Matrix models as conformal field theories: genus expansion

We obtain the topological expansion of the hermitian matrix model using its representation as a CFT on a hyperelliptic Riemann surface. To each branch point of the Riemann surface we associate an operator which represents a twist field dressed by the modes of the twisted boson. The partition function of the matrix model is computed as a correlation function of such dressed twist fields. The perturbative construction of the dressing operators yields a set of Feynman rules for evaluating the free energy and the loop observables at any genus.