The regularized determinant of the Laplacian on n-differentials on a hyperbolic Riemann surface is studied. The main result is an intrinsic characterization of the connection form for the determinant line bundle, endowed with the Quillen metric, over the Teichmüller space, in terms of the Green’s function of the Cauchy-Riemann operator. Further, an explicit series representation of that Green’s function, on a Schottky uniformization of the surface, is established. This is a rigorous version of physical heuristics due to Martinec and Verlinde & Verlinde, relating the determinant to the stress-energy tensor of Faddeev-Popov ghost fields on the Riemann surface. One corollary is a simpler proof of the rigorous hyperbolic Belavin-Knizhnik formula, due to Zograf and Takhtajan, which is an intrinsic characterization of the curvature form of the determinant line bundle with Quillen metric. Another corollary is a proof of an explicit holomorphic factorization formula for n = 1 and genus greater than 1, due to Zograf, which generalizes the well known formula for n = 1 and genus 1 relating the determinant of the Laplacian to the Dedekind eta function.