The continuous Anderson operator H is a perturbation of the Laplace-Beltrami operator by a random space white noise potential. We consider this ‘singular’ operator on a two dimensional closed Riemannian manifold. One can use functional analysis arguments to construct the operator as an unbounded operator on L2 and give almost sure spectral gap estimates under mild geometric assumptions on the Riemannian manifold. We prove a sharp Gaussian small time asymptotic for the heat kernel of H that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called Anderson Gaussian free field, and prove that the law of its random partition function characterizes the law of the spectrum of H. We also give a simple and short construction of the polymer measure on path space and prove large deviation results for the polymer measure and its bridges. We relate the Wick square of the Anderson Gaussian free field to the occupation measure of a Poisson process of loops of polymer paths.