I am confused as to why a regularised determinant is used in the definition of Quillen's metric on the determinant line bundle defined over the space of $\bar{\partial}$ operators on a (hermitian) vector bundle over a compact Riemann surface. I mean, why not use the metric induced by the vector bundle just by itself? Is the determinant introduced so as to define a smooth metric? For the record I am referring to Quillen's paper "On the determinants of CauchyRiemann operators" over a Riemann surface.
