We investigate hexatic membranes embedded in Euclidean D-dimensional space using a reparametrization invariant formulation combined with exact renormalization group equations. An XY model coupled to a fluid membrane, when integrated out, induces long-range interactions between curvatures described by a Polyakov term in the effective action. We evaluate the contributions of this term to the running surface tension, bending, and Gaussian rigidities in the approximation of vanishing disinclination (vortex) fugacity. We find a non-Gaussian fixed point where the membrane is crinkled and has a nontrivial fractal dimension.