravity {\it qualitatively} changes the perturbative $(x_1-x_2)^{-1}(x_3-x_4)^{-1}$ singularity into a logarithmic one plus a non-singular piece. We argue that this is related to the appearence of new logarithmic operators in the gravitationally dressed operator product expansions. We also show some evidence that non-conformal but integrable models may remain integrable when coupled to gravity.