Stony Brook.

of the Dissertation Analytic torsion and Faddeev-Popov ghosts by Andrew McIntyre Doctor of Philosophy in Mathematics State University of New York at Stony Brook 2002 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