Schiffer operators and calculation of a determinant line in conformal field theory

We consider an operator associated to compact Riemann surfaces endowed with a conformal map, f, from the unit disk into the surface, which arises in conformal field theory. This operator projects holomorphic functions on the surface minus the image of the conformal map onto the set of functions h so that the Fourier series h o f has only negative powers. We give an explicit characterization of the cokernel, kernel, and determinant line of this operator in terms of natural operators in function theory.